Hamiltonian systems and celestial mechanics : proceedings of the III international symposium, PaМЃtzcuaro, MichoacaМЃn, MeМЃxico, 7 - 11 December 1998

Hamiltonian Systems and Celestial Mechanics (HAMSYS-98) World Scientific Monograph Series in Mathematics Eds. Ron Don...

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Hamiltonian Systems and Celestial Mechanics (HAMSYS-98)

World Scientific Monograph Series in Mathematics Eds.

Ron Donagi (University of Pennsylvania), Rafael de la Llave (University of Texas) and Mikhail Shubin (Northeastern University)

Published

Vol. 1:

Almgren's Big Regularity Paper: Q-Valued Functions Minimizing Dirichlet's Integral and the Regularity of Area-Minimizing Rectifiable Currents up to Codimension 2 Eds. V. Scheffer and J. £ Taylor

World Scientific Monograph Series in Mathematics - Vol. 6

Hamiltonian Systems and Celestial Mechanics (HAMSYS-98) Proceedings of the III International Symposium Patzcuaro, Michoacan, Mexico

7 - 1 1 December 1998

editors

J. Delgado, E. A. Lacomba, E. Perez-Chavela Universidad Autonoma Metropolitana-Iztapalapa, Mexico

J. Llibre Universitat Autonoma de Barcelona, Spain

World Scientific Singapore * New Jersey • London • Hong Kong

Published by World Scientific Publishing Co. Pte. Ltd. P O Box 128, Fairer Road, Singapore 912805 USA office: Suite IB, 1060 Main Street, River Edge, NJ 07661 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.

HAMILTONIAN SYSTEMS AND CELESTIAL MECHANICS (HAMSYS-98) Copyright © 2000 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in anyform or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher.

ISBN 981-02-4463-0

Printed in Singapore by World Scientific Printers

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PREFACE The papers in this volume are an outgrowth of the third "International Symposium on Hamiltonian Systems and Celestial Mechanics", HAMSYS-98, which was held in Patzcuaro, Michoacan, Mexico, December 7 to 11, 1998. The proceedings of the first two symposia were published in Volumes 4 and 8 of the Advanced Series in Nonlinear Dynamics of World Scientific, under the titles "Hamiltonian Systems and Celestial Mechanics", and "New Trends in Hamiltonian Systems and Celestial Mechanics". The scientific organizers were, in alphabetical order, J. Delgado (UAM-I), E.A. Lacomba, (UAM-I), J. Llibre (UAB) and E. Perez-Chavela (UAM-I). The local organizer was J. Mucino from IMUNAM-Morelia. The third symposium gathered together 42 researchers (15 from Mexico and 27 from abroad comprising 10 different countries) in addition to 10 grad uate students. There were 37 lectures in sessions of one hour or half an hour. The main topics of the talks were Arnold diffusion, central configurations, singularities in few body problems, billiards, area preserving maps, and ge ometrical mechanics. All speakers were invited to submit a contribution to these proceedings. All papers that were received went through the refereeing process typical of a mathematical research journal and those that were accepted form the present volume. The symposium received generous support from CONACYT (Mexico) grant 400200-5-1406PE and Academia Mexicana de Ciencias (Fundacion Mexico-E.E.U.U. para la ciencia). The organization and realization of the meeting was possible due to a lot of work on the part of a number of peo ple. Special thanks are due to Petra Seligman from CIC who took care of the logistic administration of the meeting and many other things. We also thank the economical support provided by the following institutions: From Mexico, UAM-I, FENOMEC, IMUNAM-Morelia, FC-UNAM. From Spain: UAB, DGICYT. And from the international institutions ICTP and UMALCA. And last but not least, the participants at the conference must be thanked for offering everyone a week-long intense and highly interesting mathematical experience. J. Delgado*, E.A. Lacomba*, J. Llibre** and E. Perez-Chavela*. * Universidad Autonoma Metropolitana-Iztapalapa, Departamento de Matematicas Ap. P. 55-534, Mexico D.F. ** Departament de Matematiques, Universitat Autonoma de Barcelona, 08193 Bellaterra, Barcelona, Spain.

VII

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CONTENTS

Preface

vii

The Rhomboidal Charged Four Body Problem F. Alfaro and E. Perez-Chavela

1

Planetary Rings with Shepherds L. Benet and T.H. Seligman

20

Low Reynolds Number Swimming in Two Dimensions A. Cherman, J. Delgado, F. Duda, K. Ehlers, J. Koiller and R. Montgomery

32

2-Dimensional Invariant Tori for the Spatial Isosceles 3-Body Problem M. Corbera and J. Llibre

63

The Global Flow for the Synodical Spatial Kepler Problem M.P. Dantas and J. Llibre

74

Unbounded Growth of Energy in Periodic Perturbations of Geodesic Flows of the Torus A. Delshams, R. De La Llave and T.M. Seara

Splitting and Melnikov Potentials in Hamiltonian Systems A. Delshams and P. Gutierrez

Infinity Manifolds of Cubic Polynomial Hamiltonian Vector Fields with 2 Degrees of Freedom M. Falconi, E.A. Lacomba and J. Llibre

IX

90

111

138

X

Relativistic Corrections to Elementary Galilean Dynamics and Deformations of Poisson Brackets R. Flores-Espinoza and Yu.M. Vorobjev

161

Heteroclinic Phenomena in the Sitnikov Problem A. Garcia and E. Perez-Chavela

174

Doubly-Symmetric Periodic Solutions of Hill's Lunar Problem R.C. Howison and K.R. Meyer

186

On Practical Stability Regions for the Motion of a Small Particle Close to the Equilateral Points of the Real Earth-Moon System A. Jorba

197

Variational Methods for Quasi-Periodic Solutions of Partial Differential Equations R. de la Have

214

The Splitting of Invariant Lagrangian Submanifolds: Geometry and Dynamics J.-P. Marco

229

Cross-Sections in the Planar JV-Body Problem C. McCord

250

Existence of an Additional First Integral and Completeness of the Flow for Hamiltonian Vector Fields J. Mucino-Raymundo

261

Simplification of Perturbed Hamiltonians Through Lie Transformations J. Palacidn and P. Yanguas

284

XI

Linear Stability in the 1 + N-Gon Relative Equilibrium G.E. Roberts

Analytic Continuation of Circular and Elliptic Kepler Motion to the General 3-Body Problem J. Soler

The Phase Space of Finite Systems K.B. Wolf, N.M. Atakishiyev and S.M. Chumakov

303

331

349

T H E RHOMBOIDAL CHARGED P O U R B O D Y P R O B L E M FELIPE ALFARO Departamento de Matemdticas, Universidad Autdnoma Metropolitana-Iztapalapa, Apdo. Postal 55-534, Mexico, D. F. 09340, Mexico E-mail: fel&xanum.uam.mx ERNESTO PEREZ-CHAVELA Departamento de Matemdticas, Universidad Autdnoma Metropolitana-Iztapalapa, Apdo. Postal 55-534, Mexico, D. F. 09340, Mexico E-mail: [email protected] The rhomboidal charged problem consists in studying the dynamics of four point particles endowed with masses and electrostatic charges moving symmetrically along the diagonals of a rhombus. By using blow up procedures we describe the asymptotic behavior of total collapse orbits for different values of parameters de pending on the masses and charges of the particles. We study how the shape of the potential function plays a main role in the dynamics. On the particular level of energy h = 0 we are able to give the global flow in this model.

1

Introduction

In the charged problems we consider point particles endowed with a positive mass and an electrostatic charge that can have any sign. Different authors Atela2, Atela and McLachlan3, Casasayas and Nunes 5 , have contributed in the knowledge of the main qualitative facts about the dynamics of particles moving under Coulombian forces; and of course, since Newton, celestial me chanics has played a main role in the development of mathematics. In this paper we decided to join both, Newtonian and Coulombian forces and to study how different can be the dynamic of the point particles in a particular case obtained when the particles form a rhombus at any time. We consider four point particles with positive masses mi,7712,7713 and 777.4 and charges 91,92.93 a n d 94 °f an Y s 'g n i located at the vertices of a rhombus. We fix the center of mass of 7711,7712,7713,7714 at the origin. In order to preserve the rhomboidal configuration for all time we take mi = 77i2, 7713 = 7714, 91 = 921 93 = 94 and suitable symmetrical velocities of the four particles. This model can be written as a Hamiltonian system with two degrees of freedom. We are interested particularly in the qualitative behavior of solutions that begin or end in total collision. Since the Hamiltonian is a first integral of motion we reduce our analysis to a fixed level of constant energy.

1

2

In section 2 we give the description of the problem. Originally we have four parameters, however we prove that not all of them are independent, and therefore we perform our analysis in a space of parameters of dimension 3. Because of the different nature of the forces it is possible for some choices of the parameters to avoid any sort of binary collision or even the total collision. We prove that the shape of the graph of the potential function of the system plays a main role in the analysis of the singularities of the system. As a curious fact we prove that the well known Kepler problem and the anisotropic Kepler problem can be considered as subproblems of the rhomboidal charged problem. In section 3, we study the orbits that begin or end in total collision, in this way using McGehee coordinates we replace this singularity by an in variant two-dimensional manifold over which the flow extends smoothly. This manifold is called the total collision manifold and we denote it by A. The shape and main facts about A depend essentially of the graph of the poten tial function. In section 4 we give a classification of all possible shapes of the potential with respect to the parameters. In particular we count the critical points of the potential; from here we get the central configurations for the charged rhomboidal problem. In section 5 we study the global flow on the total collision manifold for the generic cases; that is for those points in the parameter space which belongs to an open set where the dynamics given by the respective vector field is topologically the same. The knowledge of this flow allows us to determine the behavior of orbits which pass close to total collision. We can have several possibilities: approach to total collision without any binary collision, with only one kind of binary collision or with both of them. Finally using the global flow on A and the fact that for h = 0, we can give an invariant manifold at infinity which has the same properties as A, we prove that on the zero level of energy the global flow is projectable over A. In this way we can analyze the qualitative behavior of some ejecting-escaping orbits, that is, orbits which begin at total collision and escape to infinity when t —>• -I- oo. In general the invariant submanifolds associated to the hyperbolic equilibrium points on A and No connect to each other. 2

Equations of motion

We consider 4 point particles with positive masses m i , m2, ma, tru and charges q\, 92, 93 > 94 of any sign located at the vertices of a rhombus centered at the origin of the Euclidean plane. In order to preserve the rhomboidal configuration when we give symmetrical initial conditions in positions and

3

velocities and we let the particles move under the action of the Newtonian and Coulombian forces it is necessary to assume that mi = rri2, m.3 = m4, Qi = <72, and g3 = 94 • We measure the difference between Newtonian and Coulombian for ces defining the parameters Af, = rriirrij - 9^9^ for i,j = 1,2,3,4. Let us observe that if Ai:7 > 0 there exists attraction between the particles i and j and if Xij < 0 there is repulsion between these particles (the particle i means the point particle endowed with mass mi and charge qi). If we normalize the mass ratio as a = 1713 jm\, then we can suppose that Tii = »ri2 = 1 and 1713 = TTI4 = a. It is enough to study the case when the mass parameter a runs on the interval (0, 1]. The case a > 1 is obtained from the previous one changing mi by 7713. Let x be the semidistance between the particles 1 and 2 and y the semidistance between the particles 3 and 4 (see Figure 1).

(m,.q,)

C m , . <7,;

("h.fi)

(™*.<1J Figure 1. The rhomboidal charged problem.

In (x, y) coordinates the equations of motion can be written as A12 L4x2

+

2A13x (x2+y2)|J* (2.1)

y

A34

4y +' 2

2A13y

(x2+y2)i\

Where x > 0 , y > 0; x = 0 corresponds to binary collision between the particles 1 and 2; y = 0 represents binary collision between the particles 3 and 4. The total collision corresponds to x = 0 and y = 0. From here the

4

configuration space of the system (2.1) is the first quadrant in the x-y plane without the axes which correspond to the singularities of (2.1). Let us observe that if A12 = A34 = 0 and a = 1, the equations (2.1) describe a Kepler problem; Newtonian or Coulombian depending of the sign of A13. When A12 — A34 = 0 and a ^ 1, the equations (2.1) describe the Anisotropic Kepler problem, widely studied in Casasayas and Llibre4 and Gutzwiller8. We put this as a small but interesting result. P r o p o s i t i o n 2 . 1 . The Kepler problem and the anisotropic Kepler problem can be considered as subproblems of the rhomboidal charged four body problem. In Alfaro and Perez-Chavela1 we have studied all possible central configu rations for the rhomboidal charged problem, in the special case A12 = A34 = 0 whose dynamics is equivalent to the well known Kepler problem. It gives rise to a degenerate case for the potential function associated to the system (2.1), where we get a continuum of central configurations. Since in this paper our goal is to describe the dynamics of the particles, from here on we will suppose that A12 ^ 0 and A34 ^ 0. The system (2.1) can be written in Hamiltonian form by letting r = (x, y)T, M = diag{2,2a} and p = Mr. In these coordinates (2.1) takes the form r

(2.2)

dp'

P

dr

where the Hamiltonian function is given by H = i P r M - 1 p - U{v),

(2.3)

and the potential U takes the form 2x

y/x2 + y2

2j/

Since in the Hamiltonian (2.3) there appears M _ 1 and we are interested in some numerical simulations when a -> 0, we will work with an equivalent system where the mass matrix does not depend on a; in this way let: xi = x,

x2 =

(2.5)

\/ay.

5

The equations of motion (2.1) take the form Xi

=

X2

A12

2 A i 3 c*5 xi

4xf

(axl + xl)* A34 4x 2

-VQ

(2.6)

+ (ax? +xl)i

in this case we let r = (xi, x 2 ) T , M = diag{2,2} and p = M r . The system (2.6) keeps the Hamiltonian form .

dH

(2.7)

dH P =

~dr~'

where

V^-1]

H(r,p)

f/(r),

(2.8)

and U(r) = ^ 2xi

4AI3V/Q +

y/ax\

+ x\

+

A34\/» 2x 2

(2.9)

The last observation in this section is about the relation among the pa rameters of the charged rhomboidal problem; in the equations (2.6) there appear four different parameters, however they are not independent, since after our normalization of masses A34 = o 2 - qj

A12 = 1 -
A13 = a - qiq3 ,

and then A 13 ) 2 (2.10) a2 - A34 so, we will work with four parameters taking into account that only 3 of them are independent. Finally, let us observe that if A12 > 0 and A34 > 0 then Ai3>0. A12 = 1

3

(Q-

M c G e h e e coordinates

In order to study the orbits which go to total collision and the orbits which pass close to this kind of collision, and since the Hamiltonian (2.8) is a first integral for the system (2.7), from here on, we restrict our analysis to a fixed

6

value of H = h defining the level of energy Eh ■ We introduce a change of coordinates studied first by McGehee10, to analyze the singularity due to total collision by letting r = J2(x\

+ x\),

9 = arctan — ,

v = r^r,

u = r?9.

X\

Since both x\ and £2 a r e positive we have that 9 e (0, n/2); v and u are the radial and tangential velocity of the system scaled by the factor r? , we also scale the time variable by dt

3

In these coordinates the system (2.6) becomes r — rv, ■

V = U

2

«' +

J

(3.1)

-U(8),

9 = u, u =

-ui + u'(e).

The energy relation goes over to

u2 + v2 =

2U(6)+2hr,

(3.2)

where

U(0) =

/ !

A12 y/a cos 6

y/a

8A13 cos2 e +

sin2 e

A34 sin 9

(3.3)

and U'(0) means derivation with respect to the variable 6. The total collision manifold A is obtained from the energy relation (3.2) taking r = 0, i.e., A = {(r,v,6,u)

: r = 0, v2 + u2 = 2U(6),

6 e (0, n/2)}

(3.4)

Since r = 0 when r = 0 in the first equation of (3.1), we have that A is invariant under the flow; from the energy relation (3.2) we also have that A is independent of the value of the constant energy h; i.e., each energy surface has the same total collision manifold A as its boundary. Let us observe that in the equation (3.3) the potential U depends only of the angular variable 8. Here we have used the fact that U is a homogeneous function with degree of homogeneity — 1.

7

4

The potential function U{9)

In order to study the flow on the total collision manifold A, and since the shape of A depends essentially of the graph of the function U(9), it is necessary to give a classification of all possible shapes of U(6) with respect to the parameters. The case when A12 = A34 = 0 was analyzed in Proposition 2.1. Let P^IT'

^ r

1

'

*=87(a-l)-

(4.1)

The derivative of the potential function U(9) given in (3.3) is

dU dU _ d6

yf A34 fa

[I" /3sin0 fcsin#cos# 0sin9 ksin.9cos9 | 2 [j5t cos 9 ' (acos26» + sin26l)l

cos 6] cos#l 2 sin (9j '

Let

**>--£ * y/a + (a + tan

2

0)§

tan 3 9 '

then dU

1—r- .

sin#

,„.

,

(4 3)

d*=^*"^*M-

-

Prom here, the critical points of U(9) are the roots of the equation g{9) = 0, for 9 6 (0,7r/2). By straightforward computations we get 1. lim o = - o o ,

and

lim 0 =

Ba~1/2. _ 1

2. If g'{9g) = 0, then 9g = arctan aHfcs - 1J 3.

5 (0 9 )

5

, provided k > 1.

= a- 3 / 2 [/3a + (k2'b - I) 5 / 2 ].

4. For /? > 0, the function g has one zero on (0, J ) . 5. For /3 < 0, the function g can have 0,1, or 2 zeros on (0, ^ ) . Using these properties of the function g, and after some computations we have Proposition 4.1. For A12 ^ 0 and A34 ^ 0, the potential function can have 0 , 1 , or 2 critical points on (0, | ) , where

U(9)

8

i) If 0 > 0, U minimum.

has exactly one critical point; it can be maximum or

ii) If 0 < 0, k > 1, and g(6o) = 0, U has one critical point, it is an inflexion point. Hi) If 0 < 0, k > 1, and g(6o) > 0, U maximum and one minimum.

has two critical points, one

iv) In any other case, U does not have critical points. Now we are interested in the possible sign of U{9Q) . Using (3.3) and (4.2), we get after some algebraic manipulations that 1

U(90)

8A13

A34

2 cos 3 0O .(a + tan 2 90)i

tan30o.

(4.4)

From U'(6o) = 0 we put A13 in terms of the other parameters, and we sustitute this value in (4.4) getting U(90) =

a 3 / 2 - / ? tan 3 90"

^34 3

a-1

\/2 sin 0O

(4.5)

Let us observe from Proposition 4.1 that U{9) has critical points with 0 < 0 only if k > 1, which implies that a ^ l (4.1), that is a < 1, therefore signt/(0 o ) = - s i g n A 3 4 . For 0 > 0, we introduce the new function 87

+

tan#'

6 € (0, TT/2)

(4.6)

and rewrite the potential function U{6) in the form U(9)

-n

A34 cos 9

The function ip(6) has the following properties: 1.

lim u>(9) = + 0 0 ,

and

0

lim 0 .

2. If 1 and 9^ = arctan./._ 8 ^." / 5 _ 1 3.
[(-87)2/3 - I]3/2 •

(4.7)

9 Proceeding analagously like in Proposition 4.1 but now using the proper ties of the function 0 , 87 + 1 > 0, then signU(80) = signA34 . ii) If P > 0, 87 + 1 < 0, ipiOy) > 0, then sign U(90)

sign\34 .

in) If 0 > 0 , 87 + 1 < 0, ifiiO^) = 0, then U{60) = 0 iv) If P > 0 , 87 4-1 < 0,
signU(60)

- sign A34 .

v) If /3 < 0 , k > 1, then signU(6o) = —signX^ . The Table 1 summarizes the possible shapes of the graphs of the potential U{6). Before ending this section we will compute the equilibrium points of the system (3.1), which are strongly related with the critical points of the potential U(6). Since an equilibrium point (ro,v0,60,u0) cancel the vector field given by (3.1) and satisfy the energy relation (3.2), then 7-0 = 0,

u 0 = 0,

U'{60) = 0,

v0 =

±y/2U(60).

Let us observe that in order to have equilibrium points U{6Q) > 0, all of them are located on A and they are in correspondence 2 : 1 with the critical points of the potential U(6). In this sense if the potential has one of the shapes given by U2,U4,UQ,U7, or [79 (see Table 1), then there are no equilibrium points for the system (3.1). If the behavior of the potential is like Ui,U3,Us, or Uio, then one can have two or four equilibrium points (the potential can have one or two critical points). The special case when the potential has the shape given by t/ 8 is totally degenerate and we omit it here. Since the coordinate v is obtained from the energy relation (3.2), we lin earize the vector field (3.1) at the equilibrium points in coordinates (r, 8, u). The eigenvalues are given by Ai = v0,

A2 = X, —

^3

-wo + yjvl + 16£/"(0O)

4 -vo ~ \A>o + 16t/"(0o)

4

10 A34 > 0 .X34 < 0

Parameters

fl>0 8y+1>0

U

U

0

0

or

>0 8y+1<0

to (90 >0

p

2

g

Uz

U1

U '0

>0 8y+1<0

9

0

U3

U

U

fl <0

o

n

B 2

k>1 9(99)>0

fl<0

U5

0

U4

U

a

U

k<1 or

B

<0

0

2

B

o

n

k> 1 9(p9)<0

lee

U6

U7

U 0

'0 >0

B 2

0

8y+1<0 W(BV)=0

Ug U

U s1

p

n

B 2

9(e9)=0

0

U9 Table 1. Different shapes for the potential function.

n 2

UI0

0

11

Let us observe that if VQ ^ 0 and U"(6Q) ^ 0 then the respective equi librium points are hyperbolic. In these cases we can compute the local stable or unstable submanifolds associated to the equilibrium point. We denote by Wp' the local stable (unstable) manifold associated to the equilibrium point P. By checking the possible signs of the above eigenvalues and by the previous discussion we can prove the following result. Theorem 4.3. Let PQ an equilibrium point for the flow given by (3.1), where U(6o) > 0 and U"[8Q) ^ 0 for the respective variable 8$ , then PQ is located on A and i) dimWPo=2

and dimW]i0 = 1 if v0U"(80) < 0,

ii) dimWPn = 1 and 5

dimW^ = 2 if vQU"(90) > 0.

Global flow on the total collision manifold

We have seen in section 3 that the total collision manifold A is invariant by the flow given by the system (3.1); however the vector field (3.1) restricted to A still has singularities, they correspond to binary collisions. In the charged problems we have binary collisions when the resultant force between two par ticles is attractive, so in our case; if A12 > 0 and 8 —>• | , we have binary collisions between the particles 1 and 2, and when A34 > 0 and 8 —> 0 the particles 3 and 4 collide. In order to regularize the binary collisions we replace the coordinate u by 10 = "/(#) i where the function f{8) will be essentially the product f(9) = sin 9 cos 9\ if for a particular choice of the parameters we have that A12 < 0, then there are no collisions between the particles 1 and 2 , in this case we replace the function sin 8 by the constant function 1, so f(8)=cos8. After a time reparametrization ^ = f{8), the system (3.1) in coordi nates (r, v, 8, w) takes the form

dr ds = rvf(8), dv

\-j + U(8) + 2hr}f(8), ds~ = ' dO ds = w, dw [-^L + U(0) + hr]2f(B)£d7 = l

-M? + **%-

(5.1)

12 The energy relation becomes \[w2+^f2m

- U{9)f2{6)

= hrf2(6).

(5.2)

We can check easily that {fU)(8) and (f2U')(6) are analytic functions on [0,7r/2], then the above vector field and energy relation are analytic on [0,7r/2]. We also verify that the equilibrium points of (3.1) are invariant under the last change of variables and the new parametrization of time. We will use strongly the fact that the flow always has the symmetry L : (r, v, 0, w, s) —> (r, —v, 0, —w, —s). This is due to the fact that the original system (2.6) can be written in Hamiltonian form (2.8), in terms of a function which is quadratic in the momenta. This implies that the system is reversible, property which persists when we introduce McGehee coordinates and when we regularize the binary collisions. In coordinates r, v, 0, w the total collision manifold is given by A = {(r, v, 6, w): r = 0, w2 + v2f2(6)

= 2U{6)f2{6)} ,

(5.3)

where 6 € [0, § ] , and the flow on A is given by dv ds dO ds dw ds

h y + f (*)]/(*), w, [=£ + U(B)\ 2f(8)f{0)

(5.4) - ™f{6)

+

U\6)f2{6).

The flow on A is easy to study because it has nice properties. We start the description of this flow with a definition. Definition 5 . 1 . The vector field F is gradient like with respect to a function g if all non-equilibrium solutions of x = F(x) are increasing with respect to 9L e m m a 5.2. The vector field given by (5.4) is gradient like with res-pect to the coordinate v. The proof of this lemma is a straightforward computation. Prom the above lemma, the flow on A depends essentially of the shape of the graph of U(6). In this paper we restrict our analysis to the generic cases; in Alfaro and Perez-Chavela1 we study the non-generic cases. We say

13 that a given case is generic if there exists an open set H in the parameter space, such that, for any point in fi, the shape of the graph of U(8) is qualitatively the same. For the rhomboidal charged problem the generic cases are given by U\, U2, U3, I/4, C/5, Ue, U7 in Table 1. This produces four dif ferent total collision manifolds Ai,A3,A5,A 7 generated by the potentials given by Ui,U3,U5,U7, respectively (see Figure 2). The total collision man ifold does not exist for the negative potential with shape U2 , and the ones corresponding to C/4 and Us are similar to A7 .

Figure 2. Total collision manifolds Ai,A3,A5, A7 .

5.1

Global flow on A i , A3, A5, A7

On Ai there are two equilibrium points A_ and A+ corresponding to vo < 0 and VQ > 0 respectively, both of them are saddle points for the flow on

14

A i . Here we have A12 > 0, A34 > 0, A13 > 0, which means that the system of particles is attractive, giving us the possibility of both kinds of double collisions, of course we can also have total collision among the particles. This case is essentially the rhomboidal Newtonian problem, widely studied in Lacomba and Perez-Chavela9. The case A3 is obtained when A12 < 0, A34 < 0, but A13 > 0, that is, when we have repulsion between the particles 1 and 2, and also between the particles 3 and 4. The total collision is possible because there is attraction along the sides of the rhombus (A 13 > 0). The fact that we do not have binary collision implies that A3 is a compact manifold, i.e., here A3 is homeomorphic to a sphere, and the flow on A3 is very simple, the equilibrium point A- is a source, whereas the equilibrium point A+ is a sink. All non-equilibrium orbits on A3 start at A- and die at A+ (see Figure 2). The total collision manifolds A5 and A7 are non-compact, both of them are homeomorphic to a sphere minus 2 points. On A7 we do not have equi librium points and therefore the flow on it is simpler; all the orbits are un bounded, and they pass infinitely many times by double collision between the particles 1 and 2 . Since in this case A34 < 0 there are no collisions between the particles 3 and 4 (see Figure 2). The flow on A5 is the most interesting one because on it, there are four equilibrium points, A-,B-,A+,B+ ; A- and S_ correspond to VQ < 0, A- is a saddle point and 5 _ is a source. For vo > 0 we have that A+ also is a saddle point and B+ is a sink. In this case we have done some numerical simulations for some particular choice of the parameters; in these simulations we are looking for the possible connections between the invariant submanifolds associated to the equilibrium points. In all the simulations we fix two parameters and study the projection of W%_ on the 6-v plane. Vary ing the third parameter we found that for some values of this parameter W%_ escapes along the line 6 = TT/2 and for other values of the same parameter WJt_ die at Wg ; hence, at least for one value of the free parameter we have a connection between W%_ and W% (see Figure 3). After these simulations we have numerical evidence on the following result: Conjecture. On the subset fi5 of the parameter space that determines the shape t/5 of the potential function, the subspace of parameters for which we have connection between the invariant submanifolds associated to the equi librium points has codimension one in ^5 .

15

A.

W"

w-

B.

a

B.

Figure 3. Behavior of W%_ for different choices of the parameters.

6

Global flow for h = 0

For the energy relation (3.2) we can check that for h > 0, the kinetic energy plays the main role, and from here the possibilities of escape increase. If we take an inversion of the radial velocity we will see that the escapes can be studied in a similar way as the total collision, i.e., we can give a special submanifold as the boundary of each level of energy Eh , that we call the infinity manifold. For the particular case h = 0 it is possible describe the global flow on EQ , in this way first we blow up the infinity, let p = i = (qtMq)-1/2. The variables v, 6, and w are defined as above; after the regularization of the binary collisions, the equations of motion (5.1) on EQ can be written in the form dp -pvf(9),

Is

dv ds d9 ds dw

(6.1) w, -v + 1/(0)] 2f(e)f'(0) ~2~

- vw ^f(9)

+

U'{6)f\6),

and the energy relation (5.2) for h = 0 goes over

Y + r 2 / 2 w-tw 2 w „2

0,

p>0.

(6.2)

16

We define the infinity manifold No as N0 = {(p,v,e,w):

p = 0,

\

+ \v2f2{e)-U{6)f2{6)=Q},

(6.3)

where 0 € (0, § ) . Let us observe that in this case iVo is identical with the total collision manifold A defined in (5.3). Moreover, the flow on 7V0 is the same as the flow on A, so on A^o we have the same properties studied in section 5. Another important remark is that the energy relation (6.2) does not de pend on p. This means that if we know the flow on NQ , then we also know the flow on EQ U NQ since p is obtained by integration of the first equation in (6.1), in other words the orbits on EQ are obtained lifting the orbits on No along the radial direction. When this happen we say that the global flow is projectable on NQ , in other words for h = 0 we have an energy level Eo with two boundary components, that is EQ U NQ U A, where the flow is projectable on A or AT0 along the coordinates r or p. Because of projectability of the flow, we can study a very special kind of orbits on E0 , those whose projection on N0 or A are the equilibrium points given by (0,vo,0o,0), where VQ = ±\j2U{6o) and Go is a critical point of the potential U. These orbits are called homothetic, their characteristic feature is that at any moment they are central configurations. From the equations of motion (6.1) and the energy relation (6.2) we have that on the r-v plane defined through the equilibrium points (0, ±VQ, 6>O,0) , dO ds

dw _ ds

that is, this plane is invariant under the flow. Using again (6.2) we have that on the r-v plane the homothetic orbits satisfy v2(s) = U(90), i.e., the homothetic orbits join A and No , and they are the unique solutions on Eo whose projection on A and on N0 are points. For vo > 0, these orbits eject from A and escape to infinity with radial velocity | v \ = y/2U(0o) ■ From the first equation in (5.1) we get r(s) = cev°n9o)3,

(6.4)

where c is a positive constant depending on the initial conditions. Since in the charged rhomboidal problem we can have more than one central configuration, in the rest of this section we analyze the global flow for

17

just one shape of the potential U which shows how different and complicated can be the dynamics for charged problems. Suppose that the potential function U has the shape U5 , obtained when A34 < 0, A12 > 0, and A13 > 0; in this case we have repulsion between the particles 3 and 4 and attraction among the other particles. Let ;pi an orbit on Eo whose projection on A5 is given by the curve <7i (see Figure 4). Making s* = -s we have that j ^ = P1 = pvf{6)> since v(s") < 0 and p(sm) > 0 for all s" e JR., lim^.^oop^s*) = 0 ; if limj.-Kjo p(s*) = [i > 0 then p' —> /xvo/(#o) < 0 which is a contradiction; therefore the orbit (pi starts at NQ (these kind of orbits are called capture orbits). Similarly we prove that \ comes from infinity, without any binary collision and finally dies at total collision. Physically, by the combination of the different forces among the particles, there exists a time so, such that, for s > s 0 the particles 1 and 2 are closer and closer without colliding, whereas the particles 3 and 4 have a pulsating motion; when s —» 00 all the particles collide (see Figure 4). For the orbit
Figure 4. Orbits on Eo projected to A5 .

In general by the projectability of the flow, the invariant submanifolds of the hyperbolic equilibrium points on NQ and A connect each other across Eo. In Figure 5 we show one of those submanifolds whose orbits coming from A5 and escaping as t —> 00 are projectable on the orbit a € A5 .

18

Figure 5. Invariant manifold on EQ projectable on
References 1. F. Alfaro, and E. Perez-Chavela, On a Continuum of Central Configura tions in a Charged Problem, To appear. 2. P. Atela, The Charged Isosceles 3-Body Problem, Contemporay Mathe matics 8 1 , 43-58, (1988). 3. P. Atela and R. McLachlan, Global Behavior of the Charged Isosceles Three-Body Problem, International Journal of Bifurcation and Chaos 4, 865-884, (1994). 4. J. Casasayas and J. Llibre, Qualitative analysis of the anisotropic Ke pler problem, Memoirs of the Amer. Math. Soc, Vol. 52, n° 312, (1984). 5. J. Casasayas and A. Nunes, A restricted charged four-body problem, Celestial Mechanics and Dynamical Astronomy 47, 245-266, (1990). 6. I. Davies, A. Truman and D. Williams, Classical periodic solutions of the equal-mass 2n- Body Problem, 2n-ion Problem and the n-electron atom Problem, Physics Letters 99A 1, 15-18, (1983). 7. R. Devaney, Singularities in Classical Mechanical Systems, in Ergodic Theory and Dynamical Systems I, Ed. A. Katok, Birkhauser Boston, 211-333, (1981).

19

8. C. Gutzwiller, The anisotropic Kepler problem in two dimensions, J. Math. Physics 14, 139-152, (1973). 9. E. Lacomba and E. Perez-Chavela, Motions Close to Escapes in the Rhomboidal four body problem, Celestial Mechanics and Dynamical Astronomy 57, 411-437, (1993). 10. R. McGehee, Triple collision in the collinear three body problem, Inventions Math. 27, 191-227, (1974). 11. E. Perez-Chavela, D. Saari, A. Susin and Z. Yan, Central Configuration in the Charged three body problem, Contemporary Mathematics 198, 137-155, (1996). 12. A. Wintner, The Analytical Foundations of Celestial Mechanics, Prince ton Univ. Press, Princeton, N.J. (1941).

PLANETARY RINGS W I T H S H E P H E R D S L. BENET AND T.H. SELIGMAN Centro de Ciencias Fisicas, National University of Mexico (U.N.A.M.) 62210 Cuernavaca, Mixico and Centro Internacional de Ciencias A.C. Ap. Postal 6-101, 62131 Cuernavaca, Mexico E-mail: [email protected], [email protected] We shall consider the system of planetary rings with shepherds as a reduced fourbody problem neglecting interaction between ring particles. We introduce a billiard scattering model consisting of two non-overlapping rotating discs in the context of the formation and structural properties of planetary rings. We show that due to the arrangement of the symmetric periodic orbits, stable orbits are found which in the configuration space lead to the appearance of patterns qualitatively similar to planetary rings. Rings associated with different stability regions are naturally braided; different braids may overlap displaying features similar to clumps. Erosion mechanisms within the model are discussed.

Ever since their discovery by Galileo, the nature of planetary rings has been subject of discussion and research l i 2 ' 3 . The present day questions reach from formation and erosion of rings to the observed rich structure and wide variety of properties of the rings. The known ring systems (around Jupiter, Saturn, Uranus and Neptune) display some similar features: they are com posed by rather small particles, occur close to their planets and some aspects of their structure are related to the presence of a few larger bodies. Particu larly it is not clear what is the actual reason for the appearance of wide rings in Jupiter, all the narrow and faint rings of Uranus and Neptune, and the beautiful variety of Saturn rings. Some models have been proposed to explain the different particular phenomena, but the complexity and richness of the real system is overwhelming 4 ' 5 ' 6 . This talk attempts to give a working plan for a better understanding of narrow braided rings with shepherd satellites as well as results of a very simple model study. Indeed to explain the confinement of such rings, Goldreich and Tremaine 7 developed the idea that the confinement of the narrow rings was caused by shepherding satellites. These ideas were confirmed in concept by the discovery of Prometheus and Pandora on either side of the F ring of Saturn, and later by the discovery of Cordelia and Ophelia shepherding the Epsilon ring of Uranus; however, no other shepherd satellite has been seen near the other uranian rings. Considering the case of Saturn's F ring 3 , s ,

20

21 its shepherd satellites are directly related with well defined edges of the ring and induce some of its fine structure by gravitational forces. However, the braided structure of this ring, that is, the arrangement of the ring material in well defined strands that intertwined is not yet understood in these terms l'2. We shall start from the basic point of view that we are dealing with a reduced four-body problem, and neglect the interaction between ring particles. Then the problem we have on hand is a straight forward generalization of the reduced three-body problem with a very small ratio /x between the mass of the central body and the mass of any of the shepherds. This three-body problem was discussed in its simplest form by Henon 9 . The orbit is assumed circular, thus conserving the Jacobi Integral which simplifies things greatly. Henon shows that most of the structure of the reduced three-body problem for small /x can be understood starting from the symmetric collision orbits with the smaller body. This finding was confirmed by a study of the chaotic saddle of this problem 10 . In this and related studies n , it was found that some very relevant aspects of the reduced three-body problem could be related to a system consisting of a point particle among rotating hard discs 12 . The interesting aspect of this model is the similarity it has to collision orbits in the 1/r problem while the structures created are not of zero measure as in the case of collision orbits. This feature leads us to use this model as a first step in an extended chain of arguments. We therefore introduce a hierarchy of simplifications: 1. We neglect interactions between ring particles. 2. We assume a reduced four-body problem thus not only neglecting the effect of the ring on the shepherd but also those of one shepherd onto the other. This indicates small ratio of shepherd mass to planet mass. 3. We assume that the shepherds move on circular orbits. 4. We assume that both shepherds have the same angular velocities despite of the fact that they are at slightly different distances from the planet. Assumption 3 and 4 assure us of the conservation of the Jacobi integral. This greatly simplifies our arguments. On the other hand, assumption 4 is not only an idealization but also in direct contradiction with the dynamics of the system. We shall consider a simple planar scattering pinball model, i.e. a time dependent billiard system. Here, point particles (ring particles) will move freely and bounce off two rotating hard discs from time to time, otherwise escaping. Within this crude model, the hard discs represent the shepherd satellites which are assumed to rotate in non-overlapping circular orbits with the same angular velocity according to assumptions 3 and 4. The discs will be then responsible for the formation and confinement of whatever rings exist.

22

At first sight rotating discs and 1/r potentials seem to have nothing in common. But as mentioned above the limit of disc radius zero gives the same reflection behavior as the mass zero limit for 1/r potentials, which is known to contain the relevant information for the restricted three-body problem 9 . Indeed, the m class periodic orbits of Henon have quite similar properties to the periodic orbits for a rotating disc, as we have shown in a previous work 10 . Our construction will be based on the structure of the chaotic saddle of the scattering system, and thus on the arrangement and stability of the fundamental periodic orbits of the system 13 . We shall focus on the sym metric periodic orbits 14 formed by collisions of the ring particles with the outer disc, which may also interact with the inner disc. We shall see that there are intervals of the Jacobi integral (the only relevant physical quantity) where these orbits happen to be stable. The small KAM regions found around them are dynamically confined and initial conditions starting on them will re main trapped on a bounded region of phase space. An ensemble of particles with such initial conditions will then form ring like structures in configura tions space that display properties similar to the braided F Saturn ring. Our assumption of collision-less ring particles will be crucial for the treatment. Collisions with the inner disc will sweep away ring particles depending on the configuration of the billiard, representing this a mechanism of erosion of the rings 4 - 15 - 16 . In this model we can relax assumption 4 in an adiabatic approximation. The sweeping will be more extensive and can still be calculated. As we shall find that this sweeping action is the principal effect of the inner shepherd, we shall first see what ring structure we obtain from a single rotating disc by analyzing the stability of its periodic orbits. Indeed we construct a whole family of rings in configuration space, which display a structure similar to the braids of Saturn's F ring. Then we consider the role of the inner disc in the resulting arrangement of the symmetric periodic orbits. Finally we discuss how this first step relates to the entire program and what will be the next steps towards its completion. To establish notation consider a point particle in an open billiard which consists of two non-overlapping discs on circular orbits (Fig. 1). We consider the discs of radius d< whose center is at distance Ri from the origin (i = 1,2). For simplicity, we shall discuss the case where the discs move with the same angular velocity u), which allows us a time independent Hamiltonian formulation, although the dynamics of the billiard indeed allow us to relax this condition. The (constant) angle formed by the vectors pointing to the centers shall be denoted by 0. In what follows, we shall refer to the inner and the outer discs as disc 1 and disc 2 respectively. In these terms, the

23

Figure 1. Geometry of the two-rotating discs scattering model: Ri (t = 1,2) define the position of the center of the disc and d< their radius; 0 defines the relative positions of the discs.

non-overlapping condition reads as R\ + d\ < R2 — d.2In a rotating (synodic) frame, the well-known Jacobi integral plays the role of the Hamiltonian J

= ^(vl+V2y)-u(xpy-yVz)

+ V{x,y)

(1)

with V(x,y)=0 outside the discs, and infinite inside them. On a fixed coordinate frame (sidereal frame), the dynamics are straight forward. A particle moves on a straight line with constant velocity until it hits one disc, which changes both the magnitude and the direction of the velocity vector 12 . After the collision, a point particle wins (looses) energy if it bounces off the discs on its front (back) side, i.e. against (in favor of) the rotation of the colliding disc. This naive observation is precisely the key point for the construction of the symmetric periodic orbits of the system. If a particle bounces radially, that is at one of the intersection points of the disc and the line that joins the origin and the center of the disc, then the energy is conserved and incoming collision angles are equal to outgoing ones. This leads to symmetric periodic orbits whose importance for the construction of the chaotic saddle is well-known 9 ' 14 . Matching the time between consecutive radial collisions for the particle and one disc, the periodic orbits obtained from bounces off the same single

24 1.51.00.50.0-

to

^v

_n.,\ <&x« * ^ £&«

-1.0-1.5-

.0 -1.5 -1.0 - 0 . 5

0.0

00

0.5

1.0

1.5

'

.

0.25-

-

A

-0.5-

i

-i—

0.50

-

Y n"2

-0.25

} n-1

2.0 -1.5 -1.0 - 0 . 5

2.C

0.0

0.5

1.0

1.5

2

Figure 2. Chart of symmetric periodic orbits for the two disc billiard (Ri = 1.8, d\ = 0.1, R.2 = 3 , da = 1). (a) Periodic orbits J-n (continuous curves) and J\i for /3 = 0 (dotted curves) for various values of n. (b) Periodic orbits of J\i for 0 = 0 (continuous curves) for various values of n; some periodic orbits J22 are shown as dotted curves.

ith-disc (i = 1,2) are given by

12

J{Ri-di)2

-.2 [2 cos2 on - ((2n + 1)TT - 2a*) sin 2a*]. (2) l(2n + 1J7T — 2ajJ Here, a* G [—5r/2, TT/2] is the outgoing angle for the radial collision with the disc (negative values correspond to a particle running against the rotation), and n = 0 , 1 , 2 , . . . denotes the number of full turns that the disc completes before the next collision. These curves are plotted for some values of n in Fig. 2a as continuous lines. Symmetry considerations show that similar periodic orbits bouncing be tween the two discs only occur for 0 = 0 and 0 = n. These orbits satisfy Jii = T-r—.

J12 = 2VP~

w

^

2

~ d2^vPsintt2

(3)

where the (constant) velocity between collisions is (for j3 — 0, n) _ p

u)(R
(4)

and, in these cases a i and 0:2 are related by sinQ2 _ sinai Ri + d] i?2 — d%

(5)

Notice that eq. (5) defines the limits |ct21 < a™** for which the orbits J\2 exist, where sin(a 2 " ax ) = (R\ + di)/(i?2 - ^2)- The case where the equality holds correspond to tangent collisions with disc 1.

25 In Fig. 2b we have plotted for some values of n the symmetric periodic orbits which involve disc 2 for /? = 0, as given directly by eqs. (2) and (3-4). It follows from eqs. (3-4) that for (3 = IT the n curves correspond to the n + 1 curves of f3 = 0. As it can be appreciated in Fig. 2b, the curves J\2 display cusps at the edges of the interval where they are defined (|c*2| = a™*") and decrease monotonically. iFrorn the definition of n it follows that there is no branch of the n = 0 family of symmetric periodic orbits for negative values of CK2 in the case /3 = 0. In fact, this is related to the singularity observed in J\2 for n = 0 and /3 = 0 as a 2 —► 0 (Fig. 2a), which represents the infinite velocity that the particle needs to perform an c*2 = 0 periodic orbit, which bounces on both discs before they complete a full turn. On the other hand, we notice that the curves J^i (continuous lines in Fig. 2a) are convex or concave depending only on the sign of Q2, and then display local maxima and minima. The important fact to observe here is that the derivative of the J\% families is strictly negative for any n, while dJu/da2 (i = 1,2) vanishes at least in one point for all n. This vanishing derivative is essential for the occurrence of saddle-center bifurcations which are responsible for the appearance of stable islands 1T. For simplicity, we consider first the case where the scattering billiard is composed only by one rotating disc, the outer one. A saddle-center bifur cation scenario associated with the maxima and minima of the curves Ju is responsible for the appearance and destruction of the single-disc symmetric periodic orbits for the rotating billiard 12-18. Therefore, just below each max ima or above the minima, there exists an interval of Jacobi integrals where one of the corresponding periodic orbits is stable and is surrounded by invariant tori; the other fixed point corresponds to a hyperbolic one whose manifolds form an incomplete horseshoe. These maxima and minima are defined by the condition d Jyzl&to'i — 0) which can be rewritten as 2 tana* = — — ± 1. (2n + l > - 2 a *

v(6)

'

independently of the specific geometrical parameters considered in the model. Considering the n = 0 curve in Fig. 2a near the maximum, there are only two fixed points, one is stable and appears surrounded by KAM-tori. These invariant tori as well as some regions of chaotic motion are kept bounded dynamically by the invariant manifolds of the partner hyperbolic fixed point. This scenario persists under variations of the Jacobi integral until the formal parameter describing the binary horseshoe's development attains the value 2/3 (see 19 ' 20 for details), which indeed defines a continuous interval for J where bounded trajectories exist. Beyond this interval, the stable fixed point

26

turns into inverse hyperbolic; the incomplete hyperbolic structure formed is thus sufficient to make that generic initial conditions escape in scattering trajectories. In the present context, we shall concentrate in the set of initial conditions that remain trapped and where small variations of these initial conditions are also trapped. That is, we consider all those initial conditions which are dynamically bounded by the manifolds of the hyperbolic fixed point, without considering if the orbit is periodic, quasi-periodic or chaotic. This set of initial conditions has strictly positive measure 2 1 . We shall stress that, for a generic ensemble of initial conditions denned on the entire phase space, most initial conditions will escape to infinity. How ever, particles whose initial conditions are chosen within the bounded regions described above will be trapped by collisions with the disc. This observa tion is fundamental for understanding the formation of spatial patterns by particles whose initial conditions are started precisely on those dynamically bounded regions found in distinct intervals of the Jacobi integral. An ensem ble of particles all of which are within the regions bounded dynamically will form definite structures in configuration space or rings that will not escape (Figs. 3). Notice that the particles within a ring can have different velocities, since their corresponding value of the Jacobi integral may be slightly different, and accelerations or decelerations due to not perfect radial collisions (<j> fa 0) will be common. Moreover, in the sidereal frame these structures will rotate with the period of the discs. In Fig. 3a we present in the sidereal frame the ring formed by an ensemble of initial conditions whose Jacobi integral belongs to a neighborhood of the maximum of J22 for n = 0. Figure 3b displays the set of rings formed by the maxima and minima of J22 for n = 0,1. In these figures, each point represents at fixed time t the position of a single particle which moves on a rectilinear trajectory. These particles have collisions with the disc at distinct times and in any point along the disc's circular orbit. The loops shown in Figs. 3b are associated with single rings corresponding to families of symmetric periodic orbits where n > 1, i.e. when the disc completes one or more full turns before having the next collision with the same particle. They result from particles on different rectilinear trajectories which approach during some interval of time. In connection with the properties observed in Saturn's F ring 2 , features like braids or clumps can be interpreted within this simple model. First and more important, braids arise as different stable strands where the ring particles move (Figs. 3). They are the natural consequence of the structure of the symmetric periodic orbits as defined by the parameter n. In turn, the regions where braids intersect can be associated with clumps, since the local

27

i

4 -

i

i

i

(b)

2^

0-

G ^

•

;

;

■

■

-2-4- 4 - 2

0 x

2

4

Figure 3. Rings for the one-disc scattering model (sidereal frame), (a) Ring associated with the n = 0 maximum of the J22 periodic orbits; (b) corresponding rings for n = 1 maxima and n = 0 minimum.

density of particles increases there. The observational fact that clumps have a finite life-time has no analogue in this model; this might be related to the assumption of neglecting collisions among particles, the planar character of the billiard or the fact that we neglect all perturbations, which may even

28

cause the rings to have a finite life-time on a much longer time-scale 4 i 1 5 - 1 6 . Finally, we notice that the thinness of the rings is a consequence of the very small region occupied in phase space by the invariant tori and thus of the dynamically bounded regions. Clearly the appearance of rings as described above requires but one ro tating disc. Hence, for the full two disc billiard such independent rings may appear associated with each one of the discs; we shall neglect the rings as sociated with the inner disc. The only situation influenced by both discs is related to the periodic orbits which display bounces with both discs as given by J12. As mentioned above, the J12 families decrease monotonically as a function of a? and it can be shown that d J\2/&a2 is strictly non-zero. There fore, these families of periodic orbits are all unstable and do not contribute to the formation of rings. Yet the inner disc has another important effect on the rings generated by the outer disc. It may interfere with the stable periodic orbits that define a ring and thus eliminate this ring through collisions of the ring particles with the inner disc. To see this, consider particles with initial conditions belonging to some symmetric periodic orbits J22- Depending on the geometry of disc 1, some of these will collide with the inner disc. Particles that collide with the inner disc will loose the needed correlations that bounded their motion. Such particles will be knocked out of the disc system either directly or after a few more collisions. In Figs. 4 we illustrate this plotting the actual characteristic curves for the symmetric periodic orbits J22 for various angles /3, where the size effects of the inner disc are taken into account (R\ = 1.8, d\ — 0.1). The gaps shown in some families of periodic orbits are due precisely to defocusing collisions with the inner disc. We recall that the angle /3 defines the relative initial position of the discs. The dependence of the symmetric periodic orbits on this angle is impor tant since it allows to relax condition 4 mentioned above. If we consider that the discs move at different and incommensurable speeds this implies that all rings that cross the trajectory of the inner disc are eliminated. Thus only rings pertaining to large angles a will survive. This mechanism confines the rings to exist between the trajectory of the inner edge of the outer disc and the outer edge of the inner disc. It remains to be seen under what conditions any stable orbits survive this sweeping procedure. This depends on the de tailed geometry of the system. The existence of such a geometry implies the persistence of stable rings for the two disc billiard, even in the case of different angular velocities, i.e. in the absence of a Jacobi invariant. In terms of the geometry defining the billiard, an important parameter is Q 2 nsx , which defines the range of angles for which the periodic orbits J22

29

7

-"

(b)

1

1

1

o

1

V

(d)

_

(e)

~

(f)

^ \

- A., A , i

i

- 2 - 1 0

i

i

1

i

i

i

i

2 - 2 - 1 0

i

1

i

i

i

2 - 2 - 1 0

i

i

i

1

2

a Figure 4. Periodic orbits of the J22 family (n = 0,1...4) for different values of 0. (a) 0 = 0°, (b) 0 = 5°, (c) 0 = 10°, (d) 0 = 45°, (e) 0 = 90° and (f) 0 = 135°.

are influenced by the inner disc. One can thus define the geometry of the billiard in such a way that the associated a™** is so small that none maxima or minima of the curves J22 will be affected. That is to choose the orbit and the radius of the inner disc in such way that no stable periodic orbits are affected for all values of /3. This is trivially fulfilled for a™** less than the absolute value of the angle where the maximum of the J22 family with n = 0 is found, which ensures that all the rings will be preserved. This intuitive idea can be generalized, since the stationary points of the families J22 satisfy eq. (6), and the absolute value of the solutions of eq. (6) are arranged in a precise way. Then, for a™** bounded from above by the absolute value of a precise solution of eq. (6), all the corresponding solutions with \a*\ > a™*" will not be affected by the inner disc. The associated rings will exist for all values of 0, and therefore, the rings will persist even for different disc's angular velocities. However, in the situation where a™*" > "max there

30

is no such upper bound. As (3 changes in time, the gap induced on the curves J22 by the inner disc changes continuously and destroys all stable regions and therefore all rings. Summarizing, we have studied a scattering billiard, which allows to dis cuss some qualitative aspects of shepherding processes in planetary systems with rings, with particular consideration of the braided structure observed in Saturn's F ring. In our model, stable rings are consequence of the ex istence of stable periodic orbits and their associated KAM islands, which define phase space regions of bounded motion. We choose an ensemble on phase space of non-interacting point particles whose initial conditions belong to these regions. This ensemble generates definite patterns or rings in con figuration space. These have a natural braided structure as a consequence of the hierarchical arrangement of the symmetric periodic orbits of the system. Different strands of the ring may overlap resulting in structures that suggest qualitatively the clumps observed in Saturn's F ring. The model includes mechanisms, which may destroy some of the rings or even the totality of them; these mechanisms are specified by the position and radius of the inner disc. These erosion processes are indeed a starting point to explain the existence of a finite number of braids in Saturn's F ring. The same concepts, using a different geometry, could also explain the absence of observed braids in Epsilon ring of Uranus. Certainly, our model is unrealistic. We are neglecting entirely the \/r potential of the central body as well as the interactions between ring particles. The interaction with the shepherds at least resembles the collision orbits which Henon has shown to be the skeleton of the restricted three-body problem for small mass parameter \i as given by the /i = 0 case 9 . Based on the known relation between the stable periodic orbits responsible for the braids in our model with relevant families of orbits in the restricted three-body problem a next step toward understanding the braided structure would be to analyze a restricted four-body problem with a large central body and two small shepherds. Based on the same symmetry considerations used here we can conjecture that the \i = 0 collision orbits form the skeleton of the stable structure supporting the braids. We could also study the effect of small eccentricities of the shepherd orbits in our disc model. The authors want to express sincerely their gratitude for the useful dis cussions and critical remarks to F. Bonasso, C Jung and F. Leyvraz. This work was partially supported by the DGAPA (UNAM) project IN-102597 and the CONACYT grant 25192-E.

31 References 1. A. Brahic, Planetary rings, in Formation of Planetary Systems (A. Brahic, ed.), Centre National D'Etudes Spatiales, Cepadues-Editions, Toulousse (1982). 2. Planetary rings, R. Greenberg and A. Brahic (eds.). The University of Arizona Press, Tucson (1984). 3. B.A. Smith, and the Voyager imaging team, Science 212 (1981) 163; Science 215 (1982) 504. 4. M. Henon, Nature 293 (1981) 33. 5. J.-M. Petit and M. Henon, Icarus 66 (1986) 536. 6. S.F. Dermott, C D . Murray and A.T. Sinclair, Nature 284 (1980) 309. 7. P. Goldreich and S. Tremaine, Nature 277 (1979) 97. 8. S.F. Dermott, Nature 290 (1981) 454. 9. M. Henon, Bull. Astron. (serie 3) 3 (1968) 377generating Families in the Restricted Three-Body Problem, Lecture Notes in Physics m52, Springer-Verlag, New York (1997). 10. L. Benet, T.H. Seligman and D. Trautmann, Celest. Mech. Dynam. Astron. 73 (1999) 167. 11. L. Benet, D. Trautmann and T.H. Seligman, Celest. Mech. Dynam. Astron. 66 (1997) 203. 12. N. Meyer et al., J. Phys A: Math. Gen. 28 (1995) 2529. 13. P.H. Richter, H.J. Scholz and A. Wittek, Nonlinearity 3 (1990) 45. 14. L. Benet, C. Jung, T. Papenbrock and T.H. Seligman, Physica D 131 (1999) 254-264. 15. R.A. Kolvoord and J.A.Burns, Icarus 95 (1992) 253; Icarus 99 (1992) 436. 16. J.-M. Hertzsch, H. Scholl, F. Spahn and I. Katzorke, Astron. Astrophys. 320 (1997) 319. 17. M. Henon, Ann. Astrophys. 28 (1965) 992. 18. H.R. Dullin, Nonlinearity 11 (1998) 151-173. 19. B. Rueckerl and C. Jung, J. Phys A: Math. Gen. 27 (1994) 55. 20. C. Lipp, C. Jung and T.H. Seligman, in Proc. of the Fourth Int. Wigner Symp., N.M. Atakashiev, T.H. Seligman and K.B. Wolf (eds.) World Scientific, Singapore (1996). 21. A.J. Lichtenberg and M.A. Lieberman, Regular and Stochastic Motion, Applied Mathematical Sciences 38, Springer-Verlag, New York (1983).

LOW REYNOLDS NUMBER SWIMMING IN TWO DIMENSIONS ALEXANDRE CHERMAN Fundagao Planetdrio Av. Pe. Leonel Franca, 240, Rio de Janeiro, Brazil, 22451-000 JOAQUfN DELGADO Departamento de Matemdticas, Universidad Autdnoma Metropolitana-Iztapalapa Av. Michoacdn y la Purisima, 09340, Mexico FERNANDO DUDA Departamento de Mecdnica da UFRJ Cidade Universitdria, Rio de Janeiro, 21944-970, Brazil KURT EHLERS California State University, Monterey Bay 100 Campus Center, Seaside CA, 93955-8001, USA JAIR KOILLER Visiting Position. Instituto de Matemdtica da Universidade Federal Fluminense Av. Mario Santos Braga, s/n Niteroi, 24020-140 Brasil RICHARD MONTGOMERY University of California, Santa Cruz CA, 95064, USA A geometrical approach for low Reynolds number swimming was introduced by Shapere and Wilczek1. Here we pursue some developments for the two dimen sional theory. The outer membrane or the ciliary envelope of the planar organism is represented by the conformal image of the unit circle. Power expenditures and velocities can be computed using complex variable techniques. As an example, we present the calculations for a self deforming ellipse. The results compare well with observations for the nematode Turbatrix aceti. We also compute the most efficient swimming stroke, using the criterion efficiency = velocity/hydrodynamical power. A pattern noticed by SW for the circle and the sphere is confirmed: effi ciency is optimized around certain high order geometric modes. For the case of a deforming membrane, these modes require great mechanical stress. However, such high order geometric modes are easily emulated by ciliary envelopes without extra (mechanical) power expenditure. Therefore, coordinated spatio-temporal ciliary movements, besides providing an inherent maneuverability, have the added advan tage of saving energy.

32

33

1

Introduction

We have been interested in nonholonomic motion and in microswimming since two of us (JK and RM) listened an inspiring talk by Frank Wilczek at Cornell, about ten years ago. Our general program was presented in 3 and in particular, we described the collective "N-body probem" of microswimming. Here N can be very large! (We hope this catchword to be somewhat related to Celestial Mechanics.) Microscopic organisms propel themselves in an inertialess environment. Shapere and Wilczek 1,2 , 1989, henceforth SW, showed that the rotation and translation of a swimming circular organism can be described by the holonomy of a connection on a principal bundle (whose structure group is the group of Euclidean motions). Actually at this point the mathematical structure is only formal: The base space is infinite dimensional and consists of all possible shapes of the microorganism (the shape space) and could be modelled by a convenient functional space of embedings. If the amplitude of the swimming motions consist of small amplitude surface oscillations (i.e a stroke), then the holonomy can be computed using the curvature elements of this connection, computed at the average shape, the base shape. We report here our findings in the two dimensional context, the first re sults appearing in 4 . There are very powerful techniques for solving the Stokes equations in the plane and it is possible to compute the Stokes curvature (the covariant derivative of the Stokes connection) for any base shape that is the conformal image of the circle. We present some corrections to SW (concern ing only the rotational part of the connection). We also compute the power expenditure operator, details appearing in 5 . We discuss the case of an elliptical swimmer. Two of the major modes of swimming can be studied with this model: that of an organism with a self deforming outer membrane, and that of a slender organism that uses undulatory body motions. The results for the slender body using undulatory cycles provide a good model for the nematode Tarbatrix aceti. We numerically compute the most efficient axially symmetric swimming (the efficiency notion here is simply that efficiency = velocity/power). We verify that they are governed by certain high order geometric modes, following the same pattern as SW found for the circle and the sphere. Comparisons between different efficiency concepts are given in 15 . It is most important to notice that for a deforming membrane, high or der geometric modes require great mechanical stress. On the other hand, such high order geometric modes are easily emulated by ciliary envelopes,

34

without extra mechanical power expenditure! Coordinated spatio-temporal ciliary movements, besides providing an inherent maneuverability, have the further advantage of saving energy". In an article to be submited elsewhere6, the curvatures for the three di mensional case of an ellipsoidal organism are presented, extending the results of Lighthill8 , Blake9 , and Shapere and Wilczek2 . With little extra work, one can also make a direct comparison of a sphere with an infinite swimming sheet for the swimming velocity, power output, and optimal combination of tangential to normal components of the surface distortions. 2

The planar swimmer

The fluid domain is represented by the complex z-plane. A fluid velocity field will be given by the real and imaginary parts of complex valued functions v:C —» C. The outer membrane or the ciliary envelope C of the planar swimmer will be represented by the unit circle 7 = {£ 6 C | |£| = 1} under the conformal map r r z = u;(0 = R^+ f + --- + p), where R is a real number. While the organism's physical shape is represented by a closed curve in the z-plane, we will make our computation of the curvature in the £-plane where the organisms shape is parametrized by the unit circle. As a basis (modes) for the vector fields on C we will choose the push forward under composition (not using the jacobian matrix), of the basis of Fourier modes on the unit circle: V =
Stokes Equations in the plane

For the planar problem there is a very powerful technique, developed by Mkhelishvili11 in the context of planar elasticity theory, that uses Cauchy inte°This suggests that Nature has found, by inventing cilia, a way to circumnvent a "mathe matical obstruction" for the efficiency of motion. 'Although the notations is coumbersome, we follow SW1 notation for pourpouses of com parison.

(1)

35

grals to solve a functional equation which is equivalent to the Stokes equations in the plane. It is well known that the Stokes equations in the plane are equivalent to the biharmonic equation A4\P = 0, see Bachelor12 for instance. It is an elementary fact from complex function theory that biharmonic functions have a representation in terms of analytic functions. SW use this representation to show that any solution can be written as v{z,z) = 4>(z) - zWJT) +W)

(2)

where and ip are analytical functions called complex potentials by similitude with Laplace equation in the plane. The boundary value problem for the Stokes equations is therefore equivalent to solving a functional equation with boundary condition

V{8) = 4>(s)-sW)+Jta)

(3)

where s is a coordinate on C. The point is to insert a boundary vector field into the lefthand side of (3), then find the analytic extensions of 4> and rp. For boundary values on the unit circle SW matched power series coefficients on both sides of (3). For conformal images of the circle the algebra becomes much more complicated, and the use of Cauchy integrals seems to be the best method of solution. The boundary data (3) pulled back to the unit circle under the transformation (1) takes the form

V{o) = (a) - ^ & > ) + W)-

(4)

u)(a) The main tools for solving this functional equation are the following versions of Cauchy's theorem 11 , Corollary 1 (Cauchy's theorem for infinite domains)

j _ r f{z)dz 2ni Jcz-£

=

r _/(£) + /(cc), 4 outside c \ 0,

(5)

otherwise

for f analytic outside and continuous on C, a simple closed curve oriented counterclockwise. Corollary 2 A necessary and sufficient condition for the function f, contin uous on the unit circle 7, to be the boundary condition of a function analytic outside 7 is: 2iri J 7 z - i for all £ outside 7.

(

'

36 The basic strategy of Muskhelishvili's method is to use Cauchy's theorem for infinite regions to eliminate one of the analytic functions

, in order to determine the other function. 2.2

Shapere-Wilczek's Connection and Curvature Forms

Appealing to the Stokes paradox, a form of which states that the only solution to the Stokes equation corresponding to a rigid translation of a cylinder is a rigid translation of the fluid as a whole, Shapere and Wilczek asserted that the rotation and translation of the organism associated with a vector field V(a) on the boundary of the organism that generates a velocity field v(z, z) can be determined from the asymptotics of that velocity field. Specifically Arot ■^SW

dz

— Im d> Jlim\z\ !^oo

27r

.

z),

(7)

rz)

(8)

*

and Atr

dz , 2nizViz<

-f

Jlim\z\—»oo One of the purposes of our paper is to provide some corrections and clar ifications to SW. It turns out that while formula for Atr is correct, Arot is not (unless C is a circle). We provide the corrections in section 4. By the general theory of connections, the curvatures are the infinitesimal rigid motions 1Z associated with traversing infinitesimal rectangles centered at C in shape space, spanned by the vectors nV^ and eV£. More precisely,

n{T)v*,tv*)c =e^C

+b

(9)

where w and b are written as U) -=

Arot([Vv*,ev?n}) cnFrot +

« n

+ « n+Wfl.

(10)

and

6 == Atr{[nvl «&]) = «?^mn + « » + « l + « * • (11) The reason for choosing this particular decomposition of the Stokes curvature form, as used by Shapere and Wilczek, will become evident in the examples. For the calculations, one makes the change of variable z = w(f), in which Atr becomes Atr

-y

»M0, "(0)-

(12)

37

The expression for the Lie bracket pulled back to the £ plane is [V„,Vm] = (Vm ■ V)vn -- (Vn ■ V)vm

-{-dzTz^Vm ,dvm 9 ( dz^

l

3

+

{

lzTz^Vm~ _ ( a ^ dz]^Vn

y„.

(13)

T h e Elliptical Swimmer (following t h e S W recipe)

We now compute the SW curvatures for the elliptical cylinder and apply the results to study swimming strategies for a nearly elliptical swimmer (we call this kind of swimmer a self deforming ellipse). It turns out that the rotational component according to SW's recipe is not correct. We present it here for the sake of comparison with the corrected result in section 4). The ellipse is given by the simplest conformal mapping, mi = M, all the others m.i = 0. In fact, the transformation 2=

W

(0=fl«+y)

(14)

maps the region exterior to the unit circle 7 in the f-plane to the region exterior to the ellipse C with semi-axis R(l + M) and R{\ — M). The hodographed basis vn(z(o-)) =

\on+1.

has a complicated explicit expression in the z-plane, namely, for s = R(exp(i6) +

Mexp(-i6)),

we have vn(s)=

expWn + 1)*)

+

=

x(Lt^EEj

\

Fortunately, the calculations in the £ plane are feasible. The boundary condition for the ellipse corresponding to formula (4) takes the form 0(CT)

1 a2 + M -r 0>)+

~ ar ^

^ = ACT"+1'

{15)

38

Because the details of determining the solution to this equation parallel those found in Muskhelishvili in his computation of the stresses on a plate with an elliptical hole, will only state the results:

0 n> 1 0 n= -1 A£ n+1 n < - 1 ,

(16)

and n > -1 n = -l

X

V>(0 = {

(17)

e^$(n + lWn<-l-

Substituting these into the formula for the v n (£) we obtain

Av„tf,0 = * ( f l -

z (e - M) n

(18)

4>' (0 + ^(0

n > -1 n = -1 = < A, Ae+1-(n + l ) ^ i | ^ A r + ^ ^ ^ ( n + l)Ar,n<-l

xt ~\

Expressions (7) and (8) for the connection may now be used to decompose the boundary vector fields into horizontal and vertical parts. Boundary Value n+1

Horizontal Projection Xan+1 0 0 ACT-1 - ImMXa-*

Xvn(a) = A<x ,n>0 Xvo(a) = ACT AU_I (cr) = A Xv-2(cr) = ACT-1 Xvn(a) = Xan+\n< - 2 ACT n + 1

Vertical Projection 0 Arot = ImX Atr = X Arot = ImMX 0

(19)

Caveat: the results for the rotational part are not correct, see Proposion 3 in section 4. To compute the Stokes curvature we compute the Lie bracket of horizontal vector fields, re-expand the resulting vector field in terms of the Fourier basis, then use the above table to determine the corresponding rigid rotation and translation. As an example if m,n > — 1, [VVn,evm} = (er)(m + 1)CT m—n-f 1

oo

erj(n + 1)CT'n - m + l \) ]V"* >JMCTY, fc=0

(20)

39 comparing this with the above table to determine the vertical components we find that for m,n > —1

{

(m + VjM^r0

n-m=j>0 and j is odd otherwise. (21)

(n + 1)M V f%n = ptr

u

_ ptr

- n = k>0 and k is odd otherwise.

m

.= 0

(22)

(23)

For the circle M = 0 and the result of Shapere and Wilczek is recovered, the only modes that couple are those that differ by one. If M ^ 0 all modes that differ by an odd number couple, although the strongest coupling takes place between ones that differ by just one. The effects of coupling between distant modes becomes more pronounced the further the shape is from a circle. 3.1

Examples and discussion

Two examples will be presented, using only the translational part of the con nection (so that the previous calculations are correct). The first, an analog of the example of Blake's swimming circular cylinders 10 , under symmetrical deformations. The second example will be that of a long thin organism that swims utilizing the undulatory mode. The results of this model will be com pared with observations of the swimming motions of nematodes by Gray and Lissman. For the first example consider an ellipse with semi-axis 1 — M and 1 + M, and swimming stroke parameterized by: M S{a,t) = (a+ —) + (.025 cos 2irt)vu + (.025sin27rt)ui 5 (7

+(.015 cos 2wt)vls + (.015 sin 2irt)vig

(24)

To calculate the net translation of the swimmer the curvature components corresponding to the coupling of modes 14 and 15,14 and 19,15 and 18, and 18 and 19 are needed. After one swimming stroke the swimmer's net translation

40

4

5

6

Figure 1. Ellipse (M = 3) at t=0, 0.25, 0.5, and 0.75

is given by: / •7ri4j5ai4
+ / Jo

Fi5Tga,i5ai8dt

(25)

^14 15014019^

Jo = .043 + .017M 2 - .019M

(26)

See Figure 1. Next, the undulatory mode of swimming will be studied by taking the ec centricity of the ellipse to be large and the deformations to have purely imag inary coefficients. Because this strategy is used by swimmers of all Reynolds numbers, this two dimensional model may provide a starting point for build ing models where the inertial forces are not negligible. J. Gray and H.W. Lissmann7 conducted an extensive study of the various locomotion modes of Nematodes. They made observations and measurements of nematodes creep ing on top of gelatin and damp glass, gliding through densely packed suspen sions, and swimming through fluids. The present model will be applied to the latter class of nematodes, the Turbatrix Acetii. While swimming in an open fluid is clearly a three dimensional problem, the quantitative similarly

41

between the two and three dimensional cases suggests that we might obtain a reasonable model for the Tubatrix whose body motions were observed to be essentially planar by Gray and Lissman. The Turbatrix Aceti is a nematode which appears occasionally in domestic vinegar. It has a body length of about 840/im and a diameter of 28/im. Its ratio length/width ranges 13 about 45. It swims by passing undulatory waves down its body. The observed wavelength was 712/im, the frequency is 5.2 s e c - 1 or about one complete stroke every 1.9 seconds. The amplitude was measured to be approximately 107^m and observed speeds were 718/xm per second, or 138/xm per stroke. Thus the foreword progress was was about 16.5% of the worms length per stroke. To model the Tubatrix pick as the base shape an ellipse with semi-axis .06 and 1.94. Frequently the amplitude of the undulatory wave was observed to increase as it passed down the body of the nematode, the tail moving nearly four times as much as the head. This characteristic is related to the ability of the nematode to swim without yawing. The yawing motion associated to a particular swimming stroke is not detected by the present model because the side to side motion is zero on average. As a simple model, consider undulatory swimming motions with a constant amplitude. The specific modes are chosen to best represent the observations of Gray and Lissman (1963), see Figure 2. The amplitude and ellipse size were chosen based on measurements of Gray and Lissman. Parametrize the swimmer by:

S(a,t) = a+^1

+ .17icos2irt(v3(cr) + V-5(
dvnd£

- HtLo(Ma2)k(n 0,

+ l)tf}Mam-n-\

n<

-1

n > -1

42

t-

i

r-

i

k

• •

k—

Figure 2. Locomotion of Turbatrix aceti at t = 0,0.2,0.4,0.6,0.8 and 1.0

' [Er= 0 (^ 2 ) fc ] 2 dvnd[. d£ dzVm

I ~)

((n + l)lf)Ma-n~m-1

- (n + l ) e ^ o - n - m + 1 ) , n < - 1

£7j(m + l ) ( j m - , l + 1 ,

n>-l.

The resulting curvature coefficients are: F3<% = 4,F_ 6 i _§ = 4,F_5i4- = 4M, F_5_4 = 4, all other non-zero components being obtained by antisymme try. The net translation is then found by computing:

L

F3ia3(t)d.4(t)dt

+ I F_6_-5a-6(t)d.-5(t)dt+ Jo + f Jo

f Jo

F-5<4a-5(t)a4(t)dt

F_sAa-5{t)d4(t)dt. (28)

The result of this integral is -0.71. After scaling, the predicted translation per stroke is 154pm, or 18% of its length. Recall that the observed translations were approximately 16.5% of the length. The result seems quite good, in spite of the simplifications, namely: the model is two-dimensional; only four

43

modes were excited, with constant amplitude; the correct Reynolds number is of order 1, in the limit of applicability of Stokes approximation; effects of order > 3 in the amplitude were not taken into account. 4 4-1

Mathematical details and corrections to SW The canonical vs. the hodographed Fourier basis

Suppose that in Goursat's representation for the solutions of planar Stokes equations:

v(z) = W) + (
(29)

we take ip and 4> holomorphic in the exterior of the closed curve C. Without loss of generality, a_i a_ 2 4> — a0-\ 1 7T + ... • z z* * = *=i + ^ + ... (30) z z2The presence of a 0 is a "confession of debt" to Stokes' paradox, since there is no solution vanishing at infinity for the translation of a cylinder. Write

/KO

\e=i

)

with (1.2)

V

*

= MZ)

jvrkeal=^

T T T

- Z(P^Z) = 1 4

m

- Hz" ~ k*zk-X)

Thus for each k < 0 we have four terms, taking for xp and <$> either zk or izk. This is called the canonical basis. Observe that Vk have good symmetry properties with respect to conjugation z —> z and reflection z —> —z. However, Wk do not have these symmetries. This is already one good reason to suspect that the canonical basis is not the best choice. In Appendices A and B we obtain the curvatures for the circle, using the canonical basis, in Appendix C we show that in the case of the ellipse, the canonical basis does not allow a practical computation.

44

Let z = z(£) the conformed map taking the exterior of 7 = S 1 to the exterior of C. We transport "hodographly" the Fourier basis {cr n + 1 } in 7 to C, that is: Vn(s) = ( £ ( s ) ) n + \

n€Z.

We adopt the indexing n —» n + 1 just to maintain the convention used by SW. In contradistinction with the canonical basis, the symmetry properties are preserved by the hodographed Fourier basis. Suppose, for instance that C is symmetric with respect to the x-axis. Taking real coefficients, we pro duce symmetric deformations for all n. With imaginary coefficients, we get ondulating deformations. The Fourier basis has other advantages. For any conformal map, half of the problem to find the Stokes extensions is trivial: for n > — 1 we take always ip = fn+1 and 4> = 0. As we will see in section 5, there are also remarkably simple properties for the power expenditure operator. 4-2

Total force and total torque

A simplified version of Stokes' paradox is given by the following Lemma 1 If the potentials ij) and <j> have no singularities at 00, as in (30) then the total force vanishes:

I fds = 0 Proof. This follows from the identity fds — —2nidU where

U = 4>(z) + zp-xp . In Appendix E we present a derivation of the force field / = 4fiRe(' n) - 2/i( z W - W) n . (31) Actually, Stokes' paradox implies that in order that the total force be ^ 0, the potential U must contain a logarithmic term. Following Blake 9 , Shapere and Wilczek1 avoided the logarithmic singularities altogether, using the following Criterion 1 Expand a vectorfield v(z) along C with respect to the canonical basis (without logarithmic singularities). Then the translational component of the connection 1-form is the constant term ip0.

45

For the rotational component of the connection, we now present a correction1 regarding the rotational part. Criterion 2 In dimension two the rotational part (as in dimension 3) is de termined by the condition "total torque T = 0 ". In fact, we have Lemma 2 Using the canonical basis, T = 47r/j/m[6_i] Proof. A simple calculation gives tds = Im[z/]ds = -2fj.Re[zdU] = -fidF where F = zz(& +4>')-z7p-zlp

+^ + l

e Z=

I il>{z)dz JZo

Observe that £ is multivalued if b-\ ^ 0. Remark 1 The reader may feel unconfortable with the presence of the con stant rigid translations of the fluid as a whole, which do not decay at infinity: 1 and i as allowable vectorfields. Can one do differently? the answer is yes. Translations of the shape can also be induced by stokeslets. However, these have even worse behaviour, logarithmic at infinity14. Thus, in two-dimensions, one can still define translation-horizontality us ing the criterion "total force = 0", but this requires replacing the constant vectors 1 and i by the two stokeslets, on the x and y directions. In a future work we plan to check if the criterion using stokeslets yields the same results for the connection. 4-3

The 1-form of the connection

Given a velocity field V along C, we denote (a, b) the translation and u>k the infinitesimal rotation such that V — uiz — (a, b) is horizontal. First, we introduce terminology. Definition 1 If(a,b) = (0,0) we say V is translationally-horizontal. 0 we say V is rotationally-horizontal.

Ifu> =

46

Along C, we expand V in terms of canonical components oo

4

«w=^+EE«* • k
SW argue that vh(s) = v(s) — a(iz~l)\c

dZ 2 n i z

v(z,z)

(32)

with

a = Imao

7r~.v(z,z)

(33)

2 n l

is rotationally-horizontal. In other words, the rotational component of the connection, would be

Arsoo

^-Mz,z),

(34)

ll{%

This is not correct. Note that an infinitesimal rotation of C is (except for the circle), a combination of i/z with other elements of the canonical basis. Hence, for the rotational part, it is not sufficient to consider the single element i/~z in the basis, corresponding to xj>-\ with imaginary coefficient. In other words, the flaw stems from the fact that, unless C is a circle, i/z\c does not represent an infinitesimal rotation Re = iz\c ■ 4-4

Infinitesimal rotations: corrected formula

Consider a conformal transformation

z = u(t) = R{£+Y + --- + ^ ) from 7 to a curve C (we can also take n = oo). The following lemma is obvious.

47 Lemma 3 The infinitesimal rotation vAth angular velocity w = 1, is given by i z ( 0 = iR(v0 + miV-2 +

h mn u_( n + i))

(35)

where, in the right-hand side, v^k = exp(i(—k + 1)6) are elements of the hodographed Fourier basis. For each k = 0, k = —2, k = - ( n + 1 ) (k = - 1 is absent), we must compute the Stokes extensions of the hodographed Vk(s). Actually, for each k we need to find "only" the coefficient 6* j of ipk in the expansion of v\ in terms of the canonical basis. In the proposition below we omit the subscript - 1 in the 6fc's. Proposition 1 For indices k = 0,k = -2,---,k

= - ( n + 1),

the rotational part of the connection is given by: Arot,

A

x _ {Vk)

1

Mb*)

,, fi x W

~ R Re{b" + m, 6-2 ■ • • +mn &-(»+D) '

In fact, the vectorfields along C h Vk = t,

Imbk

*-flite(6P + m l 6 - » - - H n B 6 - ( ^ ) )

n, I

*(l''+mi

^ w

- » + ~ + m « "-(»">) (37)

are rotationally-horizontal. Conjecture 1 Following Ken Meyer's talk in this Conference, our bet is 95%: All the remaining elements of the hodographed Fourier basis are rotationallyhorizontal (that is, for k > 0 and for k < — (n 4-1) )■ This involves finding the asymptotics of the Stokes extensions for the corresponding elements of the hodographed Fourier basis. We now give a more intrinsic version for (36). In order to obtain Arot(v) = a we require that (v — aiz)\c produces zero total torque. De note T the operator yielding the total torque. From T(v — ctiz) — 0 we get a = T(v)/T(iz) . Theorem 2 rot _ -^Ep lim Im — & Airot(, (v)A = v{z)dz u(x :

oo \ liri Jtz\ = R * J\t\

where Tc = T(iz) is the torque associated to the infinitesimal rotation iz(s) ofC.

48 4-5

The ellipse. Rotational components.

We saw, in section 3 the expressions for the Stokes extensions 1, a_l

<M0 =--4 +

+ •••,

bki

MO --• ^

+

-

We are interested in finding which A; have nonvanishing coefficients 6iLj. We claim that only UQ A and U_2,A do have them. In fact, since ( A _

Afi

,n = 0

• ,n= -2 £ € 2 -M (omitting the superscripts k) it follows that the only nonvanishing 6*i occur for k = 0 and A; = —2. Hence L e m m a 4 The following expression for the total torque holds I

Ann T(v-2,x)

= Im{R\) =

-RIm(\),

■MRIm{\),

in particular the rotational components of the connection are

A^(Xv0) = - ^ M A ) ,

A-(A,_ 2 ) = - ^ M A ) .

lc J-c To compute Tc, we consider the infinitesimal rotation iz\C = iz(C). The Stokes extension is given by: iz(C) = iR{a + Mo—1) = iRa + iRMa~l

= v0,iR(o-) +

V-2}IRM((T).

Thus the infinitesimal rotation of the ellipse requires two elements from the hodographed Fourier basis, precisely those having coefficient s 6* x ^ 0. Now, iR

m =

iRM2i2

+ M~l

-T—^-^TAT iR2 iR?M2 +■ RZ R£ iR2(l + M2) + ■•

49 Lemma 5 The total torque associated to the infinitesimal rotation T(iz) of the ellipse is T c = -47r/ifl(l + M 2 ). substituting the above expression in the expression for the rotational compo nents of the connection yields Proposition 2 The rotational components of the connection at the basis el ements VQ,\ and i>-2,A are

Arot(vo,x) = Arot(v.

-2,A) =

Im(X) = U)Q, R{1 + M 2 ) MJm(A) = u.-2R{l + M2)

in particular the following decomposition into rotationally-horizontal purely rotational components holds

. Im(X) \

/m(A)

,

MJm(A)

«* J .XW = ( A --l

l + M2

) a --%1 +

and

j

M^

Summarizing: T h e o r e m 3 Given

V( . A . > ) = £ A » < T B + 1 along the ellipse. Then Atr(v) = X-i,

Im(X0) M/m(A_2) Arot{v) = " R(l + M2) ' R{l + M2)' The rotational components of the curvatures can be computed in a similar fashion as in section 3, finding the 6_i residues in the Lie brackets. 5 5.1

Power expenditure The differential form fds = gdO

Forces can be represented by the same space Vc of velocity vectorfields along C, with the L2 norm. Intrinsically, however, it is an abuse to consider 2fif = 2/x (2Re(' - z ") n)

(38)

50

as an element of Vc- Actually, the stress force is a differential form fds, where s is arclength parameter along the curve C. It belongs to the dual space V*. In what follows we use the notation / = f(v(s)) for the vector of forces associated to the vector of velocities v(s) along C. We now explore (to our benefit) the possibility of not identifying these spaces. We use the pull-backs of the forces as forms to 7, under the conformal mapping z = z(£) sending the exterior of the 7 to the exterior of C. Proposition 3 Let s be the arc length parameter along the curve C, and 6 the polar angle in the unit circle corresponding under the conformal mapping z = z(£). Let f = f(v{s)) be the vector of forces associated to the vector of velocities with complex potentials , ip, and let g d9 denote the pullback of the differential form f ds to the unit circle, so f ds = g dd, then I + II + III + IV + V

(39)

where

i = d4o-' d

in=m j y

_

jma

\dtj dz/d£ z{£)a-1 dP4>ldt? dz/d£

z{0a-^_m£zMe 2 (dz/dO

(40)

The proof is a long but straightforward calculation, using the chain rule and taking into account that £ = a = exp{i9) along 7. Since we are using hodographed velocity fields v(a) from 7 to C, the power expenditure is given by the usual inner product V = 2fi I {v,g)i where now we identify gd0 with the vector field g 6 V 7 , the space of vector fields along 7.

51

Lorentz reciprocity 14 carries over to this representation, in other words, the operator ^:V7->V7,

v^g.

(41)

is self-adjoint and positive. If we could determine the spectrum of V1, then the task of computing power expenditures would be completely solved. We assert that one of the advantages of the hodographed basis is that half of the spectral problem is a free lunch! In fact, for n > - 1 we know that ip = £-( n+1 > , 0 = 0 . hence V = a - ( n + 1 > = <7 n + 1

and drj)

9=~(J-1

= - ( n + l ^ - ^ + ^ a - 1 = - ( n + 1) an+2 a'1 = - ( n + l)c7 n + 1

Theorem 4 For every conformal mapping z = z(£)> ifn+1 > 0, then <jn+1 is an eigenvalue ofV"1, with eigenvector — (n + 1). It is interesting to look what happens geometrically in the physical bi plane. The force field / G Vc> is parallel (with opposing sense) to the velocity field v(z(a)) = c n + 1 . The scale factor depends on the conformal mapping, ds/dd To compute the power expenditure V, the scale factor disappears when we change the integration to 7. The difficult part of the spectral problem involves only the indices n + 1 < 0. The following proposition shows that they form an invariant subspace. Teorem 5 The operator Vy(v) = g leaves invariant the subspace of nega tive Fourier modes an+1. Proof. Use Lorentz reciprocity. Let m > 0; then (V(a " + 1 ) , am) = (V(am),

an+1) = f -maman+ld9

= 0

52

5.2

Computing g for the ellipse; negative Fourier modes

Recall that for p < —1

4"-H'*',. *

= A (

P +

1 ) ^ - ^

We insert in (39-40). In the denominators appear factors 1 — Ma2, coming from the derivatives of z(£) and £ 2 — M, followed by conjugation (remember a = o-1). Well, for 0 < M < 1

n=0

We know, however, that in the final result for g, only negative powers of C do appear. The term /J

=

^

( 7

= A(p + l K + 1

is in accordance, but the terms. I, 777, IV, V are problematic. Nonetheless, a "miracle" occurs: Lemma 6 For the conformal mapping z = R(£ + M/£) of the exterior of unit circle onto the exterior of the ellipse C, let f{Xap+1) denote the vector of forces associated to the hodographed vector field Xvp+\(o-) = Xap+1, let gd6 denote the pullback of the differential form f ds to the unit circle, so f ds = gdO (see Proposition 3), then I + III + IV + V = 0, in particular for p + 1 < 0, f{Xap+i)ds

= X{p+ l ) ^

1

d6.

Proof. A direct calculation. Theorem 6 For the ellipse, the hodographed vector fields v(a) = ap+1 for p + 1 < 0 are also eigenvectors ofVy : v —> g, with eigenvalue p + 1. We think that this is indeed remarkable: when we vary the eccentricity 0 < M < 1, using the hodographed Fourier basis, the power expenditure functional does not change! 5.3

Hypotrochoids

The degree of algebraic difficulty increases, but we can still calcultate, g(Xap+1)d6 for the hypotrochoids 11

—(«♦£).

(42)

53 with m = 1,2,..., and M a positive constant satisfying 0 < mM < 1. We present the results: Theorem 7 Consider the hodographed Fourier basis vn(a) = 0 vn(a) = a n + 1 is an eigenvalue of the Lorentz operator V (see 41) with eigenvalue, — (n + 1). 2. For m = 2 and n + 1 < 0 the "miracle" (see Lemma (6)) still holds, namely if f ds = gd9 is the pullback ot the form of forces and g = I + II + III + IV, asi in (40), then I + III + IV + V = 0. 3. For m = 3 and p + 1 < 0 the same "miracle" holds, except for p = —2, here for the hodographed vector field Au_2(c) = ACT-1 ,

g(Xo-') = (-/? + Mfta-1, /? =

^ ^

so the eigenvalue is different from the others. 4- For m — 4 we found g(Xa-1) = -Xa-1+

ACT"2

Here the hodographed Fourier basis is not an orthogonal basis for the power expenditure operator. 6

Efficiency of an elliptic microswimmer

There are basically two competing efficiency notions for microswimming. Con sider a swimming stroke of period r, in shape space S and the ratio X E

X/T E/T

mean velocity mean power

.

Froude 's efficiency is an non-dimensional quantity which results from mul tiplying this ratio by a "characteristic force" T, and in the Stokesian realm it is equivalent (up to a shape dependent factor) to Lighthill's efficiency (mean velocity) .„,. — . (44) mean power The other notion, which we call SW efficiency (because Shapere and Wilczek essentially use it) is given by efz, =

etsw = —£ ,

(45)

54

where the presence r in the denominator is equivalent to consider all swimming strokes with period T = 1. In this paper we will consider the latter notion, and consider a variational problem first introduced by Shapere and Wilczek leading to the most efficient strokes. For a comparison of the two notions, see 15 . 6.1

The variational problem

We expand the time-dependent, small shape deformations in terms of a basis, so that the problem is linearized to a Lagrangian £ = ^{K4A) ~ v-(Tq,q)

.

(46)

where the matrix K encodes the power expenditure, F the curvatures, v is a Lagrange multiplier. The Euler-Lagrange equations are Kq = vTq .

(47)

In other words, C describes the problem of minimizing the energy expenditure E =\ j\Kq,q)dt

,

X = \ f

.

for a prescribed holonomy (Fq,q)dt

Inserting q = eVeint

,

(48)

, A= — v

(49)

, S = AT * .FAT * skewsymmetric,

(50)

where V is an eigenvector, gives TV = \KV Define W = fCW so that: BW = \W Taking into account that

T

=

2TT/Q,

ef-—

.

(51)

the efficiency can be written as ,

(52)

55

where we omit the factor 27r. Some simple manipulations with E and X give E = ^Q2€2(fCV, V)T X =-l-ne2(FV,V)T

.

Since TV = XK.V, we get e f = ^ - = -iA,

(53)

Recall that A = iQ./v is an eigenvalue of (51), and as expected, it is purely imaginary. In short: Finding SW's efficiency is equivalent to find the supremum of the purely imaginary spectrum. 6.2

Numerical results

For the elliptical swimmer, undergoing symmetric swimming strokes, matrices K and F are given in Appendix F. We found the spectrum using MathLab, for M = 0 (circle), M — 0.1 e M = 0.2 (ellipses with small eccentricity) and M = 0.8 e M = 0.9 (high eccentricity). Regarding the number of Fourier modes, we solved the problem exciting the first five, the first ten, the first 30, and finally the ten modes between 30 and 39. Tables of results are available upon request. Our observations are as follows. As expected, the absolute values increase as more modes are considered. Interestingly, they do not diminish significantly as we exclude the lower Fourier modes. This indicates, as shown by SW for the circle and the sphere, that the more efficient strokes involve essentially the high geometric modes. In the table below we list the highest eigenvalues (51), for small and for high eccentricities. First 5 modes First 10 modes First 30 modes 10 modes, from 30 to 39

M = 0 1.9906i 2.4008i 2.7016i

M = 0.1 1.9664i 2.2845i 2.4884i

M = 0.2 1.9431i 2.1826i 2.3139i

A* = 0.8 2.4541i 3.31621 4.91681

M = 0.9 2.7878i 4.2483i 7.2448i

2.6529i

2.4717i

2.3134i

4.0299i

5.6556i

We have also considered very high order modes, between 30000 e 30009, and we verified that there seems to exist a higher bound for all efficiencies, as conjectured by SW 2 .

56 The advantage of high order geometric modes supports Nature's choice for ciliary motion rather than deformations of a membrane. The mechanical stress would be too strong for a membrane undergoing high order deformations. On the other hand, it is quite easy to a ciliary envelope to emulate these motions. Acknowledgments Thanks to LNCC/CNPq for supporting a scientific visit of one of the authors (J.D.) to Rio de Janeiro, February 1998. Appendix A. The Lie bracket Let z = x + iy,z = x — iy. Denote d_ _ 1 fd_ _ .d_\ % dz ~ 2 \dx dy)

d__l(d_ ' dz~ 2\dx+

.d_\ dy) '

l

A vector field v = (a(x, y), b(x, y)) = a— + b— is represented as v = f

dz

+ f

a=z

where / = a(x, y) + ib(x, y) = f(z, z). Lemma 7 The Lie bracket w = [u, v\ can be computed as follows:

w = [u, v] = 9z £;

9z 9z 9z.

or as

7" 7.

fz fz Jz fz.

-

9 9 .

(54)

L d Td w = h— + h—, az oz

with h = 9zf + 9z-f ~ fz9 - h9Consider, for n, m < 0, the vectorfields vn = zn , um = zm-

razz™-1

(55)

57 written as differential operators as fi

ft

Vn=7l—+Zn az

— az

u m = tzm - mzz™-1)—- + (zm - rrizz™-1) — . az az We obtain [vn, Vp] = pznzp~1

- nzPz"-1.

(56)

Curvatures for the circle For the circle Sa : r = a, the translational component of the curvature is the constant term in the Fourier expansion of these brackets alongo Sa- For n < p < 0 we get Ftr(vn,vp)

= an+pFtr(a^,aM) 1

=

=

2

f*

^ L f

2 n

(pe(n-^Dtf _

ne
dg

Just the first integral can contribute, and only if p = n + 1, so that F"VnUM) = — a

.

(57)

B. Curvatures for the ellipse using the canonical basis The ellipse x2 v2 2 + a b2 is described in polar coordinates by r = r(0), with /cos :2 8

sin20\"1/2

- +-£r)

•

( 58 )

It is possible, but not practical, to compute directly the curvature coef ficients using the canonical basis. Let's see why. To compute Ftran3(vn,vp), we substitute (58) in (56). The two terms have the same absolute value, /cos 2 fl V a2

+

an^\"^b2 )

1 ) / J

58

but different phases exp(i(n - p + 1)0) and exp(i(p - n + 1)0). As a working hypothesis, suppose we will have only p + n odd (recall p + n is < 0 ). The situation is not completely desperate, since the absolute values as well as the phases are trigonometric polynomials. We must now expand the bracket in the basis {vn>wn}. Assuming that we need just the even negative n's, we mould get finite trigonometric expressions on both sides. However, finding the coefficients of the expansion becomes a very complicated combinatorial-matricial problem. C. Ellipse using the Fourier basis. More details. Lie brackets In the text we only gave the case m,n < — 1. For m < —1 and n > — 1 we have: [A„t/„, Amum] = ^T, " ^ T ^ " 1 ~

n Mk(T n-m+1+2*; bliYJ

)

\J

k

_ J2 ^Hm

+

1)Mkan+m+1-2k

k

+ J2 ~^.(m + l)(jfc + l)Jlf

fc

<7-n-m+1+2fc -

k _ J2

*!^»l( m

+

l)(fc

+

1)M*+lCT-n-m-l+2fc

H

k

For m,n < — 1 one gets - m)Mkan+™+l-2k

[Kvn, Xmvm] = £ ^r(n k

+

K

h

J2 ^-(n

- m)(k +

i)Mk+1a~n-m-1+2k

k

+ J2 ^-(m

- n)(k + l)Mka~n-m+1+2k

K

k

+2k k

+

59

A„A

£ ^ ( n + l)MVn+m+l+2fc D. Formula for the Force f

Given a closed curve C, we denote the exterior normal by dz as where s is the arclength, traversed counterclockwise. Proposition 4 n = —i —

1. The pressure is given by p = -4/xfle [<*>']

(59)

f = 4nRe(4>' n) - 2/x( z W - W) n .

(60)

2. The stress force at z € C is

3. The differential form fds can be written as fds = -2t/i {(' +J')dz+

{zW -W)dz)

(61)

Proof. We start with %l> = / -rig. The velocity components are u — f and v = —g. Since / and g are harmonic, Au = Av = 0, Stokes equations are satisfied with pressure p = 0. We compute the stress tensor: en =fx =

^ifx+9y)

ei2 = o ( / y - 9x) = fy = ~9x

e22 = ~9y = - g C / x + S y )

We write the normal vector as n = p + iq. Then

fenei2\fp\

\e21en)

\q)

=

Hxfy

\fy-fx)

'fx-fy\

\.(P\ \q)

[P

Jv fx J \-q, = (en + ien)(p - iq) = V n

60

Now we compute with = F + iG. The components of u + iv = <j> — z' are u = F - xFx - yGx v = G + xGx - yFx A short calculation gives ux = -xFxx

- yGxx

Uy = Fy — lFXy — GX — J/GXy vx = 2GX + xGxx - yFxx =

Vy

X(jTXy

y+Xy

using one the the Cauchy-Riemann equations, namely Fx = Gy. Using the other one, Fv = —Gx we obtain: Au = - 4 F « , Av = ~4Fxy . Hence, the pressure is P=-^Fx

= -2fi(<j>' + W),

and the components of the stress tensor are: e n = -xFxx

- yGxx

e22 = xGxy - yFxy

(en =

ei2 = 2 ("x + uy) = -xFxy

-eu)

- yFxx

In complex variables notation e n + ie\2 = —" z so that the force associated to 4> is - 2 / i i p zn + 2fi((f>' + 4>')n Everything together yields / = 2nW n - 2 # z n + 2/x(0' + ~ft)n

61 E. Matrices K and F for the elliptic swimmer We consider symmetric deformations, so all A„ = 1.

Curvatures n\m

0 0 0 1

I R

-2 0 2 0 -3 0 3

M R

-4 0 4 0

1 1 R

0 M+l R 2 R

0 0 0 2M R

2

-2

2 R

0

0

0 oM±i

1 R 2M*+M-1

1 R o M+l z R

0

0

0

2M R

0

0

2M'+M-1

n

3 R

0

0

3 R

0

0

0

0

0

0

2 R 2 o2M +M-l

R

4

0

0

0

-4

M R

0

0

0 M+l R

3

-3

R

0 1 0 0 0 0 0 0 0

1 0 2 0 0 0 0 0 0

-2 0 0 1 0 0 0 0 0

2 0 0 0 3 0 0 0 0

-3 0 0 0 0 2 0 0 0

3 0 0 0 0 0 4 0 0

R oM+1

R 4 R

Power expenditures ■n\m 0 1 -2 2 -3 3 -4 4

0

2 •* R

QM + 1

"^

0

-4 0 0 0 0 0 0 3 0

4 0 0 0 0 0 0 0 5

4 R

0

0

0

0

62

References 1. A. Shapere and F. Wilczek, Geometry of self-propulsion at low Reynolds number, J.Fluid Mech. 198, 557-585, (1989). 2. A. Shapere and F. Wilczek, Efficiencies of self-propulsion at low Reynolds number, J.Fluid Mech. 198, 587-599, (1989). 3. J. Koiller, R. Montgomery and K. Ehlers, Problems and progress in Microswimming, J. Nonlinear Science 6, 507-541, (1996). 4. K.M. Ehlers, The Geometry of Swimming and Pumping at Low Reynolds number, PhD Thesis, Univ. of Calif., Santa Cruz (1995). 5. A. Cherman, M.Sc. thesis, Centro Brasileiro de Pesquisas Fisicas, (1998). 6. K. Ehlers, in preparation. 7. J. Gray and H.W. Lissmann, The Locomotion of Nematodes, J. Explt. Biol., 4 1 , 135-154, (1964). 8. J. Lighthill, On the squirming motion of nearly spherical deformable bod ies through liquids at very small Reynolds number, Commun. Pure Appl. Math. 5, 109-118, (1952). 9. J. R. Blake, A spherical envelope approach to ciliary propulsion, J.Fluid Mech. 46, 199-208, (1971). 10. J.R. Blake, Self propulsion due to oscillations on the surface of a cylinder at low Reynolds number, Bull.Austral.Math.Soc.3, 255-264, (1971). 11. N.I. Muskhelishvili, Some basic problems of the mathematical theory of elasticity, P.Noordhoff, Groningen-Holland, (1953). 12. G.K. Batchelor, An introduction to Fluid Mechanics, Cambridge Univer sity Press, (1970). 13. D.B. Dusenbery, Life at Small Scale, Scientific American library, (1996). 14. C. Pozrikids, Boundary Integral and Singularity Methods for Linearized Viscous flow, Cambridge Texts in Applied Mathematics, 1992. 15. J. Koiller and J. Delgado, On efficiency calculations for nonholonomic locomotion systems, Reports Math. Phys., 42, 165-183, (1998).

2 - D I M E N S I O N A L INVARIANT TORI FOR T H E SPATIAL ISOSCELES 3 - B O D Y PROBLEM MONTSERRAT CORBERA Departament de Fisica i Matematica Aplicades, Universitat de Vic, Sagrada Famtlia 7, 08500-Vic, Barcelona, Spain E-mail: [email protected] JAUME LLIBRE Departament de Matematiques, Universitat Autdnoma de Barcelona, 08193 - Bellaterra, Barcelona, Spain E-mail: [email protected] We consider the circular Sitnikov problem as a special case of the restricted spa tial isosceles 3—body problem. In appropriate coordinates we show the existence of 2-dimensional invariant tori that are formed by union of either periodic or quasiperiodic orbits of the circular Sitnikov problem, these tori are not KAM tori. We prove that such invariant tori persist when we consider the spatial isosceles 3—body problem for sufficiently small values of one of the masses. The main tool for proving these results is the analytic continuation method of periodic orbits.

1

Introduction

The main objective of this work is to prove the existence of 2—dimensional invariant tori filled of periodic or quasiperiodic orbits for the spatial isosceles 3—body problem. We note that in particular these tori are 2—dimensional invariant tori for the general spatial 3—body problem. We start reducing, with the help of appropriate coordinates, the dimension of the phase space of the isosceles problem, obtaining in this way the reduced isosceles problem. We see (in Theorem 2) that our tori filled of periodic or quasiperiodic orbits come from periodic orbits of the reduced isosceles problem. Using the analytic continuation method, we prove (in Theorem 6) the existence of symmetric periodic orbits of the reduced isosceles problem, for sufficiently small values of one of the masses, near the known periodic orbits of the reduced circular Sitnikov problem (a particular reduced restricted isosceles problem). Finally we analyze the 2—dimensional invariant tori of the isosceles problem that come from those periodic orbits. In this paper we present results without proofs. The proofs can be found in Corbera and Llibre2, and they constitute the main results of the Ph. D. of the first author.

63

64

2

Equations of motion of the isosceles problem

We consider three particles P\, Pi and P 3 of masses mi, 777.2 and 7713 respectively, such that mi = m.2- Their initial positions and velocities in the inertial coordinate system (X,Y,Z,X,Y,Z) are (X, Y, Z 2 , X^Y^Zi), {-X,-Y,Z2,-X,-Y,Z2) and (0,0,Zi,0,0,Zi), respectively. Of course the dot denotes the derivative with respect to the time t. The spatial isosceles 3-body problem, or simply in this work the isosceles problem, consists of de scribing the motion of these three particles under their mutual Newtonian gravitational attraction. We note that the name of isosceles comes from the fact that the particles P\, P2 and P3 at any instant form an isosceles triangle, eventually degenerated to a segment. We introduce cylindrical coordinates {r,6,z} 6 R + x S 1 x R as follows. First we put the origin of the coordinate system at the center of mass of mi, 77i2 a n a " "13. Then we define a new variable z = Z\ — Z2 G R which denotes the distance between the third particle P3 and the plane that contains the particles Pi and Pi with the convenient sign (positive in the upper position and negative in the down position). Finally we consider polar coordinates, (r, 0) € R + x S 1 , in the plane that contains P\ and P2. We choose the unit of mass in such a way that TTZI = 7712 = 1/2 and T7i3 = fx, and the unit of length is chosen so that the gravitational constant to be one. Then the kinetic energy in the coordinate system (r, 6, z, r, $, z) is given by

r =■K" + rH2

+

1+M " ) '

and the potential energy goes over to U =

1

fi

(z 2 + r 2 ) 1 / 2 '

8r

Therefore the Lagrangean eqxiations of motion for the isosceles problem are

d (dL\ 3L for q € {r, 6, * } . dt\dq) = ~~ dq where the Lagrangean is L = T — U; that is 1 | ( r ) = # - 72

Hr

" I T " " (Z1 4. r 2\3/2 '

2

d

It{T

9)-- = 0 ,

( " iU

dt V l + M J

(1) /iz

(22

+ r 2)3/2

•

65

We note that the second equation of system (1) leads to the angular momentum first integral C = r26 .

(2)

Of course, system (1) also has the first integral given by the energy H = T+U. 3

Equations of motion of the reduced isosceles problem

In this work we only consider solutions of the isosceles problem (1) having nonzero angular momentum (i.e. we do not consider solutions with collision between the masses). Under this assumption it is sufficient to consider solu tions of (1) having a fixed value of the angular momentum C = c for some c ^ 0, because the phase portrait of the isosceles problem on each angular momentum level c ^ 0 is the same, as it is shown in the following proposition. Proposition 1 Let i(t) — (r(t),r(t), z(t), z(t),8(t),8(t)) be a solution of the isosceles problem (1) with angular momentum C = c for some c ^ 0. 7/ we take a = c^/c2, then 7(t) = (ar{a3/2t),a-1/2f{a3/2t),az{a3/2t), 1 2 3 2 3 2 _3/2 3 2 a - / i ( a / t ) , 0(a / t.),a 0(a / *)) is a solution of (1) with angular mo mentum c. Assuming that the value of the angular momentum of the isosceles prob lem is fixed to C = c for some c ^ 0, we reduce in two units the dimension of the phase space obtaining the reduced isosceles problem .. _ c 2 r

1

~:i*~8r1~'

nr (22 + r 2)3/2 • ( )

(1 + /0* {z2 + r 2 ) 3 / 2 ' 4

Relationships between the reduced isosceles problem and the isosceles problem

Any solution j(t) — {r(t),r(t),z(t),z(t),0{t),9(t)) of the isosceles prob lem (1) with angular momentum c ^ 0 is given by the solution (p{t) = (r(t),f(t),z(t),z(t)) of the reduced isosceles problem (3) by taking 6{t) = ^fy

and

6(t) = ^ - ^ d r + e0

(mod 2TT) , (4)

where 9Q — 6(0) is an arbitrary integration constant. On the other hand given a solution tp(t) of the reduced isosceles problem (3), we obtain infinitely many

66 solutions 7e 0 (t) of the isosceles problem (1) with angular momentum c, one for each choosing of the integration constant 9Q € S 1 . Moreover these solutions are given by -ye0(t) = (
°(t)=

I -^dr

+ e0 = F(t) + e0,

Jo r*(T) 1 for a fixed 9Q & S . Then the set {Ugoesi7e0} " diffeomorphic to one of the following manifolds. (a) A circle S 1 C Sc formed by a periodic orbit of (1) with period S/J,)2, if'

128TTC 3 /(1 +

(b) A 2-dimensional torus S 1 x S 1 C £c, if if is a T-periodic orbit. Moreover this torus is formed by union of (i) periodic orbits of period mT, if F(T) — 2nl/m and I, m coprime;

with I G Z, m € N

(ii) quasiperiodic orbits, if F(T) = u2ir with u> an irrational number. (c) A cylinder S ' x R c ^ , if(fis neither the equilibrium point nor a periodic orbit. 5

Symmetric periodic orbits of the reduced isosceles problem

It is easy to check that equations (3) are invariant under the symmetry (r, f, z, z, t) —► (r, - r , - z , z, -t) . This means that if
67

This symmetry can be used to find r—symmetric periodic solutions of the reduced isosceles problem through the following well known result. Proposition 3 Let
(5)

i.e. the time reversibility symmetry, that will be denoted in what follows by t—symmetry. This symmetry is also used to find t—symmetric periodic solutions. Proposition 4 Let ^o + T/2), then ip(t) is a t—symmetric periodic solution of period T. A periodic solution of the reduced isosceles problem is said to be a doublesymmetric periodic solution if it is simultaneously a r—symmetric and a t—symmetric periodic solution. This kind of periodic solutions are charac terized by the next result. Proposition 5 Let tp(t) = (r(t),f(t),z(t),z(t)) isosceles problem (3).

be a solution of the reduced

(a) Iff(t) andz(t) are zero att = to andr(t) andz(t) are zero att = t0+T/4 but they are not simultaneously zero at any value of t S (to, to + T/4), then f(t) is a double-symmetric periodic solution of period T. (b) Ifr(t) and z(t) are zero att = to and r(t) and z(t) are zero att = to+T/4 but they are not simultaneously zero at any value of t € (to, to + T/A), then (fi(t) is a double-symmetric periodic solution of period T. 6

Restricted isosceles problems

To obtain the restricted isosceles problems we assume that the value of the mass 7713 is infinitesimally small (i.e. u = 0). Then the equations of motion

68

of the restricted isosceles problem become 1 r -= r62~8r2 ' 2 Jt{r o)- = 0 ,

(6)

z z — {z2 + r 2 ) 3 / 2 '

We observe that the first two equations of (6) are the equations of motion of a 2—body problem in polar coordinates. Then the particles P\ and P? (the primaries) move on the plane 2 = 0 describing a solution of this 2—body problem. Moreover the particle P3 that lies on the straight line orthogonal to the plane that contains Pi and P2, moves under the gravitational attraction of the previous two, but does not influence their motion. Thus for every value of the angular momentum c ^ 0 and every solution (r(t),6(t)) of that 2—body problem, system (6) defines a different restricted isosceles problem; it can be a circular, elliptic, parabolic or hyperbolic restricted isosceles problem depending on the nature of the solution (r(t),6(t)). As above if we assume that the value of the angular momentum is fixed to C = c for some c ^ O , then we can reduce in two units the dimension of the phase space obtaining the reduced restricted isosceles problem

c2 _ J_

r = r° z =

8r 2 ' z

(7)

(22+r2)3/2

In this work we only are interested in solutions of the reduced restricted isosceles problem (7) for c ± 0 that are periodic. So we will consider only the reduced circular and elliptic restricted isosceles problems, that we will call reduced circular Sitnikov problem and reduced elliptic Sitnikov problem respectively. 6.1

The reduced circular Sitnikov problem

Without loss of generality we can assume that the primaries are describing a circular orbit of radius 1/2 (or equivalently, a circular orbit of period 27r). This corresponds to fix the value of the angular momentum to c = 1/4. Then the equation of motion for the infinitesimal mass becomes Z =

~ ( * 2 + 1/4)3/2 '

W

69 which is the equation of the known circular Sitnikov problem. This equation can be integrated and we know analytic expressions of all its solutions (see for instance Belbruno et al.1 or Szebehely6). We note that the knowledge of an analytic expression for the solutions of the circular Sitnikov problem plays a key role in our analysis, because it allows us to prove our results analytically. Assume that (z(t), z(t)) is a solution of (8) with arbitrary initial conditions z(0) = ZQ and i(0) = z 0 . Then it is clear that
1_

~~ ToT3" ~ 8 r 2 '

Z =

~(z2+

z r 2 )3/ 2

;

having initial conditions r(0) = 1/2 ,

r(0) = 0 ,

2(0) = z0 ,

i(0) = i 0 .

That is, n/y/2 (see for instance Belbruno et al.1). Moreover all these periodic orbits are double-symmetric periodic orbits. We note that the periodic orbits of the reduced circular Sitnikov problem correspond to periodic orbits of the infinitesimal mass, but in general they are not periodic orbits involving the three masses; that is they are not periodic orbits of the circular restricted isosceles problem. More precisely, each periodic orbit of the reduced circular Sitnikov problem gives a 2—dimensional invariant torus of the restricted isosceles problem filled with periodic orbits with period r = 27rp if T = 2np/q for some p, q G N coprime and p > q/(2y/2), and it gives a 2-dimensional invariant torus filled with quasiperiodic orbits if T = 2itw for some irrational w > l / ( 2 v ^ ) (see Theorem 2). 6.2

Reduced elliptic Sitnikov problem

We assume that the primaries are describing an elliptic orbit with period 2ir and eccentricity e. This corresponds to fix the value of the angular momentum to c = \ / l — e 2 /4. Then, choosing conveniently the origin of time, a solution of the reduced elliptic Sitnikov problem is a solution ip(t) = (r(t),r(t),z(t),z(t)) of the reduced restricted isosceles problem

•■ = 1~ei3

r

~

Z =

16r

- J 2_

8r ' z 2 3 2 ~(z*+ r ) / '

70

having initial conditions r(0) = (1 - e)/2 ,

f(0) = 0 ,

z(0) = zo ,

i(0) = i 0 ,

for some zo. A) € R. Since r(t) is 27r-periodic, the periodic solutions of the reduced elliptic Sitnikov problem must have period multiple of 27r. Therefore if
Continuation of periodic orbits

We use the analytic continuation method of Poincare 4 (see for instance Siegel and Moser5 or Meyer and Hall3) to continue periodic orbits of the reduced circular Sitnikov problem (which are double-symmetric periodic orbits) to symmetric periodic orbits of the reduced isosceles problem for /x > 0 suffi ciently small. These periodic orbits are continued in two different ways. The first way goes directly from the reduced circular Sitnikov problem to the re duced isosceles problem. The second way uses two steps, first we continue the periodic orbits from the reduced circular Sitnikov problem to symmetric periodic orbits of the reduced elliptic Sitnikov problem for small values of the eccentricity e, and after we continue those symmetric periodic orbits of the reduced elliptic Sitnikov problem to the reduced isosceles problem for fj, > 0 sufficiently small. Using direct continuation (the first way) we can continue all periodic orbits of the reduced circular Sitnikov problem except the ones that have period multiple of An. In particular, we can continue the periodic orbits with period 27ru; and u> irrational, of the reduced circular Sitnikov problem, which are quasiperiodic orbits of the circular restricted isosceles problem. The continuation in two steps (the second way) allows us to continue only periodic orbits of the reduced circular Sitnikov problem with period T = 2np/q for all p, q € N coprime and p > q/(2\/2), which are also periodic orbits of the circular restricted isosceles problem. We note that the periodic orbits of the reduced circular Sitnikov problem that cannot be continued directly can be continued in two steps. Moreover the rest of the periodic orbits with period T = 2-Kp/q can be continued in both ways obtaining different periodic orbits for the reduced isosceles problem.

71 The main results about continuation of periodic orbits of the reduced circular Sitnikov problem to symmetric periodic orbits of the reduced isosceles problem for /x > 0 sufficiently small are summarized in the following theorem. Theorem 6 Let if be a periodic orbit of the reduced circular Sitnikov problem with period T > ir/\/2. Then

0 sufficiently small. (a) Case T =

2TTUJ

with w > 1/(2\/2) an irrational number.

(i) ip can be continued to an 1-parameter family on /j, of doublesymmetric periodic orbits with period near T. (b) Case T = 2-npjq for some p, q € N coprime with p > q/{2\/2). (i) p odd: (1) ip can be continued to an 1-parameter family on p. of doublesymmetric periodic orbits with period near T. (2)

q/{2\/2), can be continued to:

72

(a) two I—parameter families on e of r—symmetric periodic orbits and two 1—parameter families on e of t—symmetric periodic orbits (that are not double-symmetric) of the reduced elliptic Sitnikov problem, with period 2wp = qT, for e > 0 sufficiently small, when p is odd; (b) two 1—parameter families on e of double-symmetric periodic orbits of the reduced elliptic Sitnikov problem, with period 2irp = qT, for e > 0 sufficiently small, when p is even. 8

2-dimensional invariant tori for isosceles problem

Let YT denote the 2—dimensional invariant torus of the circular restricted isosceles problem that comes from a periodic orbit of the reduced circular Sitnikov problem with period T (see Section 6.1). Then we have the following result. Theorem 8 The torus YT can be continued to the following families of 2—dimensional tori of the isosceles problem with fi > 0 sufficiently small. (a) Case T = 2iru) with LJ > l/(2\/2) an irrational number. (i) YT can be continued to an 1-parameter family on fi of sional tori. (b) Case T = 2ivp/q for some p,q € N coprime with p >

2-dimen

q/(2\/2).

(i) p odd: (1) YT can be continued to an 1-parameter family on fj. of 2—dimen sional tori. (2) YT can be continued to four 2-^parameter families on n and r of 2—dimensional tori. (ii) p even and q ^ 1: (1) YT can be continued to an 1-parameter family onu of 2—dimen sional tori. (2) YT can be continued to two 2-parameter families on n and r of 2—dimensional tori. (Hi) p even and q = 1: (1) YT can be continued to two 2-parameter families on n and r of 2—dimensional tori.

73

Remember that the phase portrait of the isosceles problem on each angu lar momentum level c with c ^ 0 is the same (see Proposition 1). Therefore we have obtained 2—dimensional invariant periodic and quasiperiodic tori on each angular momentum level c ^ 0. Acknowledgments The authors are partially supported by a DGES grant number PB96-1153. References 1. E. Belbruno, J. Llibre and M. Olle, On the families of periodic orbits which bifurcate from the circular Sitnikov motions, Celestial Mechanics and Dynamical Astronomy 60, 99-129, (1994). 2. M. Corbera and J. LLibre, Periodic and quasi-periodic motions for the spatial isosceles 3-body problem, preprint, (1999). 3. K. R. Meyer and G. R. Hall, An Introduction to Hamiltonian Dynamical Systems, Springer-Verlag, New York, (1991). 4. H. Poincare, Les Methodes Nouvelles de la Mecanique Celeste, 3 Vols. Gauthier-Villars, Paris, 1892-1899, reprinted by Dover, New York, (1957). 5. C. L. Siegel and J. K. Moser, Lectures on Celestial Mechanics, SpringerVerlag, Berlin, (1971). 6. V. Szebehely, Theory of orbits, Academic Press, New York, (1967).

T H E GLOBAL F L O W F O R T H E SYNODICAL SPATIAL K E P L E R P R O B L E M MARCIA P. DANTAS Departamento de Matemdtica, Universidade Federal de Alagoas, 57032-160 - Maceid, AL, Brasil E-mail: [email protected] JAUME LLIBRE Departament de Matematiques, Universitat Autdnoma de Barcelona, 08193 - Bellaterra, Barcelona, Spain E-mail: [email protected]

1

Introduction

The knowledge of the global flow of the synodical spatial circular restricted three-body problem for fi = 0 (i.e. the synodical spatial Kepler problem) allows to describe the flow for fj, > 0 sufficiently small during a finite time. The description of this flow for n = 0 uses the integrability of the system as a Hamiltonian system. 2

Topological study of Hamiltonian systems

It is well known that a Hamiltonian system has at least one first integral, the Hamiltonian H. Every first integral F defines, through their inverse image F~l (/) = / / , sets which are invariant under the flow of the system. The topological study of a Hamiltonian system which has only one inde pendent first integral is the topological study of the map H; more precisely, the topological classification of the sets //, and of the points h € R for which the topology of //, changes. If the Hamiltonian system has more than one independent first integral that are in involution, then we study the topology of the intersections of the sets defined by the inverse image of the first integrals and the foliation of the phase space by these sets. In other words, for a Hamiltonian system with n degrees of freedom, we study the topology of the subsets //,, Ihjlt •••, Ih,fi,...,fr-i of the phase space where H,F\,„.,Fr-i axe r independent first integrals in involution. In short, we describe how the phase space is foliated by the energy levels //, and also by the subsequent levels defined by the other first integrals Fi, i = l,...,r — 1. Of course, r = n if the Hamiltonian system

74

75

is integrable. Furthermore we are interested in the change of topology of the subsets //,, //;,/,, •-., Ihji fr-i, more precisely, in the bifurcation sets of the maps H, ( # , > , ) , ..., (Jf ) Fi J ...,F r _ 1 ). During this last century several mathematicians have used the topological study for understanding the dynamics of Hamiltonian systems, integrable or not. Thus Poincare was the intellectual mentor of this idea which was a kind of new approach to study the flow of Hamiltonian systems. This topological method was used by Birkhoff5, in the study of the three-body problem and some general dynamical systems, and by Kaplan 14 , in the two-body problem. However a great interest in it appears after the publication of Smale 22 ' 23 on the topology of mechanical systems. These works of Smale were followed by many authors who extended the results of the topology of the n-body problem, for n > 2, i.e. the motion of n bodies which are moving in the space under the action of their gravitational forces. See for instance the papers [2], [6-7], [11-12] and [15-20] due to the authors Albouy, Cabral, Casasayas, Easton, Fomenko, Llibre, Martinez Alfaro, McCord, Meyer, Nunes, Pinol, Soler and Wang. The main interest of the topological study is that the intersections of the invariant sets are invariant by the flow of the Hamiltonian system; i.e. if an orbit has a point in that intersection, the whole orbit is contained in the intersection. In other words, an orbit is contained in some Ihj1,...,/r_1 and therefore, from the global point of view, the topology of 7/,,/j j r _ 1 is a very important invariant of the orbit (see Smale 22 ). Thus the topological study of a Hamiltonian system allows us to give a first topological classification of its orbits. More precisely, for integrable Hamiltonian systems under good conditions (see the Liouville-Arnold Theorem) the topological study allows us to know how the flow evolves and where the orbits are moving. This is a way to understand the global flow of integrable Hamiltonian systems. Liouville-Arnold Theorem (Arnold, 1963) Consider an integrable Hamil tonian system X # with n degrees of freedom defined by the C2 Hamiltonian H -.UxW1 - » R , where UxRn is an open subset of R2n andletH,Fi,...,Fn_i be n independent first integrals in involution. If //,,/, / n _, ^ 0 and (h, f\,..., fn-1) is a regular value of (H, Fi,.. .F n _ i ) , then the following state ments hold. (a) /*»,/,,...,/„_! is a n-dimensional the flow ofX-H-

submanifold of U x R" invariant under

(b) If the flow on a connected component J£ , ,_ of I/lj1,...,fn_1 is com plete, then Ihji,...,/„_! " diffeomorphic to Tk x R n ~* for some k £

76

{0, ...,n}. We note that i///J/i .../„ i ^ compact, then the flow on it is always complete. (c) Under the hypothesis (b) the flow on //J,/,,...,/„_, ** conjugated to a linear flowonTkxRn~k. For more details about Hamiltonian systems and the proof of the previous theorem see Abraham and Marsden 1 and Arnold 3 ' 4 . The Liouville-Arnold Theorem shows that, for integrable Hamiltonian systems, the invariant sets associated with the intersections of all indepen dent first integrals in involution are generically submanifolds of the phase space. Moreover, if the flow on such submanifolds is complete, then these submanifolds are diffeomorphic to the union of generalized cylinders and the flow on them is conjugated to a linear flow. We note that the invariant sets are not always manifolds. The topological study is important for Hamiltonian systems whether they are integrable or not. The non-integrable Hamiltonian systems with symme try (as the n-body problem for n > 2) have more than one independent first integral in involution. Thus for such systems many authors consider the foliation of the phase space by the known first integrals. For integrable Hamiltonian systems (as the two-body problem) we consider the foliation of the phase space by all the independent first integrals in involution. Once we know the global flow of an integrable Hamiltonian system, we can understand the flow of a perturbed Hamiltonian system which is sufficiently close to the integrable one during a finite time. This is achieved by using the continuous dependence of the solutions of differential equations with respect to initial conditions and parameters. In the last fifteen years there has been an intense development in the topological study of integrable Hamiltonian systems. We can divide these studies into three types. The first one is Fomenko's study of foliations of the phase space of arbitrary integrable Hamiltonian systems which satisfy some additional assumptions. He constructed a Morse type theory for Boot integrals defined on isoenergy surfaces of integrable Hamiltonian systems. By developing these ideas, he obtained general results about some aspects of the foliations stating topological invariants which, under additional assumptions, allow to conclude if the system is integrable or not (see Fomenko 12,13 and Casasayas et al.8). The second way to study the foliations of the phase space of integrable Hamiltonian systems is to study the foliation for a particular integrable Hamil tonian system without using any additional assumption (see the papers [7] and [15-19] due to the authors Casasayas, Llibre, Martinez Alfaro, Nunes, Piiiol

77 and Soler). The Liouville-Arnold Theorem helps in the study of this foliation but it is very far from solving it. In fact, from the point of view of obtain ing the foliation of the phase space by the invariant sets Ih and /A,/,,...,/,. for r = l,...,n — 1, this theorem only establishes the possible topology of a connected n-dimensional submanifold Jj!,/, /„ i when (h,/i,...,/n-i) is a regular value of the map {H,F\, ...,Fn-i) and the flow on them is complete. Thus, for a given value (h, f\, . . . , / n - i ) € R n , the Liouville-Arnold Theorem does not give the number of connected components and their topology. It only says that, for regular values under assumptions which are in general difficult to verify, as in the case of the completitude of the flow on a connected compo nent, its topology is one of the following generalized cylinders Rk x (S 1 ) , for some k = 0,1, ...,n; but it does not give the exact A;. Also the LiouvilleArnold Theorem does not give any information about the foliation of /;, by 7/ij,, or about the foliation of 7/,,/j by Ih,fi,h an( ^ s o o n It is known that the set of regular values (h, f \ , . . . , / n - i ) has full measure in R", but in general it is different from R n . In order to study the foliation of the phase space by the invariant sets -f/i,/,,...,/n_,, we must consider the critical and the regular values (h,/i,...,/n-i) • Thus when ( / i , / i , . . . , / n - i ) is a critical value, the invariant set /fc,/ ll ...,/ n _ l is called a separatrix level. Of course on a separatrix level the flow does not need to be conjugated to a linear flow. Finally, the third type of studies is about foliations of the phase space for Hamiltonian systems which are not integrable, but they have some inde pendent first integrals, as for instance the n-body problem with n > 2, see Albouy 2 , Cabral 6 , Easton 11 , McCord et al.20 and Smale 22 - 23 . In this work, we study the topology of a particular integrable Hamiltonian system, i.e. the synodical spatial Kepler problem which plays a main role in Celestial Mechanics. 3

Synodical spatial Kepler problem

It is well known that the two-body problem can be reduced to the Kepler problem. We can consider two kinds of referential (the inertial and the rotat ing) and two kinds of motion according to the dimension n of the configuration space, the planar when n = 2, and the spatial when n = 3. Of course each spatial motion takes place in a plane, but it continues to be a spatial motion once the phase space has dimension 3. Thus there are four different models of the Kepler problem: two with two degrees of freedom, the sidereal (synodical) planar Kepler problem, and two with three degrees of freedom, the sidereal (synodical) spatial Kepler problem.

78

These Kepler problems have been studied by several authors (see the papers [7] and [15-19] ). In this work we present the topological study of the synodical spatial Kepler problem. In fact a topological study of the four Kepler problems including the regularizations of the singularities can be found in [9] (where there are the proofs of the results here presented). By considering a coordinate system that rotates with frequency 1 around the X3-axis, the sidereal spatial Kepler problem becomes the synodical spatial Kepler problem. Thus, the synodical spatial Kepler problem is the integrable Hamiltonian system with three degrees of freedom associated to the Hamiltonian H

=

IYI2 1 2 " jx[ "

XlY2

+ X Yl

(1)

* '

with phase space {(X,Y) € (R 3 \ {(JJ.,0,0) ,(fi-1,0,0)}) independent first integrals in involution

x K 3 }, and three

C = —2H (Jacobi integral), r, lY|2 1 / -J . E = '—- ■—- (sidereal energy), M = |M| = | X A Y | (modulus of the sidereal or synodical angular

momentum).

This model of the Kepler problem is a limiting case of the circular re stricted three-body problem in rotating coordinates when the mass parameter of one of the primaries tends to zero. Thus, knowing the global flow of the synodical spatial Kepler problem we can obtain information about the flow of the circular restricted three-body problem during a finite time when the mass parameter of one of the primaries is sufficiently small. The first work on the topology of the Kepler problem was carried out by Kaplan 14 , in 1942. There he studies the foliation of the planar sidereal Kepler problem for negative values of the energy. In 1982 Llibre 15 studied the synodical planar Kepler problem describing the foliation of the phase space by //,, and the foliation of //, by Ihe, where h and c are the values of the Hamiltonian and the angular momentum, re spectively. He used Devaney's regularization 10 of the binary collision and McGehee's regularization at infinity21. In 1984 Casasayas and Llibre 7 , in their study of the anisotropic Kepler problem, considered the sidereal planar Kepler problem. They obtained the foliation of the phase space by //,, and the foliation of //, by 7/,c, where h and c are the values of the sidereal energy and the angular momentum, respectively.

79 Their study also includes the separatrix levels and the regularizations of the origin and infinity. In 1993 Llibre and Soler19 improved the foliation of the phase space by //,, and the foliation of Ik by Ikc for the synodical planar Kepler problem. In 1994 Llibre and Nunes 17 did a general study of integrable Hamiltonian systems with two degrees of freedom defined by a central force. Using the topological method, the synodical spatial Kepler problem was studied by Llibre and Martinez Alfaro16 in 1985. There the regularization of the collision singularity was considered. Subsequently in 1990 Llibre and Pinol 18 , in their study of the elliptical restricted three-body problem, included the regularizations of the collision and infinity singularities and so improved the results on the synodical spatial Kepler problem. In both papers the au thors stated the topology of the invariant sets Icm3f> i-e- they considered the independent first integrals in involution C, M$ = E + C/2 — XiY? — X2Y\ and F = M2 + Mf = M2 - Mi where M = (Mi, Af2, M 3 ). In this paper we study the synodical spatial Kepler problem taking as independent first integrals in involution C, E and M. We extend the previous results by stating the foliation of the phase space by Ic, the foliation of Ic by J ce , and the foliation of by Icem- In fact, in [9] we also give the topology of ■»m, -*c,mi -*e a n d

4

Jew

Topology of Ic, Ice and Icem

For presenting our results we need some notation. We define R + = (0, oo) and denote by Sm the m-dimensional sphere in R m + 1 of radius 1 with center at the origin 0. We note that S° is formed by two points. Throughout this paper manifold means differentiable manifold. If A and B are topological spaces the product space Ax B has the product topology. We use the symbol « to denote a diffeomorphism between two manifolds. Thus we present without proof the following diffeomorphisms: S ' x R « S2 \ {one point}, 5 1 x R 2 « S 1 x (S2 \ {one point}) 52 x R 3 « S2 x [S3 \ {one point}) S'xS'xR 2

«53\51, » S5 \ S2,

« S 1 x (R 2 s {0}) « S 1 x (S2 \ 5°) « 5 3 N { S 1 U 5 1 } ,

2

S x S x R « S2 x (R 3 \ {0}) « S2 x (S3 x 5°) » 5 5 x (S2 U S2). The results about the synodical spatial Kepler problem are stated in the following theorems. From them we obtain the foliation of the Jacobi levels Ic by the first integrals E and M. The first theorem concerns the topology of the Jacobi levels Ic.

80

Theorem 1 The Ic level of the synodical spatial Kepler problem is diffeomorphic to S5 \ S1) R i ^ x R ^ S 1 x R 4 « 5 2 X R 3 N S ' S2US1' 2 2 «S2xS2xR S US )

i/c>3, i/0
If c = 3, Ic is obtained by identifying a circle of S5 \ S2 with another circle of S 5 \ S 1 . This circle is formed by equilibrium points of the system. The next result is about the topology of the /«, levels. In this case we need to distinguish between the motion on the plane that does not contains the X3-axis (e ^ —c/2) and the motion on the plane that contains the X3-axis (e = - c / 2 ) . Theorem 2 The following statements hold for the Ice levels of the synodical spatial Kepler problem. (a) Ife^

- c / 2 , then S1 ife < 0 and |e + c/2| = m , , S1 x S 3 if e < 0 and je + c/2| < m», S 1 x R 3 « S 1 x (S3 x Poo) ife > 0.

(b) Ife=

- c / 2 , then c e

/ Sx x S 1 x R 2 « 5 1 x S 1 x ( 5 2 \ P 0 ) i/e < 0, * \ 5 l xS1 x S ' x R i/e>0.

Here PQ and P ^ denotes points of S2 and S3, respectively and m, =

l/J=Te. We remark that the circular spatial orbits are contained in S1 x S3 if m = m . > |e + c/2| > 0, and in S 1 x S 1 x R 2 if m = m . > |e + c/2| = 0. Theorem 3 77ie following statements hold for the I^m levels. (a) Ifrnj^O,

then

Icxm. *

' S1 5 1 x S1 S1 x S1 x S1 ' S1xSl S'xR S1 x R x S1

i/e<0 and0< ife < 0 a n d 0 < i/e < 0 and 0 < ife<0and0< ife>0and0< i/e > 0 and 0 <

|e + |e + |e + |e + |e + |e +

c/2| c/2| c/2| c/2| c/2| c/2|

= m = m„, = m <m„, <m<m., <m = m„ = m, < m.

81 (b) Ifm = Q, then

'52xR S2 xRxS0

i/e<0, ife>0.

Here m» = \fsj-2e. For finishing we state the results about the change of topology of the sets hem, i-e. for the n i p G = (C, E, M) . For this we introduce the bifurcation set E G formed by the points (c, e, m) in R 3 for which the map G fails to be locally trivial (see Llibre and Nunes 17 , and Smale 22 ). For an integrable Hamiltonian system with 3 degrees of freedom, we can characterize E G Proposition 4 For an integrable Hamiltonian system with 3 degrees of free dom, the bifurcation set is given by Ef = {y : y is a critical value off}

U {y : Iy changes its topology} ,

where f = ( / i , / 2 . / 3 ) , and fi, for i = 1,2,3 are three independent first inte grals in involution. For the synodical spatial Kepler problem we have the following results. Proposition 5 The set of critical values of the map G = (C, E, M) is given by Uc,e,m)

£ R 3 : - 2 e f e + ^\

< 1 and m=

, e + ^ I.

It is well known that, for the Kepler problem, the parabolic orbits are those for which the energy is zero (e = 0) and the collision or ejection orbits those for which the total angular momentum is zero (m = 0). Thus we have the next two results. Corollary 6 The parabolic orbits of the synodical spatial Kepler problem that are not collision or ejection are associated to regular values of the map G. Corollary 7 The orbits of the vertical plane containing the X3~axis of the synodical spatial Kepler problem that are not collision or ejection are associ ated to regular values of the map G = (C,E,M). From Theorem 3, and by Corollaries 6 and 7, we have the following results for the synodical spatial Kepler problem. Corollary 8 There is a change of topology in the Icem levels associated to regular values in the following cases. (a) When m = 0, i.e. at collision or ejection orbits. (b) When e = 0, i.e. at parabolic orbits.

82

Finally, by Theorem 4 we obtain the next result. Theorem 9 The bifurcation set of the synodical spatial Kepler problem is given by E G equal to { ( c , e , m ) G R 3 : c = 3} U {(c,e,m) G R 3 : m = 0 , j e + f | } u {(c, e, m) G R3 : e = 0, - 1 / (2m 2 )} . 5

Foliation of the phase space

From Theorems 1, 2 and 3 we state the foliation of the synodical spatial Kepler problem by the first integrals C, E and M. We note that the domain of definition of c, e and m is delimited by the bifurcations of the circular orbits e = —l/2m 2 , the planar motion, m = |e + c/2|, the collision orbits m = 0 (and consequently e = — c/2), and the parabolic orbits e — 0 (see Theorem 9). Corollary 10 The foliation of the phase space of the synodical spatial Kepler problem by the first integrals C, E and M is given in Tables 1, 2, S, 4 and 5. There m 3 = e + c/2, m» = l/\J—2e means the maximal value of the angular momentum that exists for e < 0, and e* satisfies \e + c/2\ = 1/y/—2e. From now on we give the description of the flow. We call direct space (DS) and retrograde space (RS) to the region of the Jacobi levels with e > —c/2 (m-3 > 0) and e < —c/2 (7713 < 0), respectively. If c > 3, then e £ [ei,e2] U [e3,oo) and each of the Jacobi levels Ic has two components I\c and /2c, see Table 1: (1) In Iic we have that e G [ei,e2] and 0 < |e + c/2| < m < m»: (a) If e = Ci, then m = m* and there is one retrograde circular planar orbit in the level Ice « S1. (b) If e\ < e < —c/2, then formed by the levels Ice planar orbits when m = when — (e + c/2) < m 771 =

the orbits are in the retrograde space RS « S1 x S3. There is a set of: 5 1 elliptic — (e + c/2); S1 x S 1 elliptic spatial orbits < m,\ and S 1 circular spatial orbits when

771,.

(c) If e = - c / 2 , then the orbits are in the level Ice « S 1 x S 1 x R 2 that contains the collision or ejection orbits. There is a set of: S2 ejection-collision elliptic orbits when m = 0; S1 x S 1 elliptic spatial orbits when 0 < m < m«; and S 1 circular spatial orbits when m = m».

83 (d) If —c/2 < e < e2, then the orbits are in the direct space DS formed by the levels Ice « S 1 x S 3 . There is a set of: S1 elliptic planar orbits when m = e + c/2; S 1 x S 1 elliptic spatial orbits when e + c/2 < m < m»; and S1 circular spatial orbits when m = m». (e) If e = e2, then m = m, and there is one direct circular planar orbit in the level Ice « S1. (2) In he we have that e € [e3,oo), 0 < e + c/2 < m < m» if e < 0, and 0 < e 4- c/2 < m if e > 0. The orbits are in the direct space DS : (a) If e = e3, then m — m, and there is one direct circular planar orbit in the level 7 ce « 5 1 . (b) If e3 < e < 0, then the direct space DS is formed by the levels Ice « S 1 x S 3 . There is a set of: S 1 elliptic planar orbits when m = e + c/2; S1 x S 1 elliptic spatial orbits when e + c/2 < m < m»; and S1 circular spatial orbits when m = mt. (c) If e = 0, then the orbits are in the direct space DS formed by the levels Ice w Sl x R 3 . There is a set of: S 1 parabolic planar orbits when m = c/2, and S 1 x S1 parabolic spatial orbits when m > c/2. (d) If e > 0, then the orbits are in the direct space DS formed by the levels Ice « S 1 x R 3 . There is a set of: S 1 hyperbolic planar orbits when m = e + c/2, and S 1 x S 1 hyperbolic spatial orbits when m > e + c/2. If c = 3, Ic is foliated as in the case c > 3 except for the value e = e2 = e3 at which J c e m is a circle of equilibrium points that are common to both components I\c and he- See Table 2. If 0 < c < 3, we have that e € [ei, co), 0 < |e + c/2| < m < m, if e < 0, and 0 < e + c/2 < m if e > 0. Each of the Jacobi levels Ic has one component, see Table 3: (1) If e = ei, then m = m, and there is one retrograde circular planar orbit in the level Ice ^ S1 which coincides with the retrograde space RS. (2) If ei < e < —c/2, then the orbits are in the retrograde space RS formed by the levels Ice « S 1 x S3. There is a set of: S 1 elliptic planar orbits when m = - (e 4- c/2); Sl x S 1 elliptic spatial orbits when - (e + c/2) < m <m»; and S 1 circular spatial orbits when m = m„.

84

h

Ice

1

s

5ax5

3

e € [ei,e2)U[e3,oo)

*cem

|e + c / 2 | < m

ei = e

1 s 1 S x5' S ' x 5 ' x Sl Sl x S 1 S'xR S ' x 5 ' x 51 S'xS1 S1 x S 1 S ' x S ' x Sl S1 x 5 '

—7713 = TTl = 771* —777,3 = m < 771, —77i3 < m < m« —m.3 < m = m« 0 = m < m, 0 < m < m, 0 < m = m, 7Ti3 = m < m , 7713 < 771 < 771, 7713 < m = m , 77i3 = m = m . T713 = m — m« 7713 = m < 77i« 7713 < m < m* 7713 < 771 = 771, 7713 = "^ 7713 < m

ei < e < - c / 2

he

s2

S1

xS'x R2

X R3

S^S3

sll s

he S1 X R4

s3

5

x

Sl

x5

e = -c/2

- c / 2 < e < e2 e = e2 e3 = e

3

xRJ

e3 < e < 0 0<e

S1 Sl 1 S'xS1 S " L l 1 S xS S " x 5 1 S1 x 5 ' S1 x R S'xRxS1

Table 1. The foliation of Ic for c > 3.

(3) If e = - c / 2 , then the orbits are in the level Ice « 5 1 x S1 x R 2 that contains the collision or ejection orbits. There is a set of: S2 ejectioncollision elliptic orbits when m = 0; 5 1 x 5 1 elliptic spatial orbits when 0 < m < m«; and 5 1 circular spatial orbits when m = m,. (4) If —c/2 < e < 0, then the orbits are in the direct space DS formed by the levels Ice f s S ' x S 3 . There is a set of: S1 elliptic planar orbits when m = e + c/2; S1 x S 1 elliptic spatial orbits when e + c/2 < m < m,; and S 1 circular spatial orbits when m = m,. (5) If e = 0, then the orbits are in the direct space DS formed by the levels Ice w 5 1 x R 3 . There is a set of: S 1 parabolic planar orbits when m = c/2, and S 1 x S 1 parabolic spatial orbits when m > c/2. (6) If e > 0, then the orbits are in the direct space DS formed by the levels Ice « S1 x R 3 . There is a set of: S1 hyperbolic planar orbits when m = e + c/2, and S1 x S 1 hyperbolic spatial orbits when m> e + c/2. If c = 0, we have that e G [ei, 00), 0 < |e + c/2| < m < m , if e < 0, and 0 < e + c/2 < m if e > 0. Each of the Jacobi levels Ic has one component, see

85

h

e € (ei.oo) ei = e

i-ce

S1

h

SxxS3 s3

ei < e < - c / 2

S 1 xS1 x R 2

e = -c/2

3

S'xS3 s

- c / 2 < e < e2

Sl

e = e2 = e 3 3

S'xS3 s

e2 < e < 0

Sl x R J

0<e

^cem

S1 1 S xS1 l S x 5 ' x S1 51 x S 1 S2xR 1 5 x S 1 x Sl 1 SlxSl S " l l 1 S xS S " Sl x S 1 x S 1 S'xS1 S1 1 S xS1 1 5 x 51 x 51 51 x S 1 5'xR S'xRxS1

|e + c/2| < m -7713 = m = m , — 7713 = 771 < 771, —7773 < 777 < 777, —7713 < 771 = 771,

0 = m < m» 0 < m < m» 0 < m — m, 7Ti3 = m < m « 7773 < 771 < 771, 7773 < 777 = 777* 7713 = 7 7 1 = 771, 7713 = m < m , 7713 < m < m , 7713 < 771 = 771* 77I3 = 771 77I3 < 771

Table 2. T h e foliation of Ic for c = 3.

/c

e £ [ei,oo) ei = e

■"ce

S

1

S1 x 5

3

ei < e < - c / 2

S2 X

S1 xS1 x R 2

e = -c/2

R3 S1

S'x53

-c/2 < e < 0

S1 x R J

0<e

*cem

1 s l S xS1 l l S xS x 51 1 S xS' S'^xR S ' x S ' x 51 SIx51 51 x S 1 1 5 x S1 x 5 1 5'xS1 S'xR S'xRxS1

|e + c / 2 | < 771 —7713 = m = 771. —7713 = m < m » —7713 < m

< m ,

—7713 < m = m . 0 = 771 < 771, 0 < 771 < 771, 0 < m = 771, 7713 = m < m , 7713 < 771 < m , 7773 < m = 771, 7713 =

m

7713 < 771

Table 3. The foliation of Ic for 0 < c < 3.

Table 4: (1) If e = ei, then m = m, and there is one retrograde circular planar orbit

86

h

s2

eG [e ls oo) e\ = e

Ice l

s

1

3

X

S xS

X R

Sl xS'x 5'xR Sl x R J

s2

ei < e < 0 e= 0 0<e

■•cem

S1

S'xS1 S ' x S ' x 51 Sl xSl SzxRx5u 5'xRxS1 S'xR S'xRxS1

|e + c/2| < —m3 = m = —m3 = m < -7Ti3 < m < —T7i3 <m = 0= m 0< m T7l3 = m

m m» m, m» m,

7Tl3 < 771

Table 4. The foliation of Ic for c = 0.

in the level Ice « S 1 which coincides with the retrograde space RS. (2) If ex < e < 0, then the orbits are in the retrograde space RS formed by the levels Ice K S1 X S3. There is a set of: S1 elliptic planar orbits when m = - e ; Sl x S 1 elliptic spatial orbits when —e < m < m»; and S 1 circular spatial orbits when m — m,. (3) If e = 0, then the orbits are in the level I*e % Sl x S 1 x S1 x R that contains the collision or ejection orbits. There is a set of: S 2 ejectionparabolic orbits and S2 parabolic-collision orbits when m = 0, and S1 x S1 parabolic spatial orbits when m > 0. (4) If e > 0, then the orbits are in the direct space DS formed by the levels Ice ss S1 x R 3 . There is a set of: S 1 hyperbolic planar orbits when m = e + c/2, and S1 x S1 hyperbolic spatial orbits when m > e + c/2. If c < 0, we have that e € [ei, oo), 0 < |e + c/2| < m < m , if e < 0, and 0 < e + c/2 < m if e > 0. Each of the Jacobi levels Ic has one component, see Table 5: (1) If e = ei, then m = m* and there is one retrograde circular planar orbit in the level Ice ss S 1 which coincides with the retrograde space RS. (2) If ei < e < 0, then the orbits are in the retrograde space RS formed by the levels Ice ss S 1 x S3. There is a set of: S1 elliptic planar orbits when m = —e; S 1 x S 1 elliptic spatial orbits when —e<m<m.\ and S 1 circular spatial orbits when m = m«. (3) If e = 0, then the orbits are in the retrograde space RS formed by the levels Ice «s S 1 x R 3 . There is a set of: S 1 parabolic planar orbits when m = —c/2, and S 1 x S1 parabolic spatial orbits when m > - c / 2 .

87

h

he l

s 2

S1 x S 3

s

X

s2

S1 x R J

X

R

S1 x S ' x S'xl S1 x R J

e e [ei,oo) e\ = e

-*cem

1 1s S x5' ex < e < 0 S ' x S ' x 5 1 SlxSl 0 < e < -c/2 S'xR 5'xRxS1 e = -c/2 S'xRxS1' S'xRxS1 S'xR -c/2 < e 5'xRxS1

|e 4- c/2| < m —77i3 = TTI = m » - m 3 = 771 < 771, —7713 < 771 < 771. —7713 < TTl = m « —777,3 = 771 —77I3 < 771 0 = 777 0<77l 7773 = 777 7773 < 771

Table 5. The foliation of Ic for c < 0.

(4) If 0 < e < —c/2, then the orbits are in the retrograde space RS formed by the levels Ice « S 1 x R 3 . There is a set of: S 1 hyperbolic planar orbits when m = — (e + c/2), and 5 1 x S1 hyperbolic spatial orbits when m > - ( e + c/2). (5) If e = - c / 2 > 0, then the orbits are in the level Ice w S 1 x S 1 x S1 x R that contains the collision or ejection orbits. There is a set of: S2 ejectionhyperbolic orbits and S2 hyperbolic-collision orbits when m = 0, and Sl x S 1 hyperbolic spatial orbits if 0 < m. (6) If e > —c/2, then the orbits are in the direct space DS formed by the levels Ice « 5 ' x R 3 . There is a set of: S1 hyperbolic planar orbits when 777 = e + c/2, and S1 x S 1 hyperbolic spatial orbits when m > e + c/2.

References 1. R. Abraham and J.E. Marsden, Foundations of Mechanics, Benjamin, Reading, Massachussets, (1978). 2. A. Albouy, Integral manifolds of the n-body problem, Inventiones Math. 114, 463-488, (1993). 3. V.I. Arnold, Mathematical Methods of Classical Mechanics,, SpringerVerlag, (1978). 4. V.I. Arnold, V.V. Kozlov, A.I. Neishtadt, Dynamical Systems III, Ency clopaedia of mathematical sciences, Springer-Verlag, Berlin, (1978). 5. G.D. Birkhoff, Dynamical Systems, New York, (1927).

88

6. H. Cabral, On the integral manifolds of the n-body problem, Inventiones Math. 20 ,59-72, (1973). 7. J. Casasayas and J. Llibre, Qualitative analysis of the anisotropic Kepler problem, Memoirs of the Amer. Math. Soc, Vol. 52, n° 312, (1984). 8. J. Casasayas, J. Martinez Alfaro and A. Nunes, Knots and links in integrable Hamiltonian systems, J. Knot Theory Ramification 7,123-153, (1998). 9. M. P. Dantas and J. Llibre, The foliation of the phase space for the 3-dimensional Kepler problem, preprint, (1999). 10. R. Devaney, Singularities in Classical Mechanics Systems, in Ergodic The ory and Dynamical Systems I, Proceedings Special Year, Maryland 197980, A. Katok (ed.), Birkhauser, Basel, 211-333, (1981). 11. R. W. Easton, Some topology of n-body problems, J. Differential Equa tions 19, 258-269 (1975). 12. A. T. Fomenko, On typical topological properties of integrable Hamilto nian systems, Izv. Akad. Nauk (SSSR) Ser. Mat. 52 ,378-407, (1988); English Transl. in Math. USSR Izv. 32 (1989). 13. A. T. Fomenko, Differential equations and applications to problems in Physics and Mechanics, Mechanics Analysis and Geometry: 200 years after Lagrange, M. Francaviglia (editor), Elsevier Science Publishers B. V., (1991). 14. W. Kaplan, Topology of the two-body problem, American Mathematical Monthly 49, 316-326,(1942). 15. J. Llibre, On the restricted three-body problem when the mass parameter is small, Celest. Mech. 28, 83-105, (1982). 16. J. Llibre and J. Martinez Alfaro, Ejection and collision orbits of the spa tial restricted three-body problem, Celest. Mech. 35, 113-128, (1985). 17. J. Llibre and A. Nunes, Separatrix Surfaces and Invariant Manifolds of a Class of Integrable Hamiltonian Systems and Their Perturbations, Mem. of the Amer. Math. Soc, 107, n° 513, (1994). 18. J. Llibre and C. Pinol, On the elliptic restricted three-body problem, Celest. Mech. 48 , 319-345, (1990). 19. J. Llibre and J. Soler, Global flow of the rotating Kepler problem, in Hamiltonian System and Celestial Mechanics, Advanced Series in Non linear Dynamics, Vol. 4, World Scientific, Singapore, 125-140, (1993). 20. C. K. McCord, K. R. Meyer and Q. Wang, The integral manifolds of the three body problem, Mem. Amer. Mat. Soc, 132, n° 628, (1998). 21. R. McGehee, A stable manifold theorem for degenerate fixed points with applications to celestial mechanics, J. Differential Equations 14, 70-88, (1973).

89

22. S. Smale, Topology and mechanics, I, Inventiones Math. 10, 305-331, (1970). 23. S. Smale, Topology and mechanics, II. The planar n-body problem, In ventiones Math. 11, 45-64, (1970).

U N B O U N D E D G R O W T H OF E N E R G Y I N P E R I O D I C P E R T U R B A T I O N S OF GEODESIC FLOWS OF T H E T O R U S AMADEU DELSHAMS Departament de Matematica Aplicada I, Universitat Politecnica de Catalunya, Diagonal 647, 08028 Barcelona, Spain E-mail: [email protected] RAFAEL DE LA LLAVE Department of Mathematics, University of Texas at Austin, Austin, TX, 78712, USA E-mail: [email protected] TERE M. SEARA Departament de Matematica Aplicada I, Universitat Politecnica de Catalunya, Diagonal 647, 08028 Barcelona, Spain E-mail: [email protected]

1

Introduction

The goal of these notes is to summarize the main ideas of a paper by the authors. l We will study the effects of a periodic external potential on the dynamics of the geodesic flow in the (2-dimensional) torus. This is a Hamiltonian system with two and a half degrees of freedom with Hamiltonian H{p,q,t) = ^gq{p,p) + U(q,t). Here g is the metric on the torus, and U is the external periodic potential. If U = 0, the system is autonomous and, therefore, the energy of the geodesic flow Ho(p,q) = \gq(p,p) is preserved. Note also that when the metric is flat, the geodesic flow is integrable. We will establish, using geometric methods, a result that had been es tablished by J. Mather using variational methods. 2 Namely, that for generic metrics and potentials—in particular for arbitrarily small potentials and for metrics arbitrarily close to integrable—, one can find orbits whose energy grows to infinity. More precisely, we will indicate the proof of: Theorem 1 For every r = 1 5 , . . . , oo, u, there exists a Cr-residual set in the set of metrics g on T 2 , and of periodic potentials U : T 2 x T 1 —♦ R, such that the Hamiltonian H(p,q,t) = ^gq(p,p) + U(q,t) has an orbit of unbounded energy.

90

91

The result is somewhat more surprising if we realize that, for high en ergy, the potential is a small perturbation of the geodesic flow and that since there are two time scales involved (see subsection 2.2), the frequency of the perturbation is smaller than the frequency of the geodesic flow. In this sense this result can be understood as an analogue to Arnold diffusion. The difference between this work and other approaches to Arnold diffusion in the literature comes from the fact that here, the unperturbed system, which is the geodesic flow, is a two degrees of freedom Hamiltonian system that, in general, is not integrable and it is assumed to have some hyperbolicity properties. Indeed, the hyperbolicity properties involve that the system contains hyperbolic sets with transversal intersection in an energy surface. This is somewhat stronger hyperbolicity that the so-called a-priori unstable systems in the notation of Chierchia and Gallavotti 3 . We propose the name a-priori chaotic for this kind of systems, where there are some conserved quantities, but there are hyperbolic orbits with transverse heteroclinic intersections in the manifolds corresponding to the conserved quantities. This result on instability has been proved by J. Mather using a variational approach (the differentiability requirements were much smaller). The proof presented here will rely on standard tools in the study of dynamical systems. Most of them have been used traditionally in Arnold diffusion. We also use very strongly the theory of hyperbolic dynamical systems. We hope that this can serve as a clarification of the relation between variational meth ods and more geometric methods. We also point out that S. Bolotin and D. Treschev have presented another geometric proof 4 . The main difference with the method presented here is that they rely more on KAM theory and they do not use the theory of normally hyperbolic manifolds. The generic conditions that we need can be described rather explicitly. We require for the metric that the geodesic flow possesses a hyperbolic peri odic orbit with a homoclinic orbit. Once we fix this periodic orbit and the homoclinic connection, we require on the potential that the Poincare function £ ( r ) computed on it is not identically constant (see (24)). The proof can be divided in several steps that are largely independent. As we mentioned before, each step only requires almost standard tools. Let us go over a thumbnail sketch of all these steps and they will be fleshed out in the following sections. The same method of proof allows to prove somewhat more general results. If we assume the consequences of one step, then all the subsequent ones remain valid. Moreover, we can use the same method to obtain more consequences.

92 Indeed, the proof will be established by the somewhat more general Theorem 18. Riemannian

Geometry

It was shown already by Morse—with some improvements by Mather—that a generic metric possesses a hyperbolic closed geodesic with a homoclinic orbit. Note that this shows that the geodesic flow is not integrable and is a-priori unstable in the sense of Chierchia and Gallavotti. 3 It is a crucial observation that, when we consider the geodesic flow as a 4-dimensional dynamical system, a closed geodesic lifts to a cylinder A (one periodic orbit for each energy surface). This cylinder is invariant and normally hyperbolic. Scaled variables As can be seen by elementary dimensional analysis, if we rescale the momenta and the time, we can transform the perturbation introduced by U for high energy as an small and slow perturbation of the geodesic flow. Intuitively, if the geodesies are moving very fast, in their time scale, the potential is changing slowly. For the analysis that will follow, the slowness is much more important than the smallness. Another elementary remark is that we can consider the system as au tonomous by considering one extra angle variable. We can come back from this extended phase space to the regular phase space by means of the Poincard return map. In particular, the cylinder A is invariant under the action of the unperturbed Poincar6 map, and it lifts to a 3-dimensional manifold A, invari ant under the unperturbed extended flow. In this sketch, we will be somewhat cavalier about distinguishing between the objects in the extended phase space and in the regular phase space. The theory of normally hyperbolic manifolds Since A is uniformly hyperbolic in the extended phase space, we can find a manifold close to it that is invariant for the perturbed system. The stable and unstable manifolds of this invariant manifold still intersect as transversally as possible. The importance of this invariant manifold is that it will serve as a template for many types of orbits.

93 Averaging theory Taking advantage of the fact that the perturbation is slow, we can study the motion on the perturbed invariant manifold using the method of averaging. Assuming that the system is differentiable enough, we can perform several steps of averaging so that the system on the 3-dimensional invariant manifold is a very small periodic perturbation of an integrable Hamiltonian system. KAM theory Once we know that a system is close to an integrable Hamiltonian system, we can find out that it is covered very densely by invariant tori, emerging from invariant circles of the Poincare map. Poincare-Melnikov theory It turns out to be possible to compute perturbatively the gain in energy for a homoclinic excursion. Under some generic hypotheses, it is possible to show that a homoclinic excursion leads to a jump from one KAM torus to another, giving rise to heteroclinic orbits between them. This idea can be formulated precisely by introducing what we call the scat tering map which relates the asymptotics in the future with the asymptotics in the past. This map can be constructed precisely using hyperbolicity theory and it can be computed perturbatively using the Poincare-Melnikov method (the fundamental theorem of calculus and a—somewhat delicate—passage to the limit). Under the assumption that the scattering map moves enough to jump over the gaps between KAM tori—which we have already shown are very small—we can construct pseudo-orbits whose energy grows unboundedly. These pseudorbits can be described as follows. When the potential is in such a condition that produces a jump of energy in the direction that we want, we execute the jump, following a heteroclinic orbit, and in the situations when it is not favorable, we just bid our time near a KAM torus. Shadowing lemma We need to construct orbits that follow these pseudo-orbits closely. This can be done by a topological argument very close to the original one invoking the obstruction property. 5 ' 6 Now, we turn to the task of fleshing out this sketch.

94 2 2.1

The m a i n s t e p s of t h e proof Riemannian geometry and dynamics of the geodesic flow

Let us begin by describing the dynamics of the geodesic flow. The phase space of the geodesic flow is T*T 2 = R 2 x T 2 . We will denote the coordinates in T 2 by q and the cotangent directions by p. The geodesic flow is a two degrees of freedom Hamiltonian, with Hamiltonian Ho(p,q) = ^gq(p,p), with respect to the canonical symplectic form Q, which, moreover, is exact: Q = dd. In local coordinates, 9 — J^Ptdgt, Cl — Y^i^Pi A dqi. We will denote by $(*;Pi9) = $t(Pi<7) this geodesic flow, and we remark that, even though this flow is not integrable, the energy E = HQ is preserved. For each E, we will denote £ E = {(p,q) : Ho(p,q) = E}, which is a 3dimensional manifold diffeomorphic to T 1 x T 2 , invariant under the geodesic flow. We observe that, for any £0 > 0, tBo = UE>E0^B - [E0, 00) x T 1 x T 2 , that is, the phase space is foliated by the energy surfaces. Given an arbitrary geodesic "A" : R —» T 2 , parameterized by arc length, we will denote by Ae(t) — (XE(t), X%(t)) the orbit of the geodesic flow that lies in the energy surface E#, and whose projection over q turns along the range of "A". Moreover, we fix the origin of time in XE so that it corresponds to the origin of the parameterization in "A". (Formally Ho(XE(t)) = E, and Range("A") = Range(A|), "A"(0) = A«,(0).) Note that

(A|(*).A^(t)) = (V2EXp1/2 (yZEt) ,A«/2 (V2Et)) .

(1)

So that, in this situation, the role of E is just a rescaling of time. The following theorem is due to Morse and Mather: Theorem 2 For a Cr generic metric g,r>2, and for any value of the Hamil tonian Ho(p,q) = E > 0, there exists a periodic orbit Ag(t), as in (1), of the Hamiltonian flow associated to Ho whose range AE is a normally hyperbolic invariant manifold in the energy surface. Its stable and unstable manifolds W? ,u are 2-dimensional, and there exists a homoclinic orbit 7B(<) such that its range 7 B satisfies *

Vc(w^\A*)n(wx a \A*). Moreover, this intersection is transverse as intersection of invariant manifolds in the energy surface along 7£. For E — 1/2, we have that dist(Ai/ 2 (* + a±),7i/2M) —► 0

as t —> ±00.

(2)

95 We suppose that the length of the unit geodesic "A" on the metric g is 1. Note that the speed of a unit geodesic is 1 and, therefore, its energy is 1/2. Then "A" as an orbit of the geodesic flow has period 1 and energy 1/2. Because of this we use the notation Ai/2The quantity A = a + - a_ is called phase shift. In this situation the energy is preserved and this makes the periodic orbit non-hyperbolic in the whole space. In order to obtain some hyperbolic object it is natural to discuss what happens for all energy surfaces considering A = U B > E ^-B for all values of the energy. Lemma 3 Define A = U B > B &■£• This is a 2-dimensional manifold with boundary which is diffeomorphic to [EQ, CO) X T 1 , and the canonical symplectic form on T*T 2 Q restricted to A is non-degenerate and invariant under the geodesic flow $ t . The stable and unstable manifolds to A, W\, W%, are Z-dimensional man ifolds diffeomorphic to [EQ, co)xT 1 xR, and7 = U E > £ 0 1E is 0,2-dimensional manifold diffeomorphic to [£0,00) x R that verifies 7c(WX\A)n(wx\A).

Let us note that the intersection of the stable and unstable manifolds along 7 is transverse, because TxW\nTxWl

=

Txl.

This meaning of transversal intersection has several important consequences. For example, 7 will be a locally unique intersection. Since the manifold A is a cylinder, we are going to take a coordinate sys tem in A, given by one real coordinate (momentum) and one angle (position). The real coordinate will be J — \/2Ho > T/2EQ. For the angle coordinate, we will take (f € T 1 , which is determined by dJ A dtp = fi|A and ip = 0 corre sponds to the origin of the parameterization in "A". We can identify a point in A with the values (J, (p) taken by these coordinates. The coordinate repre sentation of the periodic orbit AE(<) in these coordinates is given by (J, Jt), where J = \/2E. 2.2

The scaled problem

In order to make the perturbative structure of the problem more apparent we will scale the variables p and the time ( in the Hamiltonian H(p, q, t) = \gq{p,p) + u{q,t). We pick a (large) number Em, and call e = l/y/E^. Since e2H(p,q) = ^gq(ep,ep) + e2U(q,t), we introduce p = ep and consider the symplectic form

96 fi = dp Adq = eQ., noting that q,p are conjugate variables in fi. Next, we introduce a new time t = t/e, and we obtain a Hamiltonian system with Hamiltonian H€(p, Q, eT) = ±gq(p, p) + s2U(q, ^) = #o(p,
(3)

We also introduce E = E/Et, and we are going to see that, for sufficiently large E,, there exist hyperbolic whiskered tori with heteroclinic orbits between them whose energies are between E = 1/2 + 6 and E = 2 — 6 for some small 6 > 0. Since the result is for all sufficiently large Em, we will be able to take a larger E* and continue the transition chain. From now on and until further notice, we will drop the bar from the problem. We will refer to the bar variables as the rescaled variables and the original ones as the physical variables. Then the Hamiltonian He and all the functions derived from it will be 1/e periodic in time. In order to do this more apparent we will use the notation given in (3). 2.3

The inner and outer maps

We will define two maps in A which describe, respectively, the inner dynamics in A and the outer dynamics given by the homoclinic connections that lie in 7. In the unperturbed case the dynamics is known, and we will be able to give explicit expressions for these maps. Later on, when we consider the problem with external potential, the perturbative computation of these maps will be the main tool to control the behavior of the orbits. The inner map FQ will be the time l/e map of the geodesic flow, also called Poincar£ map, restricted to A. As we are dealing with the autonomous case, the phase portrait of this map is identical to that of the geodesic flow. For simplicity we will consider the Poincare map associated to the initial time 0, that is Fo(x) = $ i / e ( i ) , where $t(x) denotes the geodesic flow on the torus. We note that the behavior of this unperturbed map is exactly the same as the unperturbed flow, and then all the possible choices of the initial time give rise to the same map. Even though this map is defined in all the phase space, we are interested only on the dynamics in A, which, in the coordinates (J, (p), is given by: F0{J,
(4)

We call attention to the fact that Fo is a twist map and this will be useful to study the effects of the perturbation on it. We also note that the periodic orbits of the geodesic flow are invariant curves for this map, and they are given in these coordinates by J = const.

97

The outer map or "scattering map" So : A —» A, associated to 7, will transform the asymptotic point at —00 of an orbit homoclinic to A into the asymptotic point at +00. Thus we define x+ = S 0 (x_) if there exists z 6 7 C T*T 2 , such that dist ($t(x±), $t(z)) —» 0 ,

as

t —► ±00.

From the results of Morse and Mather of Theorem 2: dist(A 1/2 (t + a ± ) , 7 1 / 2 ( 0 ) -» 0,

as

t -» ±00,

(5)

and the rescaling properties give

dist (AE (t/y/W + a±/VwYjE

(t/VzE))

Hence, expressing the map So in the coordinates

^>0 as t -► ±00.

(6)

(J,f),

5 0 (J,a_ + tp) = (J,a+ + + A ) .

(7)

Now, in the unperturbed case, the dynamics is clear: if we take a periodic orbit KB, we have: FO(AE)

= AE,

(8)

SO(AE)

= AE.

(9)

These two conditions have a geometrical meaning: (8) means that A^ is an invariant curve of the Poincare map, and therefore, its orbits under the continuous flow constitute a 2-torus, which is invariant under the flow. Con dition (9) means that the stable and unstable manifolds of this curve only have homoclinic connections. After the scaling done in section 2.2, we have that for high energy, or what is the same, for small e, the external potential is an small and slow perturbation of the geodesic flow. Therefore, all the geometric structures that we constructed based on normal hyperbolicity and transversality persist for high energy. In particular, the manifold A will persist as well as the transversality of the intersection of its stable and unstable manifolds along 7. On the other hand this perturbation is 1/e-periodic in time. This will allow us to define F, S analogues of the maps FQ, SO, a n ( l to compute them perturbatively. The idea now is to use the maps F, S when the external forcing is acting. As we will look for orbits with growing energy, we will look for invariant curves

98

of F, which have heteroclinic connections between them. Then we have to look for closed curves Aj, A2 such that F(A i ) = Ai>

t = l,2,

S(A!)nA2^0. Therefore, the main technical goal will be to compute perturbatively, for high energy, the inner and the outer maps F and S, show that applying them alternatively we can construct heteroclinic orbits between several invariant curves, and then, show that these orbits can be shadowed by real orbits. Moreover, as we characterize the objects in A by the angle
The perturbed invariant manifold. Normal hyperbolicity

In order to study the dynamics with external potential, we consider the au tonomous flow associated to the Hamiltonian (3). dH0 dU P = —dT( p > q )~' 6 ~d a ~( q > £s '>

6 = e

2

q=^(p,q),

> (10)

5 = 1,

defined on the extended phase space T*T 2 x ^T 1 . The reason why we introduce the notation 6 = e2 above is so that we can study more conveniently the limit e —► 0. The normally hyperbolic pertur bation theory can make statements for the above system for all 6 < 60 and one can choose 60 independently of e. This allows us to make statements for £ small enough, even if the scaled flow is meaningless in this limit. We note that for the unperturbed case (5 = 0), we have that A := A x i f 1 ~ [Jo, 00) x T 1 x -T1 £

£

is a 3-dimensional manifold locally invariant for this flow. This manifold is also normally hyperbolic. The next theorem assures the persistence of a hyperbolic manifold for e > 0 close to this one, and it relies in results of hyperbolicity theory. x

99

Theorem 4 Assume that we have a system of equations as in (10), and of class Cr, 2 < r < oo. Then, there exists em > 0 such that for \e\ < e», there is a C r _ 1 function T: [Jo + # e 2 , o o ) x T ' x ^ T 1 x (0,e 2 ) -» T*T 2 x -T1 £

£

such that A£ = T ({Jo + Ke2,oo) x T 1 x ^ T 1 x {e 2 })

(11)

is locally invariant for the flow (10) and is £2-close to A in the Cr~2 sense. Moreover, A£ is a hyperbolic manifold. We can find a C r _ 1 function T* : [Jo + A ^ . o o ) x T 1 x -T1 x [0,oo) x [0,£2) - T*T 2 x ^ T 1 such that its (local) stable invariant manifold takes the form Ws'loc(Ae)

= F ([Jo + Ks2, oo) x T 1 x i f 1 x [0, oo) x {e 2 }^ .

(12)

Ifx = F(J, ip, s,e2) e A,, then W^s-loc(i) = . P ( { J } x {^} x {s} x [0, oo) x {e2}). Therefore W^8,loc(A£) is e2-close to W s - loc (A) in the Cr~2 sense. Analogous results hold for the unstable manifold. Let us observe that Theorem 4 only guarantees local invariance for A£. But, in a moment, we will show that this manifold has KAM 2-dimensional tori, that will provide with invariant boundaries for it. Therefore, it is possible to take A£ invariant. This remark leads to some extra uniqueness, etc. which could be useful for future developments. However, we will not use it in the present paper. 2.5

The perturbed inner map. Averaging theory

Once we have the 3-dimensional perturbed invariant manifold A£ given by (11), we can consider the flow restricted to it, and, after some symplectic changes of variables, we can write it as a Cr~2 Hamiltonian system with one and a half degrees of freedom, and 1/e periodic in the time s. More precisely, this Hamiltonian can be written in the local coordinates (J,
(13)

The Hamiltonian ke( J, y>, es) is non-autonomous, but it has an slow, periodic, dependence on the time s. We can apply the Adiabatic Invariance Theorem, eliminating the fast angle (f by means of an averaging procedure up to any order £ m . (More details on the averaging procedure can be found in 7 or in '.)

100

Theorem 5 Let ke(J,(p,es) be the Cr~2 Hamiltonian, l-periodic in

(I, V>, s), l-periodic in (p and es, which is e2-close to the identity in the CT~2~m topology, such that transforms the Hamiltonian system of Hamil tonian ks(J,tp,es) into a Hamiltonian system of Hamiltonian Ke(I,ip,es). This new Hamiltonian is a C r _ 2 _ m function of the form: K^I^es)

= K°e(Les) +

£m+1Kl{I,tP,es),

with K®(I,es) — ^P + Oc1^2)) where the notation O c ( £ 2 ) means a function whose C1 norm is 0(£ 2 )Now, we consider the perturbed inner map in these new variables. First of all, let us note that the Poincare return maps are defined in the intersection of the 3-dimensional perturbed invariant manifold Ae given by (11) with the Poincare sections given by fixing the time s, that we denote A* C T*T 2 : (Als)

= T{[J + Ke2,oo]xT1

x{s},e2).

By Theorem 4, for every s, A| is e2-close to the unperturbed manifold A in the C~2 sense. In the variables (I,ip), we have that the perturbed inner map F, defined on A, is £m-close to an integrable map: Lemma 6 In the conditions of Theorem 5, the map F : A —» A, which is exact symplectic, can be written in the coordinates (I, xf)) as F(I,4>) = (l,iP+±A(I,e))

+emRE(I,iP),

(14)

where A(I,e) = e f*/£ D1K°(I,es)ds = I + 0(e2), and R£ is a Cr-m-4 func tion. Now, in these new coordinates, the KAM theorem provides the existence of invariant curves for the perturbed inner map F, which fill out the manifold A except for gaps which are of the order of e m / 2 + 1 : Theorem 7 Assume thatr — m — 4 > 6. Then, ifeis small enough, for a set of w of Diophantine numbers of exponent 5/4, we can find invariant curves Au for F which are the graph of C~m~7 functions uw, with \\uw\\Cr-m-7 < const e m / 2 , and the motion on them is Cr~m~7 conjugate to the rotation by w. The set of these invariant curves cover the whole annulus except for a set of measure smaller than const e m / 2 + 1 . Moreover, we can find expansions uu — u2>+£ m u^+r w , with ||r||c,—m-s < const e2m, and ||u 1 || Cr _ m _ g < const.

101

Remark 8 Note that these KAM curves with frequency w that we have pro duced for the map F are really whiskered tori for the extended flow with frequencies (u>,e). They could have been produced also by appealing to the Graff-Zehnder Theorem, but it is not direct from this theorem that the higher is the differentiability of the flow the smaller are the gaps between the tori. In any case, proceeding as in Zehnder 8 we can obtain a normal form for the Hamiltonian H£(p,q,es) in a neighborhood of these KAM tori: G{I, a, if, s, zB, z u ) = w/ + a + ^ / 2 + {z\ Q{
(15)

Such normal forms are useful later in the study of A-lemmas in subsection 2.7. We also call attention to a more general normal form due to Fontich and Martin 9 which does not require that the motion on the tori satisfies Diophantine conditions and which is also the basis of the proof of a A-lemma. R e m a r k 9 Note that KAM tori produced by Theorem 7 are of codimension 1 inside A£. If we choose a submanifold whose boundary consist on two KAM tori, this submanifold will be an invariant manifold for the extended flow. In this case the results of hyperbolicity perturbation theory can be extended to include uniqueness, as observed before. Once we have the existence of the invariant tori of system (10) it is worth while to obtain some explicit approximations in the phases { be one of the frequencies allowed in Theorem 7. Then, in the coordinate system (He,
(17)

is a I-periodic in ((p,r) function which verifies

wD1g(
+ Oc-"-«(£3),

with U given in (16), and \\g(-, -;£)||Cr-™-8 is bounded uniformly in e.

(18)

102

Furthermore, we can choose g in such a way that g = D2J1. This h satisfies (obviously)

and

uDihfa

T; e) + eD2~h{
H;-\e)

Qr — m — S

fc3).

(19)

is bounded uniformly in e.

We call attention to the fact that, due to the slow dynamics, the functions g(ip, r;e) and h(
The perturbed outer map. Poincare-Melnikov theory

Due to the fact that A£ is a 3-dimensional hyperbolic manifold and that its 4-dimensional stable and unstable manifolds intersect in a 3-dimensional "homoclinic" manifold % close to 7 = 7 x ^T, we know that any point on A£ is connected to another point through some orbit, and then it has sense to define the perturbed outer map. On the other hand, we know that A£ contains other invariant submanifolds that are the tori produced by Theorem 7. Then, this homoclinic manifold 7 £ will contain, among others, homoclinic trajectories to each of these torus as well as heteroclinic trajectories between some of them (those that axe close enough). The fact that x+ = S ( x - ) for two points x± € A£ is, by definition, equivalent to the existence of a point z 6 T*T 2 x AT1, such that the trajectory %(t) starting on it converges for t —* ±00 exponentially fast to the trajectories A*(£) started in x± respectively, i.e. A^O) = x±, 7 e (0) = z, and dist(A £ t (*),7 £ (t))
as

t -» ±00.

(20)

We want to give criteria to decide whether x + and x_ belong to diflFerent tori given by the inner map and then 7 £ (t) is a heteroclinic connection between these tori. As the points x± are in the perturbed manifold A£, they can be char acterized by the angles s,tp and the value of the perturbed Hamiltonian Hc. Then, first we will compute x + = 5(x_) in these local coordinates. Prom (7) we already know that
103

Figure 1. Illustration of the perturbed tori and the outer map: y lies in the torus of £ _ , but with the angle coordinates of x+ = S ( x _ ) .

computing the increment of the energy, which in the unperturbed system is preserved. Since the energy He is not constant in the perturbed tori, it is not enough to compare Hc(i+) with He(x-) to decide whether x + = S(x_) and x + lie, or not, in the same torus. Instead, we will look for the point y which lies in the torus of x_, but with the angle coordinates of x + = <S(x_), which are (tp_ + A,s) + 0(e 2 ) (see Fig. 1). If x + lies in the same torus as x_, then x+ = 5(x_) = y. Then the function He(S(i-)) — Hc(y), which we will explicitly compute to first order, has to be not zero if the tori have a heteroclinic connection. In general, this function is a measure of how much the homoclinic excursion gains over just staying on the torus. It gives us a quantitative measurement of the width of the heteroclinic connections. We can compute HE(y) using the local expression of the torus that con tains x_, given in Lemma 10 in terms of the energy and angle coordinates. All this is summarized in the following lemma. Lemma 11 Let x+ and x_ two points on A£, such that i+ = 5 ( x _ ) . Let y be a point on A e , belonging to the torus that contains x_, with the phases of

104

x+.

Then H€{x+) - He(y) = e3M(
lim

+ 0(e5)

/

V(T„T 4 )-OO./_ T I

dtIhU(i%(t+-22=),e8 WBV

-g(
+ 0(e2),s),

+ et)

sflEh

+ sT2)

(21)

- sTi)J + 0(£ 5 ), E = J$/2, and where g is

the function given in Lemma 10 verifying (18), associated to this torus. Then, it is clear that, if the function M((fo,£S,E;e), usually called Mel nikov function, has non-degenerate zeros, the image of the torus of frequencies (y/2E, e) by the outer map S intersects itself, or, what is the same, there exists some homoclinic orbit to it. Relatedly, this function will allow us to quantify the width of the jumps in heteroclinic connections. It so happens that the Melnikov function is the gradient of a potential—often called the Melnikov potential— 10 defined by: L(^es,E;e) = (^

hm_£

dtU (,% (t + J L )

-~h(ip0 + a+ + V2ET2,es

,£S

+ eT2)

+

et) (22)

+h(
(23)

105

0(e'm/2+1

Figure 2. Illustration of the action of the map S on a torus r: S(T) intersects transversally tori close to r.

where +T2 C(T)

=

lim

+T3+a+

dt

U(^/2(t),T)

m,T 3 )->oo

dt

U(A1/2(t),;

-Ti+a-

(24)

Proposition 13 Given a metric that satisfies the genericity conditions of Theorem 2, and orbits h-\/2-,1\/2 of the geodesic flow as in the conclusion of Theorem 2, the set of periodic potentials for which the PoincarS function £(T) is identically constant is aCl closed subspace of infinite codimension for I > 0. 2.7

Transition chains and shadowing orbits

In this section we will see that, if the Poincare function £ ( r ) is non-constant, the perturbed invariant tori have heteroclinic intersections if l + m / 2 > 3, that is, if the measure of the gaps between them is smaller than the jumps produced by the homoclinic excursions (see Figure 2). This give us the existence of a so called transition chain. After this, some results on the literature and some

106

point set topology, will give us the existence of orbits that follow an infinite chain of these tori. Lemma 14 Assume that r > 15. / / the function C(T) is not constant, we can find K > 0 such that for e sufficiently small, given a KAM torus T, we can find other KAM tori T+, T~ such that W£fhWf + , W?fhWf_, xoia He{T) > maxH£(T~) + Ke3, maxH e (T) < minH e (T + ) - Ke3. Remark 15 The lemma above does not assert the existence of transverse homoclinic orbits to any of the tori T, T~ and 7" + . The existence of transverse homoclinic orbits is related to the existence of nondegenerate critical points of the Poincare function. We emphasize that, for our purposes, what we need are transverse heteroclinic intersections. Our next goal is to see that the heteroclinic orbits that can be obtained by Lemma 14 can be shadowed by true orbits of the system. As it is usual in the literature for Arnold diffusion, the key step is to find an appropriate inclination lemma (also called sometimes A-lemma). In the literature, one can find very sharp inclination lemmas—including even some estimates of the times needed to do the shadowing—for analytic maps, when the rotation is Diophantine. n - 1 2 - 1 3 . The result that we have found best adapted to our purposes is that of Fontich and Martin 9 for whiskered tori with one dimensional strong (un)stable directions—as is the case in the problem we are considering—, which works for C1 maps and only requires that the torus has an irrational rotation. A particular case of these results 9 is: Lemma 16 Let f be aCl symplectic mapping in a 2(d + 1) symplectic man ifold. Assume that the map leaves invariant a C1 d-dimensional torus T and that the motion on the torus is an irrational rotation. Let F be a d+1 manifold intersecting Wq- transversally. Then

Wf ell'" ■*(r). i>0

An immediate consequence of this is that any finite transition chain can be shadowed by a true orbit. The argument for infinite chains requires some elementary point set topology. Lemma 17 Let { T ^ } ^ be a sequence of transition tori. Given {ei}^ a sequence of strictly positive numbers, we can find a point P and a increasing

107

sequence of numbers Ti such that

*rt(P)e N£.«(*), where N£i(Ti) is a neighborhood of size £i of the torus %. From Lemma 14 and Lemma 17, we obtain the following result, which clearly implies Theorem 1. Theorem 18 Assume that the metric g satisfies the assumptions of Theo rem 2 and that U is such that the Poincai function C is not constant. Assume moreover that both g and U are C15. Then, there exist M > 0, a > 0, such that if Ii = [Bi_,Ei+]

i = l,...

is any sequence of intervals such that EL

>M,

{E\-Ei)>M(E\)-a. Then, we can find an orbit p{t),q(t) of the Hamiltonian flow and an in creasing sequence of times t\ < t
and (p(ti), q(U)) is in a neighborhood of size M(EX_)~2 of the periodic orbit Remark 19 By assuming more differentiability in the hypothesis of the the orem, we can get a to be arbitrarily large. This is because, if the system is sufficiently differentiable, we can perform more steps of averaging theory so that the KAM tori are much closer. Remark 20 Observe that the existence of the symbolic dynamics above es tablishes that the set of orbits that have unbounded energy is uncountable. We have uncountably many choices of energy ranges to hit. By going over the argument with somewhat more care we can see that there are different orbits that realize these choices. 3

Some final remarks

We conclude with some final questions and remarks about two problems that have been asked us and that are relevant from the physical interpretation of the problem. The most important question is whether the mechanism presented here is the only one or even the dominant one. All the questions that we present

108 here have a version for the orbits constructed here and for the set of orbits whose energy grows to infinity. Q l : What is the measure of the set of orbits whose energy goes to infinity (resp. those constructed here)? We know nothing about this question except to point out that using the sym bolic dynamics, the set of orbits constructed here is uncountable. Presumably, the set of orbits constructed here is of zero measure. Q2: What is the fastest rate of growth of the energy one can achieve? The following upper bound is elementary. Proposition 21 If q{t) is a solution of the Hamiltonian flow of H(p,q,t) \9q(P,P) + U{q,t), then

=

\H(p(t),q(t))\
109 It seems to be unknown whether this is really necessary or whether it is just an artifact of the method of proof. We point out that, in a very similar problem, Chirikov conjectured in a related case that indeed one would not need to increase the time spent near the tori for a long shadowing chain. Hence, if the Chirikov conjecture were true for our case (it seems to us that in our case the conjecture is more plausible than in the case originally stated for), then we could get a rate of escape that agrees with the upper bound up to a finite constant. Q 3 : Is there a quantum mechanical analogue of the phenomena discovered here? Of course, we consider as the quantum mechanical model the Hamiltonian given by the Laplacian and a time periodic potential. We note that an analogue of Proposition 21 can be proved in quantum mechanics, from

! < * | f f | ¥ ) = ( ¥ dv * dt

One is tempted to argue that the construction presented in the previous sections is too delicate to survive the quantum mechanical effects of uncer tainty etc. We would like to decide whether there are analogues of the existence of orbits whose energy goes to infinity and also an analogue of the symbolic dynamics result. Note that in quantum mechanics, as soon as we construct one solution whose energy grows to infinity, we have an open and dense set of solutions whose energy grows to infinity.

Acknowledgements We thank J.N. Mather for communicating his results and for encouragement. This work has been partially supported by the NATO grant CRG950273. Research by A.D. and T.M.S. is also supported by the Spanish grant DGICYT PB94-0215, the Catalan grant CIRIT 1996SGR-00105, and the INTAS project 97-10771. Research by R.L. is also supported by NSF grants. We also thank TICAM, UPC and IMA for invitations that made possible these collaborations.

110

References 1. A. Delshams, R. de la Llave and T.M. Seara, A geometric approach to the existence of orbits with unbounded energy in generic periodic pertur bations by a potential of generic geodesic flows of T 2 , preprint 98-591, mp_arcOmath.utexas.edu,(1998). 2. J.N. Mather, Graduate course at Princeton, 95-96, and Lectures at Penn State, Spring 96, Paris, Summer 96, Austin, Fall 96. 3. L. Chierchia and G. Gallavotti, Drift and diffusion in phase space, Ann. Inst. H. Poincare" Phys. Theor., 60, 1-144, (1994). 4. S. Bolotin and D. Treschev, Unbounded growth of energy in nonautonomous Hamiltonian systems, Nonlinearity, 12, 365-388, (1999). 5. V.I. Arnold, Instability of dynamical systems with several degrees of freedom, Sov. Math. Doklady, 5, 581-585, (1964). 6. V.I. Arnold and A. Avez, Ergodic problems of classical mechanics, Ben jamin, New York, (1967). 7. V.I. Arnold, V.V. Kozlov and A.I. Neishtadt, Dynamical Systems III, volume 3 of Encyclopaedia Math. Sci. Springer, Berlin, (1988). 8. E. Zehnder, Generalized implicit function theorems with applications to some small divisor problems, I and II, Comm. Pure Appl. Math., 28, 91-140, (1975), and 29, 49-111, (1976). 9. E. Fontich and P. Martin, Arnold diffusion in perturbations of analytic integrable hamiltonian systems, preprint 98-319, mp_arc<8 math.utexas.edu, (1998). 10. A. Delshams and R. Ramirez-Ros, Melnikov potential for exact symplectic maps, Comm. Math. Phys., 190, 213-245, (1997). 11. J.P. Marco, Transition le long des chaines de tores invariants pour les systemes hamiltoniens analytiques, Ann. Inst. H. Poincare Phys. Thior., 64, 205-252, (1996). 12. Jacky Cresson, A A-lemma for partially hyperbolic tori and the obstruc tion property, Lett. Math. Phys., 42, 363-377, (1997). 13. Enrico Valdinoci, Whiskered transistion tori for a priori stable and un stable hamiltonian systems, preprint 98-200, mp_arcfflmath.utexas.edu, (1998).

SPLITTING A N D MELNIKOV P O T E N T I A L S I N HAMILTONIAN SYSTEMS AMADEU DELSHAMS AND PERE GUTIERREZ Departament

de Matematica Aplicada I, Universitat Politecnica de Diagonal 647, 08028 Barcelona E-mail: [email protected], [email protected]

Catalunya,

We consider a perturbation of an integrable Hamiltonian system, possessing hyper bolic invariant tori with coincident whiskers. Following an idea due to Eliasson, we introduce a splitting potential whose gradient gives the splitting distance between the perturbed stable and unstable whiskers. The homoclinic orbits t o the perturbed whiskered tori are the critical points of the splitting potential, and therefore their existence is ensured in both the regular (or strongly hyperbolic, or a-priori unsta ble) and the singular (or weakly hyperbolic, or a-priori stable) case. The singular case is a model of a nearly-integrable Hamiltonian near a single resonance. In the regular case, the Melnikov potential is a first order approximation of the splitting potential, and the standard Melnikov (vector) function is simply the gradient of the Melnikov potential. Non-degenerate critical points of the Melnikov potential give rise to transverse homoclinic orbits. Explicit computations are carried out for some examples.

1

Introduction

For more than 2 degrees of freedom, the problem of measuring the splitting of the whiskers of hyperbolic invariant tori is closely related with the existence of instability in nearly-integrable Hamiltonian systems, i.e. with the Arnold diffusion. In this lecture, the splitting is studied in a wide setting, and a general Poincare-Melnikov theory is developed.

Setup We start with a perturbation of a hyperbolic integrable Hamiltonian, with n + 1 > 3 degrees of freedom. In canonical variables z = (x, y, ip, I) £ D C T x R x T" x R n , with the symplectic form dx A dy + dip A dl, consider a Hamiltonian of the form H(x,y,ip,I;n)

= H0{x,y,J)

+ /i#i(x,y,v?,I),

H0(x, y, I) = <«, I) + 1 (A/, /> + ^ + V(x) + (A, /) y,

111

(1) (2)

112

where p is a perturbation parameter. The Hamiltonian equations associated to H are: ± = y+(X,I)

+

y = -V'{x)

- fidxHi (x, y, tp, I),

(p = uJ + AI + Xy 1=

iidyHi(x,y,
It will shown in section 2 that, under weak assumptions, the unperturbed Hamiltonian Ho has n-dimensional whiskered tori (hyperbolic invariant tori) with coincident (n 4- l)-dimensional whiskers (invariant manifolds). For a given whiskered torus of Ho, its (unique) whisker is filled by homoclinic orbits (biasymptotic to the torus). Our aim is to study the splitting of the whiskers, and the persistence of some homoclinic orbits, for fj, / 0. Main achievements To deal with this problem, the tools used are Poincare-Melnikov theory, and a geometric method based on Eliasson's approach. Our contributions can be summarized as follows: • A general Poincare^-Melnikov theory for Hamiltonian systems is devel oped, defining a scalar function L (Melnikov potential) whose gradient M (Melnikov function) gives the splitting distance at first order in (i. • There exists a scalar function C (splitting potential) such that, in suitable variables, its gradient M. (splitting function) gives exactly the splitting distance. Besides, the splitting potential £ is approximated at first order in n by the Melnikov potential L. • The results are significant for more than 2 degrees of freedom. Motivation The study of the splitting in the Hamiltonian (1-2) is closely related to the problem of Arnold diffusion in a general nearly-integrable Hamiltonian system: H(4>,J) = h(J) + ef(4>,J),

(3)

in angle-action variables (4>, J) £ T n + 1 x R n + 1 . Here, the small perturbation parameter is e. Near single resonances, it is known l i 2 ' 3 that one step of (resonant) normal form procedure can be performed and leads, under some generic hypotheses

113

and after a scaling, to a Hamiltonian of the type (1-2), taking as HQ the truncated normal form. To make this clearer, consider a selected action J* = 0, and assume that its associated frequency vector djh(0) G R n + 1 has a single resonance (this means (k\djh(0)) = 0 for a certain k' G Z n + 1 \ {0} and (k,djh(Q)) ^ 0 for n+1 any k G Z not co-linear to k*). It can be assumed that djh(0) = (0,w*), with u* G R n nonresonant. Near J*, the unperturbed Hamiltonian h in (3) can be written as:

h(J) =-- (djh(o),j) + i (djh(o)j,j)

+ o3{j).

We write = (x,
=cr

d]h(o) =

>!)■

where we have put /3 2 > 0 in order to fix ideas, A G R n , and A is an (n x n)matrix. With some scaling, we can assume 0 — 1, and our Hamiltonian written in the form H (x, y, (p, I) = h(y, I) + ef(x, y,
= + i (A/,/) + ^ - + e V ( i ) + (A,/)y,

if, (x, y, p, 7; e) = £i?(x, y,
f

f(x, 0, y?, 0)dV>

x G T.

In the normalized expression for H, note that Ho (the truncated normal form) is an integrable Hamiltonian, and then Hi can be considered as a perturbation of some size E\L where \i can be determined in terms of e. In this sense, the expression obtained generalizes the Lochak's example 4 (which, in its turn, generalizes the famous Arnold's example, 5 designed to describe the diffusion). Under generic hypotheses, it can be shown that the Hamiltonian Ho has whiskered tori with coincident whiskers associated to this hyperbolic point (see

114

section 2). Therefore, although there is no hyperbolicity in h, the perturbation / provides some weak hyperbolicity, which appears in the truncated normal form HQ- This hyperbolicity disappears for e —► 0, because the Lyapunov exponents of the whiskered tori of HQ are of the form ±i/ect. To have fixed exponents, we replace y, I by \/ey, \fil (a non-canonical linear change), and divide the Hamiltonian by e. Then the new system is still Hamiltonian, and we obtain obtain for H = Ho + Hi an expression of the form (1-2), with

-$•

" = °(£"2)'

It has to be pointed out that, after this procedure, in general the truncated normal form HQ is a coupled Hamiltonian: A ^ 0 in (2). So the motivation for the coupling term {X,I)y is that this term appears in a natural way when one studies a nearly-integrable Hamiltonian, in a region close to a single res onance. As a particular case, note that if A = 0 in (2), then the unperturbed Hamiltonian HQ is somewhat simpler because it is formed by a pendulum and n rotors: we then say that Ho is uncoupled. We will show in section 4 that the formulation of Poincare-Melnikov theory is simpler in this special case. Although the (homoclinic) splitting between the whiskers of hyperbolic tori in single resonances is very important in the detection of Arnold diffu sion (through the construction of transition chains), we point out that there are other important difficulties related with this problem. These difficulties are the study of the transition properties of the tori, the detection of heteroclinic intersections between whiskers of different tori, and jumping the gaps associated to double resonances. Regular and singular cases According to the motivation above, it is convenient in (1-2) to allow u to depend on an additional parameter e, considering fast frequencies LJ = u*/y/e. The parameters e and (i can be whether independent or linked by a relation of the type \i = ep with some p > 0; these two cases will be called, respectively, regular and singular. We have shown that, in the study of a general nearlyintegrable Hamiltonian, the actually relevant case is the singular one (with p — 1/2), and that this feature is directly related to the weak hyperbolicity of the truncated normal form. Concerning the regular situation, we recall that the strategy of keeping e > 0 fixed and letting pi —► 0 (having in this way a regular system) was introduced by Arnold 5 in order to avoid dealing with a singular perturbation problem. In this case, Poincare-Melnikov theory can be applied directly to the

115 detection of the splitting, but only if the parameter fi is taken exponentially small with respect to e. This is due to that the Melnikov integrals involved are exponentially small in e, as in the second example shown in section 5 (for the first example shown, the integrals are not exponentially small, because the perturbation is not analytic in this case). In the singular case, one assumes that the parameters e and fi satisfy a power-like relation of the type /i = e9 (the smaller p the better), and one lets e —> 0. In this case, the problem of detecting the splitting from the Melnikov integrals is much more intricate, because of the exponentially small character of the integrals involved. However, some recent works 6 ' 7 , 8 suggest that, under some weak conditions, the Melnikov integrals give the right predictions for the splitting. Nevertheless, the existence of homoclinic orbits has been established in several works. 1 - 9 ' 10 This result is valid for regular and singular systems, and we recall it in section 6. 2

T h e u n p e r t u r b e d Hamiltonian

Assumptions In this section, we take \L = 0 and study the unperturbed Hamiltonian H0 denned in (2). Note that the given ingredients of Ho are the vectors o>, A e R n , the symmetric (n x n)-matrix A, and the function V(x) of x £ T. We require the following assumptions: • The function V(x) has a unique and nondegenerate global maximum. To fix ideas, we require V(0) = 0,

V'(0) = 0,

V(x) < 0 Vx ^ 0

V"(0)<0,

(mod 2TT).

(4)

• The following nondegeneracy condition holds: det (A - AAT) = det ( * A ^ ^ 0.

(5)

• The vector w is assumed to satisfy a Diophantine condition: for some T > n — 1 and 7 > 0,

|(fc,w)|>7|itrT

VfceZn\{o}.

(6)

116

The unperturbed torus and its homoclinic whisker The integrable Hamiltonian Ho can easily be studied. Let us introduce P(x,y)

= £+V{x),

P(x,y,/) = P(x,y+(A,J));

(7)

then H0 can be rewritten as

H0 = (u,I) + ±((A-MT)l,l)

+

P(x,y,I).

We see that, on every plane / = const, the Hamiltonian Ho reduces to a 1degree-of-freedom Hamiltonian: a generalized pendulum (the standard pendu lum being given by V(x) = c o s x - 1 ) . This pendulum has (x,y) = (0, - (A,/)) as a hyperbolic equilibrium point, with (homoclinic) separatrices given by y + (A, J) = ±1/—2V(x). The Lyapunov exponents of the hyperbolic point are ± a , where we define a = y/-V"(0). Therefore, the Hamiltonian Ho possesses an n-parameter family of ndimensional whiskered tori given by the equations / = const, y = — (A, / ) , x = 0, with (n+l)-dimensional whiskers. The stable and unstable whiskers of each torus coincide, and hence all orbits on this (unique) whisker are homoclinic, i.e. biasymptotic to the torus. We will focus our attention on a concrete hyperbolic torus, that we assume located at the origin: / — 0, x = y = 0. Note that the vector u>, assumed Diophantine, consists of the frequencies of this torus: (p = u>. In view of the nondegeneracy condition (5), the neighboring tori have different frequencies. Parametrizations for the unperturbed Hamiltonian We denote % the whiskered torus of Ho having frequency vector UJ. This torus can obviously be parameterized by To-

z'o{

As mentioned above, the stable and unstable whiskers of the torus To coincide; this homoclinic whisker is given by the equations I — 0, P(x,y) = 0. We denote Wo the positive part (y > 0) of the homoclinic whisker (it is often called separatrix). To give a suitable parametrization for Wo, we consider the 1-degree-of-freedom Hamiltonian P(x,y), and denote (xo(s),yo(s)) the associated homoclinic trajectory, with xo(0) = n, yo(0) > 0. Note that Xo(s) goes from 0 to 2TT when s goes from —oo to oo. It is clear that we can give the whisker Wo the parametrization W0:

z0(s,V5) = (xo(s),y 0 (s), ¥ ) + (xo(s)-7r)A,0),

s € R,

117

where the term (io(s) — TT)A expresses the phase drift undergone by any tra jectory when traveling along Wo- This drift is associated to the coupling term. Note that, with our definition, the dynamics on Wo is given by the equations s = 1, ip = u. One has lim [ZQ(S + t,tp + wt) — ZQ(IP±7rA + ut)] = 0, t—>ioo

and this implies that that every trajectory on Wo is biasymptotic to two different trajectories on the invariant torus %. If A is an integer (a very special case) then these two trajectories on % coincide. 3

Preservation of the whiskered torus and its whiskers

The local normal form Before studying the splitting, we have to establish the surviving under per turbations of our Diophantine whiskered torus, as well as its local whiskers. Then we have to extend them to global whiskers in order to compare the sta ble and the unstable ones. The surviving of the torus and its local whiskers under a small perturbation can be ensured by means of the hyperbolic KAM theorem, a version of the KAM theorem adapted to this problem. Roughly speaking, the hyperbolic KAM theorem provides a symplectic transformation $ taking our Hamiltonian into a local normal form H = H o $ (in some domain), having a simpler expression in which the perturbed torus becomes transparent, as well as its whiskers. This kind of result follows from a convergent KAM-like iterative scheme. We are interested in a normal form defined in a whole neighborhood of our concrete torus, 1 , n according to the "Kolmogorov's approach" to KAM theory. This approach allows us to control a neighborhood of the local stable whisker, which can be ensured in this way to contain also a piece of the global stable whisker (this feature is used in section 6). On the contrary, in the "Arnold's approach" (used in other papers) the normal form only holds on a Cantor set, although a large family of surviving tori is obtained. Some more comments and references to papers following both approaches are given in a recent paper of the authors. 10 In most papers (like for instance n ) , the hyperbolic KAM theorem is dealt in terms of some local variables in a neighborhood of the torus, in such a way that the whiskers become coordinate planes. A significantly new approach was introduced by Eliasson, who rewrote the hyperbolic KAM theorem and expressed it directly in the "original variables". l This is more suitable to our

118

purpose of carrying out a global control of the whiskers in order to study their splitting (see section 6). Another key fact is the use of exact symplectic transformations to normal form in the hyperbolic KAM theorem. To recall what an exact symplectic transformation is, consider the l-form 77 = -(ydx + Idip), whose differential is the standard symplectic 2-form: dr? = dxAdy+dipAdl. Then a transformation $ is symplectic if the l-form <1?*77 — 77 is closed, and it is exact symplectic if this l-form is exact (= dS, globally, for some scalar primitive S). Eliasson J used the exactness of the normalizing transformation as a cru cial tool in order to detect homoclinic intersections between the whiskers, in both regular and singular systems (although he did not compute the split ting). A similar result was also obtained by Bolotin. 9 In a further step, in the present lecture the exactness allows us to put the splitting function as the gradient of a splitting potential (see section 6). Another paper that has influenced our version of Eliasson's theorem is a recent one by Niederman. n This paper deals with a similar framework (using the Kolmogorov's approach but not working in the original variables), and obtains more accurate estimates for the normal form. Let us introduce first some notations. Concerning the domain, we define for r > 0 the complex set Br = {(x,y, if, I) : |x|, |y|, | / | , |Im p | < r} . For a function f(x, y, y>, I) analytic on some domain D (and continuous on its closure), we denote | / | D its supremum norm. Theorem 1 (Eliasson's theorem) Let H = HQ + HH\ as described in (12) and in the assumptions (4~6), with r > n — 1. Assume H analytic on Br (r < ro). Then for |/J| small enough, there exists an exact symplectic transformation $ = $ ( - ; / J ) : Bur —► Br (analytic with respect to (x,y,
(8)

Besides, one has $ = id + 0(M)> a = 0(M)> b = 1 + O(M)The most important point about this result is that, thanks to the use of the original variables x, 7/ u the local normal form H can be put in terms of the generalized pendulum P(x,y,I). By using this feature, a "global" control of the whiskers, very useful in order to compare them and study the splitting, can be carried out. 1,1D In is not hard 10 to establish the validity of theorem 1 in the singular case, with fi — ep and w = w*/y/e, for |e| small enough.

119

Paratnetrization of the perturbed torus It is clear that the normal form H given in (8) has a whiskered torus of fre quency vector ijj. We denote this torus as T, and its associated local whiskers as W^ (stable) and W,",. (unstable). The torus T has the following obvious parametrization: f:

i »

= (0,-(A,a),^a),

^ T .

This torus can be translated to a whiskered torus T of the original perturbed Hamiltonian H:

T:

* » = * ( r M),
In section 4, it will be useful to give a first order approximation in /u for the shift suffered by the perturbed torus T with respect to the unperturbed torus To, along the /-direction. We will denote I*(f) the /-component of z*((f). To describe this approximation, we consider the (zero average) scalar function x{ + 3*7(0,0,0) = ff, (0,0, •,<)).

(9)

where the notation / denotes the (^-average of a function / . The existence of X is ensured by the Diophantine condition (6). The function x, introduced by Treschev, 12 provides a first order approx imation 10 for the perturbed torus: /*(

(io)

where we define C = - (A - AA T ) _ 1 (3/J/i - \dyHx) (0,0,0). Parametrizations of the perturbed whiskers As in section 2, we can also take parameters on the perturbed local whiskers of the normal form H: W£ c :

z{s,(f) = (xo{bs),y0{bs) - (A,a) ,

for ±s > so, € T n , with some SQ — so(r). For the original Hamiltonian H, the local whiskers can be parametrized as follows: W,oC:

*toc(*,¥>)=*(5(*,V>)),

±s>s0,

f€Tn.

In the parameters s, tp, the dynamics of H on W^c is given by s = 1, ip = u). We need to extend these local whiskers to global whiskers, in order to measure the splitting between them. The parametrizations of the whiskers

120

Wioc> vaJid for ±s > SQ, can easily be extended to further values of s in a natural way, since the whiskers are formed by trajectories associated to our Hamiltonian H. We denote W * the extended or global whiskers. These global whiskers remain at distance 0 ( M ) t o t n e unperturbed whisker Wo, for an interval of real values of the parameter s. This interval can be chosen large enough in order to make it possible to compare the whiskers W + and W~ far from the torus T. 4

Poincar6—Melnikov theory

In this section, we develop Poincare-Melnikov theory in order to give a first order approximation for the splitting of the separatrix Wo into the perturbed whiskers W * associated to the perturbed torus T . Besides, we want to de scribe the set W + n W + , i.e. the homoclinic orbits to T. Melnikov potential and Melnikov function In order to provide a first order approximation for the splitting, we introduce the (scalar) Melnikov potential L(ip) and its gradient, the Melnikov function M(ip) = d^Lfa), by means of improper integrals, /^-independent and periodic in i/3 € F . These integrals are always absolutely convergent, thanks to the fact that the phase drift along the separatrix (due to the coupling term in (2)) and the first order deformation of the perturbed hyperbolic tori are taken into account. We stress that our use of absolutely convergent integrals in the formu lation of the general Poincare-Melnikov theory for whiskered tori makes a difference with respect to some previous works, 13>14>15 where conditionally convergent integrals are used and the integration limits have to be carefully chosen. Next we define these functions in several cases, in increasing order of complexity. • The simplest case is that of a perturbation vanishing on the whiskered torus, Hi = 02{x,y,I). In this case, the whiskered torus remains un changed (T = To). We define the Melnikov potential through the follow ing integral: oo

(Hi-H^){zo{tif

/

+ ut))dt

■oo / o o

Hx(z0{t,cp + ut))dt + const. •oo

(11)

121

Note that the additive constant is such that L = 0. For the Melnikov function, it is clear that oo

/

dvHi(zo{t,

ut))dt.

(12)

-oo

The absolute convergence of the integral (12) for the Melnikov function was already pointed out by Robinson, 15 stressing that in other cases this integral is only conditionally convergent. We also recall that, for n = 1, the Melnikov potential (11) coincides with a formula given by Delshams and Ramirez-Ros. 16 • Now, we consider the uncoupled case A = 0 (note that this case intersects but does not include the previous one). In this case, we define the Melnikov potential through the following integral, also absolutely convergent: r°° . L(
/

[Hi (zo(t,

• Finally, in the general case (which includes the two previous ones), we define (tfi - ST - {*, Ho}) (zd(«, V + ut)) dr..

(13)

■oo

Recall that H\(x,y,I) denotes the average of H\, and that the function xi.xi 2/> f< -0 — xif) is t n e (zero average) function solving the small divisors equation (9). Notice that L = 0, because the function inside the integral has zero average. The absolute convergence of the Melnikov integral (13) can be ensured using that the function H\ — H\ — {x, Ho} vanishes on To, together with the fact that Wo tends to To with exponentially decreasing bounds. We remark that the formula (13) is useful in both the coupled and the uncoupled cases (in (2), A ^ 0 and A = 0 respectively). An example illustrating the uncoupled case was given by the authors. 10 Related expressions, also valid in both cases, were previously obtained by Treschev. 12 In that paper, although the Melnikov potential was not introduced, the Melnikov function was expressed with the help of some correcting terms giving rise to the absolute convergence. We have improved that expression, including the correcting terms in the integral and providing a more compact formula.

122

First order approximation for the splitting distance The following standard result 1 0 shows that a first order approximation for the splitting between the global whiskers W±, measured along the /-direction, is given in terms of the Melnikov function M = dvL. Since both whiskers are (n+l)-dimensional manifolds contained in the same (2n+l)-dimensional level of energy, it is enough to express its distance by an rc-dimensional measure. We take the difference I~ — I+ as the measure for the splitting (we denote 7 ± (s,^)) the /-component of the parametrizations z±(s,(p)). Theorem 2 Assuming |/x| small enough, one has for any \s\ < so and

(14)

We stress that the Hamiltonian character of the equations implies the fact that the first order approximation of the splitting is simply the gradient of a scalar function L. An important fact in this theorem is that I\,..., In are first integrals of HQ. Following Treschev, 12 it is not difficult to generalize theorem 2 by considering any given first integral F of the unperturbed Hamiltonian HoThus, it can be given an analogous first order approximation for F~(s,
ip) - ?+{s,
(n2),

where we define L(s,(p) = L(tp - us), which can be considered a function defined on the separatrix Wo- Using this fact^an alternative measure for the splitting could be J f - JJ 1 ",..., I~_1 — l£_1,P~ — P+ (if w„ ^ 0), instead of

/;--/+,...,/--/+. Transverse homoclinic orbits As a simple corollary of theorem 2, we see that in the regular case the simple zeros of the Melnikov function M give rise, for |/i| small enough, to transverse homoclinic intersections between the perturbed whiskers. As is well-known, if a point belongs to the homoclinic intersection, then its whole orbit is also contained in this intersection. Thus, it is enough to find the zeros of M(
123

(defined o n F ) is a Morse function (its critical points are all nondegenerate: a generic property), we deduce from Morse theory that for |/i| small enough there exist at least 2 n transverse homoclinic orbits. It is well-known that this argument does not apply in the singular case, w = w'/i/I and /x = e p , because the Melnikov function M is typically ex ponentially small in e (see the second example in section 5). To ensure that /j.M(tp — uis) dominates the O (/i 2 )-term, one has to assume \x exponentially small with respect to e. For larger values of fi, the existence of intersections cannot follow directly from (14). In fact, the study of the splitting in the singular case requires a more careful analysis, 6 ' 7 ' 8 which is not carried out here. Nevertheless, the effective existence of a number of homoclinic intersections, for both the regular and singular cases, will be established in section 6. 5

Some examples with small divisors

To illustrate the properties of the Melnikov potential L, in this section we consider some examples, showing that L has nondegenerate critical points. As a measure of the transversality of these points as zeros of M = dvL, we also estimate the determinant of the symmetric matrix dvM = d2L at the critical points. Consider the Hamiltonian H = H0 + \xH\, with 1 v2 HQ(x, y, I) = (w, /> + - (A/, /) + y + cosx - 1, Hx{x,
f{
fke*^>.

The integrable Hamiltonian HQ is uncoupled (A = 0 in (2)), and consists of a pendulum and n rotors (the standard pendulum is given by V(x) — cos x — 1). Note that the perturbation H\ depends only on the angles x,

(15)

then this is an analytic function, and p is its width of analyticity in the angles if. On the contrary, if the coefficients are polynomially decreasing:

IM ~ n^.

( 16 )

124

then the function f(tp) is not analytic but only differentiable (it is Cp for any p n — 1. Melnikov coefficients The fact that Hi = Ctefa) allows us to compute the Melnikov potential ap plying the simple formula (11). In the integral, we have to consider the wellknown (positive) homoclinic trajectory of the standard pendulum: xo(t) = 4arctane',

2 yo(t) — ±o(t) = — — .

We have: (cosxo(t) - l ) / ( y + u / t ) d t + c o n s t = 2 / -oo

J-oo

J

^

2

'dt+const.

COsh t

Taking into account that the additive constant is such that L = 0, and writing L(f) = Z)fc=4o Lkex(k'lfi\ the Fourier coefficients Lk can be computed explicitly using residue theory: J-oo coslr t

sinh (f (k,Lj))

(note that the mean value / = /o does not influence the Melnikov potential). For the Melnikov function, it is clear that M^ = ikLkUpper bounds In the analytic case (15), an upper bound for the Melnikov potential L can be given 10 from the expressions of the coefficients. The upper bound obtained holds for e > 0 small enough, and is exponentially small in e. Its size depends strongly on the small divisors properties of the frequencies:

^l^)l^^)-P(-c£-/(-+2)), where C = C(T, 7*, p) is a constant. It is an important point in this estimate to assume a perturbation with an infinite number of harmonics. As stressed by Lochak, 4 one is then forced to take into account the small divisors as sociated to the frequencies, and this leads to the exponent l / ( 2 r + 2) inside the exponential. Notice that this exponent in the upper bound is reminiscent

125

of the Nekhoroshev-like estimates. Instead, if one assumes a finite number of harmonics (like in the Arnold's example 5 ), then one obtains the exponent 1/2, but this case is highly nongeneric. In an analogous way, we can obtain an upper bound for the differentiable case (16), but then the bound becomes a power of e: max \LUp)\ < const • e r / 2 T . "" v eT"

(18)

The golden mean and the Fibonacci numbers To establish the effective existence of splitting, one has to obtain also lower bounds, giving a more precise description of the asymptotic behavior of the Melnikov potential. This requires a more careful analysis of the small divisors associated to the frequency vector w*. This analysis can easily be carried out for the golden mean, a very simple case with 2 frequencies (i.e. with 3 degrees of freedom):

«* = (i,n),

^^y11-

(is)

This case was first considered by Simo 17 and, later on, 6 lower bounds for the Melnikov function and for the splitting, in the analytic case (15), were obtained. Recently, 10 lower bounds have been obtained for the Melnikov potential L and for the determinant of d£L at the critical points (ensuring also that L has nondegenerate critical points). These lower bounds (recalled in the second example below) are exponentially small with respect to e. This implies that, in order to deduce the existence of splitting as a consequence of theorem 2, the parameter \i has to be taken exponentially small in e. The differentiable case (16) is substantially different. A concrete exam ple 18 shows that the maximum of the Melnikov function has a lower bound of finite order in £. Then taking \x as a suitable power of e is enough in order to establish the existence of splitting from theorem 2. We stress that an essential point in dealing with the singular case is to assume that, in the perturbation, at least the harmonics fk corresponding to the small divisors associated to u* are nonvanishing, because the dominant harmonic is found among these ones. Under this assumption, one can obtain 6 the largest lower bounds in the Melnikov approximation, in order to ensure that this approximation dominates the O (M2)-remainder. For the golden mean (19), the associated small divisors are directly related to the Fibonacci numbers: F0 = F1 = 1,

Fn = F n _i + F n _ 2 ,

n > 2.

126

We recall some basic facts concerning these numbers, that will be used below. Defining CF =

1

i

n + fi-1 " > / § '

we have Fn = CF ( f i n + 1 _ (-i)»+in-<«+»>),

n>0.

The best rational approximations of fi are given by the convergents Fn/Fn-\. In other words, the indexes k^ — (Fn,—Fn_i) (and also (—F n ,F n -\)) are the ones that give the dominant behavior among the small divisors (k,u*). More precisely, one has:

<*v>-*-*-.»- ^ - ^ ° ( £ ) , and also the following inequality: not a Fibonacci number, \{k,u')\

6

n > 1,

for any k = (fci, — k2) such that k2 > 0 is

= \k1-k2n\>^.

(20)

Note that the frequency vector (19) satisfies the Diophantine condition (6) with T = 1. This frequency vector is considered in the two examples that we next study. Lower bounds: An example with finite-order splitting Now we consider a concrete example in the differentiable case (16), analogous to the one of Delshams et al., 18 and obtain lower bounds for the maximum of the Melnikov potential L, and for the determinant of d%L at a critical point. For the perturbation, we consider the following function:

n>l

In this function, the only nonvanishing Fourier coefficients are the ones as sociated to the Fibonacci indexes k^n\ Since |fc(")| = Fn+i ~ fin+2, the coefficients decrease as in (16). Applying (17), the Melnikov potential L(ip) is given by the series

i(y) = E 5 « cos ( A;(n) ^)> n>l

127 with

sn

1

2TT

yTe nn(r+i ) 8 i n h (_a 7 .y

Note that all the coefficients are positive. The main part of this expression is given by CO

_

_bo

47T :

P

n

ffi — ffl(r} — n(r 1 n i n i r O 1 "" °n °n(c) — nV i i J l O g " 1 „Qn /7'

To determine the dominant behavior, we look for the minimum exponent 6°, for n > 1. This is reached for No — No(e), with ■K

nNn

2(r + 1 ) ^ ' and this gives the largest coefficient: SN0 > 5^ o ~ £ r / 2 . This coefficient itself constitutes a lower bound for the maximum of the Melnikov potential, because all the coefficients are positive and

max |L( V )| = 1(0) =

(21) n>l

We can also get an upper bound, which coincides with the one of (18). Let us break the series (21) in two parts. For n < No, note that

sS sS-i

=

<

7T

\

fir+ie*P|\2£P+lJe)

n ^ l CXP( \ '- Qr+i l 2 0 " ° + ^ ) "■ ( - 5 - )

> L

Using also Sn < §S£ (from the fact that sinhx > e x /3 for x > 1), the sum 12n No we have the inequality Sn < 4fl~ n r (simply using that sinhx > x), and we can bound Yln>N Sn as & geometric series. In this way, we obtain for the maximum value of the Melnikov potential, an upper bound and a lower bound, both of the same order:

max I^Mi ~ er/2. ve'r-1 The Melnikov potential L(ip) has
128 eigenvalues of 3 2 L(0) as a measure for the transversality. We have 3^(0) =

-£sn^)(jfcW)T, n>l

and then det^L(O) =

(Z^n)

V«>i — 2_,

fee^y w y

-

/ FnFm-\(FnFm-\

_

(X^nFn-xSn) \n>l

-fn-1

/

Fm)SnSm

n,m>l

(Wrn-l — Fn-iFm)

= Y,

SnSm — /__,

Fn_m_lSnSm,

l<m
l<m
where we have used the formula FnFm-i -Fn_xFm = ( - l ) m + 1 F n _ m _ i . Note that all terms in this series are also positive. To estimate the determinant, note that Fn_m_lSnSm

~ ft n 5 n •

ft

m

Sm,

and hence the indexes n and m can be separated. This allows us to find the indexes N\{e) and Mi(e) that give the dominant term in the series of the determinant, in the same way as before. In this way, we easily obtain an upper bound and a lower bound for the determinant: det^L(0)~er. In fact, we should estimate the minimum eigenvalue of d£L(Q). This eigenvalue can be put in terms of r = trdjL(O) and 6 = detd£L(0). Again, note that r ~ £ r / 2 (applying the same method). The minimum eigenvalue is given by T -- v V

2

2

- -46 . * r

and this has clearly a lower bound of order e r / 2 . Then it is a consequence of theorem 2 that, for (j. = o (e r / ' 2 ), the critical point
129

Lower bounds: A singular example For the sake of completeness, we also include an example 10 in the analytic case (16). For the perturbation, we consider a "full" Fourier series, with the coefficients Vfc€Z2\{0}.

\fk\=e-W<>

Note that a non-even function f(tp) is allowed, so we are not assuming that the perturbation H\(x,y) is reversible, unlike other papers. 19 ' 7 - 8 Following the method by Delshams et al. 6 (though the context is some what different), it is shown 10 that the dominant harmonics in the Fourier series of the Melnikov potential L(
2ir

(,—fn+ip

n > 1.

° ^ **(«!*)'

The main part of this expression is given by ,„n,

47T

:e-»°,

£ = £(£) = C

F

n^p+_JL_.

(22)

For a fixed e > 0, to find the dominant harmonic among the Fibonacci ones, one has to look for the minimum exponent 6°, n > 1. Let us define

" n ~ U n ° V ~ n4°B'

Do_

2CFp V2C/

The minimum exponent among the 6° is reached for an only integer NQ = No(e), such that log£Ar0 is the closest to loge, among the loge n . Then the coefficient S%0 is the dominant one among the 5 ° , and it is not hard to check that the "whole" coefficient SN0 is also dominant among the Sn- One can also check from (20) that the non-Fibonacci coefficients Lk, with k ^ k^n\ do not dominate. In terms of e, the value of the minimum exponent depends on e in the following way: _ c(loge) 0 °No ~ £ l/4 ' where C(TJ) is a continuous function, defined as the (4 log Q)-periodic extension of c(n) = C0 cosh C^^-J

.

\V-Va\<2

log Q,

130

with Co = ^ly/2nCpP,

77o = log£o-

The extreme values of this function are given by Q3/2/-Y

C 0 < c(r)) < — — - = (1.029085.. .)C 0 . In this way, the maximum value of the Melnikov potential can be approx imated by its dominant Fibonacci harmonic, and one obtains the following upper and lower bound:

™™\L^)\~^z™v\—7T74-JIt is also shown 10 that the Melnikov potential L(ip) has nondegenerate critical points. In order to detect these points, one has to consider an ap proximation given by at least the 2 dominant harmonics, because with only 1 harmonic the approximation to the matrix d^L would be degenerate. In the discussion above, it can also be considered the integer Ni(e) reaching the "second" minimum among the 6°; this integer satisfies \N\ — NQ\ = 1. Calling TV = N(e) — mm(N0,Ni), it turns out that £^+i < £ < £N, and the Fibonacci coefficients with indexes N and N + 1 give the 2 dominant harmonics in the Fourier expansion of the Melnikov potential. The two dominant harmonics give the main part of the Melnikov potential L(ip). In the trigonometric form, this main part can be written as L(,¥>) + aN) + 2\S N + l \ cos ((fc<" +1) ,
,

where a^, 07/+1 are some phases. The number of critical points is given by the determinant A „ = det (*<">, A<™)) = F „ _ , F „ + 1 - F* = ( - l ) " + \ which implies that, for e small enough, L(tp) has exactly 4 critical points. At every critical point if*, one has |detajL(v9*)|~4|5w5N+1|^0. Then L((f) is a Morse function, because all its critical points are nondegen erate. Note also that 4 = 2 2 is the minimum number of critical points for a Morse function on T 2 . To estimate the size of the determinant, we use (22) again: 4|<7°oO

I -

647f2

r-(b°„+b°„^)

ho

, ho

_ ci(loge)

131

where c\ (r?) is another (4 log Q)-periodic function, defined from Cfa) = fi3/2C0cosh (j^jk)

,

\r) - Vo\ < 21o g n,

with 7]'0 — log y/eo^T. The extreme values of this function are given by n3/2C0
fi3C0

Thus, one obtains an upper bound and a lower bound for the determinant at the 4 critical points:

de,a^-)~^e X p(-2fl2|£)) Proceeding as in the previous example, one also finds an estimate for the minimum eigenvalue. Since, in this case, 8 <; r, the minimum eigenvalue can be approximated by 6/T. This leads to an estimate of the type 1

ex

/

c2(\oge)\

iiTT P ( — i l T T - J ' where 02(77) = 01(77) — c(r]). This is also a positive periodic function, with

2

-

'w/ - V

2,

Then it is a direct consequence of theorem 2 that there exist 4 transverse homoclinic intersections, for fi = o (exp{— C2(log£)e -1 / 4 }). The estimate obtained gives a measure for the transversality of the splitting. However, it has to be recalled again that this is actually a regular situation, and a justification in the singular case /x = e p , for some p > 0, does not follow directly from theorem 2. 6

Flow-box variables and splitting potential

The aim of this section is to sketch the proof of the result 10 that, using suitable variables, the "whole" splitting distance (and not only its first order approximation) is the gradient of some function, in order to establish the existence of homoclinic orbits even in the singular case.

132

Flow-box variables In order to provide a clearer formulation for the problem of measuring the splitting, it is convenient to introduce new symplectic variables in which the Hamiltonian equations are very simple. The flow-box variables W = (S, E, ip, J) are constructed 10 with the help of the flow associated to the normal form H given in (8), from a suitable Poincare section containing the set S = z ( s i , T n ) C W,QC, with some fixed s^ > so- The new variables are then given by an exact symplectic transformation (x, y, ip, I) = ^ ( 5 , E,ip, J), defined on a real neighborhood of S = * _ 1 (5) = (0,0, T \ a). Thanks to the use of the Kolmogorov's approach to the hyperbolic KAM theory, the neighborhood where the flow-box variables are defined contains a piece of both whiskers. In the construction of the variables, one can make the local stable whisker become a coordinate plane (see (24)), and then the global unstable whisker can be seen as a graphic over the local stable one. In this way, the splitting distance and the homoclinic intersections between the two whiskers appear much more transparently. Our Hamiltonian takes, in the flow-box variables, a very simple form: H = HoV

=

E+(cj,J-a),

and hence the associated Hamiltonian equations are 5 = 1,

E = 0,

ip = u,

J = 0.

(23)

We recall that analogous flow-box variables have already been used 20,6 in some case where the symplectic change can be defined explicitly from the expression of the normal form, which is integrable. In our case, the normal form H is, in general, not integrable, and the construction of the flow-box variables is more involved (it uses implicit functions). Parametrizations in the flow-box variables Let us describe more precisely how the whiskers can be parametrized in the flow-box variables. Let us denote W^. = \I/ _1 (VVj^c) the local stable whisker (or more precisely a piece of it). This whisker becomes a coordinate plane, given by E = 0, J = a, and can be parametrized as follows: W^:

^ ( S ,

V

) = *-1(Z(

5 I

+ 5 , ^ ) ) = (S,0,VP,O),

sel,

^ F ,

(24) where I is some interval containing s = 0 (we have replaced s — si by s for a clearer notation: in this way we have s = 0 on S).

133

Now, we define W~ = # _ 1 o $ - 1 (VW~) as an invariant manifold of H, which is the equivalent in the flow-box variables for (a piece of) the global unstable whisker. Let us parametrize: W- :

W-{s,
SGI,

In components, we write W~(s,(p) =

(S-(s,
There is splitting of the whiskers when J~(s,tp) ^ o o r E~(stp) ^= 0. Never theless, it suffices to control the ./-component because the whisker is contained in the zero energy level: E~ + (w, J~ - a) = 0. The approximation given in theorem 2, expressed there in the original variables, remains true after changing to the flow-box variables. So the Melnikov function M = d^L also provides a first order approximation in /z for the splitting distance J~ (s, imply that for |/i| small enough there exists a generating function 0 (S,E,ip,J), defined in some neighborhood of S and global in the angles rl> € T", such that the map 6 : ($, E, \j), J) ■-» (5, E, V», J) is given by

S = $-d&8,

E = E-ds6,

ip = tp-dj6,

J = J-d^6.

(25)

In order to compare the whiskers W ^ and W~, it will be useful to express the splitting distance J~(s,f) — a as a gradient. This cannot be deduced

134

directly from (25), but this obstruction is easily overcome, 20 introducing new parameters that substitute the initial ones s, ip on the whiskers: S = S~(s,
tp = ip-(s,ip).

In terms of the new parameters S, rp, the unstable whisker W - appears nicely as a graphic over the stable whisker Wj£c, through the parametrization: W":

W-{S,4>)=(s,E-(S,1>),il>,J-(S,rl>j),

S el,

i> G T" (26)

(the interval I can have undergone a reduction). It is then natural to introduce the splitting potential as the following scalar function, periodic in tp: C(S,il>) = 6(S,0,rl>,a),

S el,

V € T".

(27)

This function also depends on /x, and is determined up to an additive constant. The (vector) splitting function can then be denned as the gradient of £ with respect to the angles: M(S,ip) =

d^£(S,rp).

The next theorem is easily deduced from the equations (25). Theorem 3 The functions £ and M. only depend on ip — uS: £(S, V) = £ ( 0 , 4 > - u S ) ,

M(S, rp) = M(0,tp-

uS).

Besides, these functions are related with (24) and (26) in the following way: E-(S,W

= ds£(S,i>),

J~(S,4>) - a = M{S,1>).

According to this theorem, the function M. gives the splitting distance (expressed in the parameters S, rp). It is important to stress that the fact that the splitting distance can be put as the gradient of some potential is a reflection of the Lagrangian properties of the whiskers. As a corollary of theorem 3, one can recover a result due to Eliasson: J there exist at least n + 1 homoclinic orbits (not necessarily transverse), biasymptotic to the whiskered torus T. This result, valid for both the regular case and the singular case, comes from the fact that a function on T n has at least n + 1 critical points (not necessarily nondegenerate), according to the Lyusternik-Schnirelman theory. 21 Then for a fixed 5, the splitting potential £ ( 5 , •) has at least n + 1 critical points, which give rise to respective homoclinic intersections between the whiskers W±, and hence to homoclinic orbits, contained in both whiskers.

135

First order approximation for the splitting potential Finally, using Poincare-Melnikov theory, we can obtain first order approxi mations for the splitting potential C, introduced in (27), and for the splitting function M- At first order in /i, these approximations are given, respectively, by the Melnikov potential L and the Melnikov function M denned in section 4, but they are good enough only for the regular case. Theorem 4 For S e l and ip € T", one has C{S, ip) = nL{xp - UJS) + O (M 2 ) ,

M(S, if) = /iAf (tf - uS) + O (/i 2 ) •

We finish with some remarks about the additional difficulties of the sin gular case. Note that theorem 4 provides an O (M 2 ) error term that is not small enough in the singular case fj. = ep with p > 0, due to the fact that the functions L and M are exponentially small with respect to e. (This is illustrated in the second example of section 5). Nevertheless, one can expect that, under some weak hypotheses on the perturbation, the predictions of the splitting given by the Melnikov potential L are also valid in the singular case, for some p > 0. To get better bounds of the O (/i 2 )-term for real values of the variables S, ip, one should bound this term on a complex strip of these variables. This requires some improvements of the results presented here. First, one needs a more precise version of theorem 1, carrying out a careful control on the loss 6 of complex domain in the angular variables. Such an improvement of the normal form theorem has already been performed by the authors, 10 and in fact analogous results had previously been obtained 2 ' 6 for somewhat different contexts. On the other hand, one needs an extension theorem and the flow-box vari ables extended to a suitable complex domain, which would lead to a significant refinement of theorem 4, of the type £ ( 5 , T/0 = fxL ty - Su>'/y/e) + O ( / x V " ) , for 5, %l> on a complex strip |ImS| < 7r/2 — e 1 / 4 , |Im ip\ , exponentially small upper bounds for the error term, which would be dominated by the first order approximation provided by Poincare-Melnikov theory, under some general hypotheses on the perturbation. If this is true, then the Poincare-Melnikov theory gives the right predictions for the splitting even in the singular case. The problem of giving asymptotics for the exponentially small splitting of separatrices is now being researched by the authors. In fact, the strategy de scribed above has been followed 20 ' 6 in simpler situations in which the normal

136 form is integrable and the flow-box variables can be defined explicitly. Rudnev and Wiggins 8 announced an important generalization, but their proof contains essential errors. Therefore, the problem of giving asymptotics for the splitting in the Hamiltonian (1-2), in the singular case, remains still open. Acknowledgments This work has been supported in part by the Catalan grant CIRIT 1998SGR00042. Research by A.D. is also supported by the Spanish grant DGICYT PB94-0215. References 1. L.H. Eliasson, Biasymptotic solutions of perturbed integrable Hamilto nian systems, Bol. Soc. Bras. Mat, 25, 57-76, (1994). 2. M. Rudnev and S. Wiggins, KAM theory near multiplicity one reso nant surfaces in perturbations of a-priori stable Hamiltonian systems, J. Nonlinear Sci., 7, 177-209, (1997). 3. A. Delshams and P. Gutierrez, Homoclinic orbits to invariant tori in Hamiltonian systems, in C. Jones, S. Wiggins, A. Khibnik, F. Dumortier, and D. Terman, editors, Multiple-Time-Scale Dynamical Systems, IMA Volumes in Mathematics and Its Applications, Springer-Verlag, to appear (1999). 4. P. Lochak, Canonical perturbation theory via simultaneous approxima tion, Russian Math. Surveys, 47, 57-133, (1992). 5. V.I. Arnold, Instability of dynamical systems with several degrees of freedom, Soviet Math. Dokl., 5, 581-585, (1964). 6. A. Delshams, V.G. Gelfreich, A. Jorba, and T.M. Seara, Exponentially small splitting of separatrices under fast quasiperiodic forcing, Comtn. Math. Phys., 189, 35-71, (1997). 7. G. Gallavotti, G. Gentile, and V. Mastropietro, Pendulum: separatrix splitting, preprint, 97-472, mp_arc<&math.utexas.edu, (1997). 8. M. Rudnev and S. Wiggins, Existence of exponentially small separatrix splitting and homoclinic connections between whiskered tori in weakly hyperbolic near-integrable Hamiltonian systems, Phys. D, 114, 3-80, (1998). 9. S.V. Bolotin, Homoclinic orbits in invariant tori of Hamiltonian systems, in V.V. Kozlov, editor, Dynamical systems in classical mechanics, 168 Amer. Math. Soc. Transl. Ser. 2, 21-90. Amer. Math. Soc. Adv. Math. Sci., 25, Providence, RI, (1995). .

137

10. A. Delshams and P. Gutierrez, Splitting potential and Poincare-Melnikov theory for whiskered tori in Hamiltonian systems, preprint, submitted to J. Nonlinear Sci., (1998). 11. L. Niederman, Dynamics around simple resonant tori in nearly integrable Hamiltonian systems, preprint, 97-142, mp_arcamath.utexas.edu, to appear in J. Differential Equations, (1999). 12. D.V. Treschev, Hyperbolic tori and asymptotic surfaces in Hamiltonian systems, Riissian J. Math. Phys., 2, 93-110, (1994). 13. P.J. Holmes and J.E. Marsden, Melnikov's method and Arnold diffusion for perturbations of integrable Hamiltonian systems, J. Math. Phys., 23, 69-675, (1982). 14. S. Wiggins, Global bifurcations and chaos: analytical methods, 73 Appl. Math. Sci. Springer, New York, (1988). 15. C. Robinson, Horseshoes for autonomous Hamiltonian systems using the Melnikov integral, Ergodic Theory Dynam. Systems, 8, 395-409, (1988). 16. A. Delshams and R. Ramfrez-Ros, Melnikov potential for exact symplectic maps, Comm. Math. Phys., 190, 213-245, (1997). 17. C. Simo, Averaging under fast quasiperiodic forcing, in J. Seimenis, editor, Hamiltonian Mechanics: Integrability and Chaotic Behavior, 331 NATO ASI Ser. B: Phys., 13-34. Plenum, New York, (1994). 18. A. Delshams, V. Gelfreich, A. Jorba and T.M. Seara, Splitting of sepa ratrices for (fast) quasiperiodic forcing, in C. Simo, editor, Hamiltonian systems with three or more degrees of freedom, 533 NATO ASI Ser. C: Math. Phys. Sci., 367-371. Kluwer Acad. Publ., Dordrecht, Holland, (1999). 19. G. Gallavotti, Twistless KAM tori, quasi flat homoclinic intersections, and other cancellations in the perturbation series of certain completely integrable Hamiltonian systems. A review, Rev. Math. Phys., 6, 343411, (1994). 20. A. Delshams and T.M. Seara, Splitting of separatrices in Hamiltonian systems with one and a half degrees of freedom, Math. Phys. Electron. J., 3, 1-40, (1997). 21. S.-N. Chow and J.K. Hale, Methods of bifurcation theory, 251 Grundlehren Math. Wiss., Springer-Verlag, New York, (1982). Internet access: The authors' quoted preprints are available in the P r e p r i n t s pages of http://www-mal.upc.es, or in the p u b / p r e p r i n t s di rectory of f t p : I Ittp-mal.upc.es.

I N F I N I T Y MANIFOLDS OF C U B I C POLYNOMIAL HAMILTONIAN V E C T O R FIELDS W I T H 2 D E G R E E S OF FREEDOM MANUEL FALCONI Departamento de Matemdticas, Fac. de Ciencias, UN AM, Mixico, D.F. 04510, Mixico, E-mail:[email protected] ERNESTO A. LACOMBA Departamento de Matemdticas, UAM-I, P.O. Box 55-534, Mixico, D. F. 09340, Mexico, E-mail: [email protected] JAUME LLIBRE Departament de Matematiques, Universitat Autonoma de Barcelona, 08193 Bellaterra, Barcelona, Spain, E-mail: [email protected] Let X be the Hamiltonian vector field with two degrees of freedom associated to the cubic polynomial Hamiltonian H (i,y, z,w). Using the Poincar6 com pactification we show that all the energy levels of X in R4 reach the infin ity in a surface topologically equivalent to the intersection of the 3-dimensional sphere S3 = {(x, y,z,w) 6 R4 : x2 -I- y2 + z2 + w2 = 1} with {(x,y,z,w) G R4 : H3 (x, y, z,w) = 0}, where Hz denotes the homogeneous part of degree 3 of H. Such a surface is called the Infinity Manifold associated to H. In this paper we describe all possible infinity manifolds of cubic polynomial Hamiltonian vector fields with 2 degrees of freedom. Our method is general, but since actual computations can become very cumbersome, we work out in detail only three out of ten possible cases.

1

Introduction

We consider the Hamiltonian vector field X with 2 degrees of freedom gener ated by a cubic polynomial in four variables H (x, y, z, w). This vector field can be extended to infinity in the following way. Poincare compactification allows to project the vector field onto the north and the south hemispheres of the sphere S 4 C R 5 . The extended vector field X on the equator S 3 C S 4 represents the asymptotic behavior at infinity of our original Hamiltonian vector field. The vector field in the open hemispheres is not Hamiltonian any more but it keeps having nevertheless a first integral. All the corresponding energy levels intersect the equator S 3 in the same (perhaps singular) 2 di mensional infinity surface W, invariant by X. This surface turns out to be exactly the intersection with the sphere S 3 of the set of zeroes of the cubic homogeneous part H3 (x, y, z, w) of H (x, y, z, w). We see that W is compact

138

139

and it is generically smooth and with isolated equilibrium points. In this work we describe the topological classification of surfaces W, through the computation of the index at equilibrium points of the vector field X restricted to W. This vector field Y on the cubic surface W represent the asymptotic behaviour at infinity of the vector field X. One can compute the index of the vector field Y on any connected component U of W as the sum ot its indices at the isolated equilibrium points in U. By the PoincareHopf Theorem, the Euler characteristic x of £/ is equal to the computed index. Finally, the genus g of U is related to its characteristic through the formula X = 2 — 2g. The genus of any compact connected smooth surface determines its topology, as required. In Section 2 we describe in detail the Poincare compactification of poly nomial Hamiltonian vector fields as in Delgado et al 3 . We see how the infinity manifold and its induced flow are defined. Not only the infinity manifold but its flow depend only on the highest degree homogeneous part of the poly nomial Hamiltonian. Hence, we may assume without loss of generality that the Hamiltonian is a homogeneous polynomial. Then we specialize to cubic homogeneous polynomial Hamiltonians with two degrees of freedom. In Section 3 we study general topological considerations. The most im portant conclusion when applied to cubic surfaces W is that the genus g of any connected component must be even. In Section 4 we consider the simple case of cubic reducible homogeneous polynomial Hamiltonians with two degrees of freedom. Since any such polyno mial is the product of a quadratic polynomial times a linear one, we begin by studying the quadratic case. The conclusion for cubic reducible Hamiltonians is that the corresponding cubic surface W is generically the union of three disjoint spheres. In Section 5 we begin the study of the topological classification in the cubic irreducible case. We take a normal form proposed by Sylvester and Arnold, which has only 4 coefficients. Via a change of variables we can see that the set p of parameters where the corresponding W is singular, is the intersection of 15 hyperplanes with the closed positive cone. We study the topology of the infinity manifold W — E°° in each of the connected components in the complement of p. There are too many com ponents, but one can take a quotient by the equivalence relation obtained by permutation of coordinates and other obvious symmetries. This way the number of components is reduced to 10. This is sketched in Subsection 5.1 by following a combinatorial procedure, whose details are given in the com panion paper Falconi, Lacomba and Llibre6. In Subsection 5.2 we apply the Poincare-Hopf Index Theorem to compute index and then genus for surfaces

140

W in the 3 components appearing along the diagonal in the coefficient space. This can be done for the remaining 7 components, only that the compu tation of the equilibrium points and indices although straightforward becomes very cumbersome. 2

Poincare" Compactification of a Hamiltonian Polynomial System

Let X be a polynomial vector field of degree m in R n . The Poincari. compactified vector field X corresponding to X is an analytic vector field in duced in Sn as follows (see, for instance Cima and Llibre 2 ). Let Sn = {V = (yu ■ ■ ■ 2/n+i) € R n + 1 | y\ + ■ ■ ■ + yl+1 = 1} (the Poincar6 Sphere) and TySn be the tangent space to 5 n at point y. Consider the central projec tion 0 * :T(0>...,o,i)5'n —♦ 5 n . These maps define two copies of X, one in the northern hemisphere and the other one in the southern hemisphere. De note by X' the vector field D^ o X defined on 5 n except on its equator n_1 is identified with the infinity of R". Sn-i = Sn n { y n + i = o} . Clearly S In order to extend X' to a vector field on the whole Sn it is necessary that X satisfies suitable conditions. In the case that X has degree m, X is the only analytic extension of y™+iX' to 5 " . On S n \ S n _ 1 there are two symmetric copies of X, and knowing the behaviour of X about 5 n _ 1 , we know the be haviour of X at infinity. The Poincare compactification has the property that Sn~x is invariant under the flow of X. When this procedure is applied to a Hamiltonian vector field XH = fe' ••••& - ^ ■ ■ ■' " £ £ ) w h e r e H ■■ R 2 d — > R * a Polynomial of degree m -f 1, a new compactified vector field is obtained on the sphere 5 2 d , which is denoted by XH and is given by ~ , dH* dH* dH* XH = (~ + -M/i, • • •, -x— + Aj/d, - — oyd+i oy2d ay i dH* + Ay 2 d,Ay„ +1 ), dyd where n = 2d and H'(yi,--yn+1)

x

h \yd+i, • • •,

(2)

= yZ£H(-p-,...-^)

(3)

\J/n+l

and J^f

dH'

(1)

8H*\

J/n+1

141

ir1

Figure 1. Figure 2.1: Poincari compactification.

Computing the derivatives of H* along solutions curves of XH one gets 4- PH' u' + dy„+i y "+i

*£ d

[ff («£ + **) + *£(-«£ + **«)] + *Er*^ (5)

-*(£[*»

= A(m +

, a//* 2/«+d] + a w yn+ij dyn+i

!)#*.

From this equality it follows that the level energy H* — 0 is invariant under the flow of XH . In what follows for any h € R we consider our Hamiltonian H of degree m in the form H = h-\-H\A Hm, where Hi is the homogeneous part of H of degree i for i = 1,2, • • •, m. The vector field XH satisfies the following theorem Theorem 1 Let H:R2d —► R be a polynomial Hamiltonian of degree m de pending on n=2d variables (j/i, • • • yn), and let XH be the Hamiltonian vector field associated to H. Let XH be the Poincari compactification of XH. Then the following statements hold:

142

(a) The energy level Eh = H * (0) is mapped diffeomorphically by the cen tral projection onto a subset E^ (E^) of the northern (southern) hemisphere. These sets are invariant under the flow of XH . __ (b)Let S"" 1 = Sn n {y n + 1 = 0} . Then the restriction XH \ 5 n _ 1 is given by f dHm

+

Xy

\dyd+\ where A = i=l

v

■-,

dHm a

oyid

+Ayd,

dHm

;-ddHy:+^)

(6)

W|fc)"

(c)The flow of XH leaves the subset EP° = {y £ S"" 1 \Hm{yu--yn) = 0} invariant. (d) The boundary of E^ is contained in EP°. (e) If 0 is a regular value of Hm | 5 n _ 1 , then E°°is an (n — 2) —dimensio nal smooth manifold. _ (f) the critical points of XH | 5 n _ 1 are contained in E °°. For a proof of Theorem (1) see Delgado et al 3 . In this paper we are interested in describing the topology of E°° for the case when dimension n = 4 and degree m = 3, corresponding to a cubic Hamiltonian system with 2 degrees of freedom. In this case H3 (x, y, z, w) is a homogeneous cubic polynomial and W = E°° = S 3 n {H3 = 0} is 2 dimensional. In order to get a classification of the topology using the index of the vector field (6 ) defined on E°°, we need the following very well known result. Remark 2 Since W is a compact, orientable surface, then each connected component U ofWis characterized by its genus, or equivalently by its Euler characteristic. In fact, we know \ = 2 — 2g, where x is the Euler characteristic of U and g is its genus. On the other hand, by the Poincare-Hopf Theorem, the characteristic is equal to the index of our vector field in W. 3

General Topological Considerations

First we prove by using results from duality theory some general properties about E°° = Sn n {Hm = 0} . As a corollary we will obtain that each cubic surface has even genus. We write Sn = G _ 1 (1), where G (yi, • • •,y n + i) = y\ + y\ + ■ • • yl+1 and define V+ = {yGRn+1\Hm(y)>0}

,

V. = {y \ Hm (y) < 0} ,

(7)

143

and

w+ = v+nsn, V = {y\Hm(y)=0},

w_ = v_nS",

,.

W = VnSn.

{0)

Let M be a compact orientable manifold without boundary of dimension 2r. Each homology group with coefficients in Z (the set of integers) is of the form H,(M) = L , ( M ) © T , ( A f ) ,

(9)

where Lq is a free group and Tq is a torsion group. The cohomology group of M is denoted by H' (M). From the Poincare Duality Theorem there is an isomorphism T:H r (M) -» H r (M). We define 7 : H r x H r — Z by 7{x,y) = (T(x) ,y). This bilinear form is symmetric or skewsymmetric if r is even or odd respectively, see Dold4. In a natural way T induces a unimodular form T in H r /T r (M) x H r / T r (M) which is defined in the usual way by taking the class representative. Clearly, the unimodular form F(x,y)=x.y,

(10)

is symmetric or skewsymmetric if r is even or odd, respectively. The following form of the Poincare dualitiy theorem is very well known. Theorem 3 Let M be a compact, orientable manifold without boundary of dimension 2r. Then Lq(M)^L2r_q(M):

r,(M)sr 2r - g (M).

U1

>

The symbol = denotes isomorphism between the spaces. Let i: W —> W+ be the canonical inclusion, where A denotes the closure of A, and W, W+ are given by (8). This map defines a homomorphism i„q : Hq (W) —► H9 (W+). The kernel of i*q is denoted by Kq. Unless it is not relevant, we omit the subindex q. Lemma 4 Assume that the degree of Hm is odd. Let ^»q:Hq (W) —* H, (W) be the homomorphism induced by the antipodal mapping *$?. IfO < q + 1 < n, then Hq (W) = Kq® * . , {Kq). Proof: Since m is odd, we have that S ^ W + U * ^ )

and W = W + n 4 ' ( W + ) .

(12)

On the other hand Hi(S !

'

| Z , i€{0,n} [ 0 , otherwise.

(13) v

'

144 We take q such that 0 < q + 1 < n. Since the Mayer-Vietoris sequence

.—fH^1(5B)-H,(W0->H,(i^)eH9(»(wg)-H,(S")-.(14) is exact and H,+i (5 n ) = H g ( 5 n ) = {0}, it follows that

1*

(15)

induces in homology the following commutative diagram K C H, (W) - i * H, ( H ^ ) * . | =*

S| * .

(16)

H,(W)-^H,(*TO). Since AT is the kernel of z. we get that <£, (if) is the kernel of j , . It follows that
(17)

p(tf.(JO)cH,ro©0.

(18)

and

Set (0,z) € {0} © H, (\& (W+)). Since

(y) = (0, z ) . This implies y € K. So, we have proved that
(19)

¥>(*.(*)) = H , ( w g © { 0 } .

(20)

Analogously

Therefore, we have H, (W) = K © * , (A").

■

Proposition 5 Let n=4k+3, with k in NU{0} . It is assumed that T^k (W) 0. Then there exists r such that i/2fc+i (W) = Z 4 r . Proof: Since the dimension of W is 4fc + 2, it follows from Theorem 3 that T 2fc+1 (W) = T2k (W) = 0.

=

(21)

145

Then H2k+i (W) is free of torsion and using Lemma 4 we get that H2fc+i (WO = Z 2 '.

(22)

The antipodal homeomorphism ty induces the bilinear form B(x,y)

= J " ( s , * , y ) = a:.*.y,

(23)

for (x, y) e K x K. Since ^ reverses orientation we get that x.y = —^»x.^»y. Thus x.*«y = - * , x . * , * , y = - * , x . y = y.*»x,

(24)

where the last equality is due to the fact that T is skewsymmetric. So B is a symmetric bilinear form. From the equality H2jt+i (W) = K®$!*K, it follows that there exists a basis a\, ■ ■ ■ Qj, ^ , Q I , • • • \t*aj for which the matrix (

Oti.Ctj

V * . Cti-CXj

,K

0 b\ -bO)'

corresponding to T is unimodular. Therefore B is unimodular; on the other hand B takes even values, see Lopez de Medrano 7 , then we conclude that b has a even number of rows. So, dim K = I is even. Finally H2fc+1 (W) = Z 4 r . This proves our assertion.

(26) ■

Corollary 6 If n-3, then W is not a connected surface of odd genus. Proof: Since n = 3, we have k = 0. Therefore, Hi = Z 4 r . Moreover, Hi is the fundamental group abelianazed, then we get that Hi = I?9. So, we have that 2g = 4r. It follows that g is an even number. ■ 4

Topological Classification in the Cubic Reducible Case

In this Section we study the topological classification of the infinity set E°° in the case when H3 (x, y, z, w) is homogeneous of degree 3, but also reducible. Since any reducible cubic polynomial is factorized into a quadratic times a lin ear hamiltonian, we start by studying the case of homogeneous Hamiltonians H? (x,, y, 2, w) of degree 2. Let H2 (x, y, z, w) = a n x 2 + a223/2 + 0.33Z2 + a^w2 + 2 (ai2xy + a ^ x z + auxw + 0232/2 + a24j/u> + 034.01;) the general expression of a quadratic form in 4 variables. It can be diagonalized by means of an orthogonal transformation taking it to the form

146 #2 (x,y, z,w) — p\x2 + P2J/2 + p3z2 + p4iu2,where pi, i = 1,2,3,4 are the eigenvalues of the matrix associated to H2. Generically they are all different from zero. According to Theorem 1, the corresponding E°° is invariant under the flow defined by the following system of differential equations x = p3z + (pi - p3) x2z 4- (p2 - Pi) xyw, y = PAW + (pi - p3) xyz + {p2 - pi) y2w, z = -p\x + (pi - p3) xz2 + {p2 - Pi) yzw,

. ^

W = ~P2V + (Pi ~ Pi) XZW + (p 2 - Pi) yW2.

Prom Remark 2, we need to compute the index of this vector field at each equilibrium point. This gives the Poincare characteristic, and hence the genus of the compact orientable surface E00. After some algebraic manipulations, the equilibrium points of (27) are obtained as solutions of the system of equations Pzyz - Pixw -pixw + p2yz p3z2 +pix2 PiW2 + p2y2

0, 0, 0, 0.

(28)

In order to solve (28), it is convenient to distiguish the following possibilities, according to the signs of the eigenvalues: I) All eigenvalues have the same sign. Therefore, E°° = ill II) Three eigenvalues have the same sign and one has a different sign, i.e. the signature of H<x is odd. Because of the symmetry and homogeneity of H2, it is enough with ana lyzing the case pi, P2, Pi positive and p3 negative III) There are two positive eigenvalues and two negative ones. It is enough to consider the case p\, p2 positive and p3, p^ negative. Theorem 7 Let XH the vector field induced by a quadratic polynomial hamiltonian. If E°° ^ 0, then generically we have _ a) if H2 has odd signature, the vector field XH has exactly four equilibrium points in E°° with index 1. __ b) if H2 has even signature, the vector field XH has 8 equilibrium points in E°°. Four of them have index 1 and the others have index -1. Proof: Recall that we have to intersect with the sphere S 3 , which is the level 1 hypersurface for the function G (x, y, z, w) = x2 + y2 + z2 + w2. a) This is case II). Then w = y = 0. We are left with just one equation

147 from ( 28) plus the condition for being in the unit sphere p3z2 + pix2 = 0, 2 _,_ ,2 _ ,

(29)

We get 4 equilibrium points (fc,0,fci,0), where k = ±W

p

J , fci

± w ~^* are real numbers, since p3 < 0. Computing the jacobian we get d(H2,G) d(x,z)

2pxx 2p3z = 4 (p! - p 3 ) xz, 2x 2z

(30)

so that y, u; can be taken as independent variables at equilibria. At these points y - p4w + (pi - pz) xyz + (p2 - p*) y2w, w = -p2y + (pi - Pi) xzw + (p2 - PA) yw22,

(31)

where x, z are written as functions of y, w. Hence , the jacobian matrix of the linear approximation of this system at (fci,0, k,0) is '(Pi-P3)M -P-Z

P \ u u ) . (P\-pz)kxk)

(32)

Its trace and determinant are Tr = ±2^/—P~\pl t^ 0 and Ai = p2p4 —pip3 > 0. Therefore, Index = 1 at these points. b) This is case III). Notice from the fourth equation in (28) that w = 0 is equivalent to y = 0 and we have again the 4 points from case a). Similarly, x = 0 is equivalent to z = 0. Then the equations P2y2 + PAW2 = 0, y2a +l «w.2 a=- i1, give 4 new equilibrium points ( 0 , r , 0 , r i ) , where r =

( 33 ) _£4_ ± « / P4-P2

'

ri

=

± w ~ ^ . A computation as in the case of the other 4 points shows that x, z can be taken as independent variables. The jacobian matrix of the linear approximation of the vector field at (0,r,0, r\) is (P2-P4)nr -P2

\ P3 KP2-P4)r^rJ,

with trace Tr — ±2s/—p2p4 ^ 0 and determinant A2 = p\pz — p2Pi = — A i . Now we investigate the index of these 8 points. The trace is always nonzero, but the determinant Ai at the points (fc, 0, A:i, 0) and A2 = —Ai at the points (0,r,0,r\) can have any sign. Excluding the nongeneric case

148

PiPz — P2P4, we see that when Ai > 0, the points (k,0,fci,0) have index 1, while the points (0,r,0,ri) have index - 1 ; and viceversa when Ai < 0. Hence, there are 4 points od index 1 and 4 points of index —1 in any case. Finally, from the first two equations in (28). we see that in order to have an equilibrium point with all nonzero coordinates we need that A i = 0, obtaining a nongeneric case with a closed curve of equilibria. This is a bifur cating case, where we find that the above 8 equilibrium points are contained in the closed curve of equilibria ■ Theorem 8 a) If the signature of the quadratic form H? is odd, then E°° has two connected components, each one homeomorphic to a sphere S 2 . b)If the signature is 2, then E°° is homeomorphic to a torus. c) If fyhas signature 0 or 4, then £°°=0. This means that there are no escape solutions. Proof: a) Each compact connected component of E°° has an even Poinca re characteristic \- The value 4 is imposible, since the genus g is greater than zero and x = 2 — 2g. Hence, there are 2 components with x — 2, i.e. g — 0. So, they are spheres S2. b) By a long but straightforward verification one proves that in this case E°° is connected. Since there are 4 equilibrium points with index 1 and 4 points with index —1, we have x — 0 a n d g = 1. This is topologically a torus Sl x S1. Case c) is trivial. ■ For example, if H2 = x2 +y2 - z2 — w2, under the condition x2 + y2 + z + w2 = 1, we get x2 + y2 = 1/2 and z2 + w2 = 1/2, which defines a torus S1 x S1. This belongs to case b) in the above theorem. On the other hand, if Hi = x2 + y2 + z2 - w2, we can write equivalently x 2 + y2 + z2 — 1/2 and w2 = 1/2, which define two spheres S2 x 5°. This example corresponds to case a) in the above result. Let us consider the case where i/3 (x, y, z, w) is a reducible cubic homo geneous polynomial. Any cubic homogeneous polynomial contains in general 20 coefficients as follows 2

H3 (x, y, z, to) = ax3 + a3x2w cixz 2 eixyz

by3 + cz3 + dw3 + a i i 2 y + a.2X2z+ + bixy2 + b3y2w+ + C2yz2 + czz2w + d\xw2 + d2yw2 + dzzw2+ + e2xyw + e^xzw + e^yzw.

. . * '

If Hz is reducible, the following result describes generically the topology of£°°. Proposition 9 If Hz is a reducible polynomial then EP° is generically a union of three disjoint spheres Proof: If H3 is reducible, we can write H3 = LQ, where L is linear and

149

Q is quadratic. The equation LQ = 0 is equivalent to i) L = 0: a hyperplane. whose intersection with S3 is S 2 , and ii) Q = 0: whose intersection with S 3 is S2 x S°. This gives 3 spheres for E°°. The case where Q = 0 gives a torus S 1 x S 1 is not possible generically. Indeed, if E°° is the union of a torus and a sphere, being simultaneously smooth, we would have H\ {E°°) = I? since it is the abelianized of the fun damental group. This contradicts Proposition 5 , since the range of H\ (E°°) must be a multiple of 4. ■ The example L = 0, Q = 0, given by the equations w = 0; x2 + y2 + z2 — w = 0, when intersected with S3 produces 3 two-spheres 2

w = 0,x2 + y2 + z2 = l,

,

w2 = lx2 + y2+z2

(6b

=l

>

illustrating the proposition. On the other hand, let L = 0: ax+by+cz+dw = 0 any hyperplane and Q = 0: x2 + y2 — z2 — w2 — 0. Since L = 0 contains the 2-dimensional subspace defined by ax + by = 0 and cz + dw = 0, we see that it must have points in common with the torus x2 + y2 = ^, z2 + w2 = \ obtained by intersection of Q = 0 with S 3 . Hence, the torus has always points in common with the sphere defined by any hyperplane through the origin. 5

Topological Classification in the Cubic Irreducible Case

The case H3 irreducible is considerably harder. For analyzing this case we will take the normal form proposed by Sylvester ( Segre8 and Arnold et al 1 ) H3 (x, y, z, w) = x3 + y 3 + z3 + wz + (ax + by + cz + dw)3 .

(37)

This normal form produces only irreducible cubic polynomials. Application of linear isomorphisms of R4 generate new irreducible polynomials. Any such endomorphisms can be identified to a 4x4 real matrix, requiring thus 16 entries. With the 4 coefficients of the normal form, we get the 20 coefficients that the general form (35) has. We expect that the normal form (37) generates a big open subspace of the 20 dimensional space of coefficients of the cubics. So our topological classification of surfaces E°° = {H3 = 0} n S3 will depend on 4 parameters. Notice that E°° is a smooth surface if and only if VH3 (p) is not parallel to p, for any p € E°°. By homogeneity of H3, this is equivalent to the condition V//3(p)^0, for any p G E°°. Indeed, assume that V # 3 (po) = Ap0

150

for some po € E°°. But Euler formula for homogeneous functions gives P0.VH3 (po) = 3H3 (po) = 0- Prom the above formula, we get A||pol|2 = Po.ViJ 3 (Po)=0.

(38)

Hence A = 0 and VH3 (po) = 0, as required. Let A (a, 6, c, d) = 0 be the equation which defines the set of parame ters (a, 6, c, d) € R4 such that there is a point x 6 E°° (a, 6, c, d) satisfying Vi/3 (x) = 0. We have that V//3 (x) = 0 if and only if the following equalities hold a (ax + by 4- cz + dw) =—x2, b(ax + by + cz + dw)2 = -y2, c(ax + by + cz + dw) = -z , d (ax + by + cz + dw) = — w2.

.

.

This means that the parameters a, 6, c, d are non-positive, i.e. (a, 6, c, d) be longs to (R~) with R _ = (—oo,0]. By extracting square roots in equations (39), we obtain a homogeneous linear system in x,y,z,w. Only that ± 1 fac tors have to be introduced in each of the 4 equations. We verify at once that the condition that the determinant of the system annihilates, can be writ ten as the set of zeroes of the following polynomials in the new non-negative variables A = ( - a ) 3 / 2 , B = ( - 6 ) 3 / 2 , C = ( - c ) 3 / 2 , D = {-d)3/2 . (A + e j S + ^ C + eaD) 2 = 1,

(40)

with e< G {-1,1}. The change of variables F : ( R - ) 4 -» ( R + ) 4 is the homeomorphism given by F (a, 6, c, d) = ( ( - a ) 3 / 2 , ( - 6 ) 3 / 2 , ( - c ) 3 / 2 , (-df2)

= (A, B, C, D),

(41)

transforming {A = 0} C (R~) into the subset p of ( R + ) , defined where the equations (40) are satisfied. Here R + = [0,00). The complement of p induces a partition of the parameter space R4 into open connected components. In a companion paper Falconi, Lacomba and Llibre 6 , we have proved that all surfaces corresponding to parameters in the same component are diffeomorphically equivalent. Consequently our classification problem has been reduced to determine all connected components and for each component, to identify the topological surface W.

151

5.1

Counting the Connected Components

To determine all the connected components of the partition induced by p , we use a combinatorial method, which we briefly describe below. We consider the system A + B + C + D = a0,A-B-C-D = a'0, A-B + C + D = ai,A + B-C-D = a\, A + B-C + D = a2,A-B + C-D=:o-'r A + B + C - D

= O-3,A-B-C

. . '

[

+ D =
The above hyperplanes (40) are defined by taking the a'S equal to ± 1. Each 4-tuple (A,B,C,D) £ R4 determines an 8-tuple (<J"O,<7I,<72,<73,CTO> a1,a2,o-3). Let J_ = (—oo, —1), Jo = ( - l i l ) ) J+ — (I) 0 0 )- We say that two 8-tuples (xi,...,xs) and (t/i,...,ys) of R 8 with Xi , y» ^ ± 1 , are equivalent it we have Xi £ J^ if and only if j/i 6 J^ for each i — 1,..., 8 and U S {—, 0, + } . Proposition 10 Two points (A,B,C,D) and (A\,Bi,Ci,Di) belong to the same connected component of (R + ) \ p if and only if their corresponding 8-tuples are equivalent. Proposition 10 is proved in Falconi, Lacomba and Llibre 6 . It gives a com binatorial procedure to count connected components, but we have to identify equivalent components modulo permutation of coordinates. This is due to the fact that the equations which define W are invariant under a permutation of coordinates (see Remark 11). We proved that two points (A,B,C,D) and (A\,B\,C\,D\) belong to the same connected component of (R + ) \ p if and only if their correspond ing 8-tuples are equivalent. That is, each component is identified by a word of 8 letters in the alphabet {-, 0, + } . In principle we have 3 8 words, but cer tain constraints allow us to reduce the number of possibilities. The following inequalities are easy to prove, using equations (42) °0 ^ °'l)^2,0'3iffii o 2) o 3 ^ °0,

(43)

cr0 > 0.

(44)

The sum of c?i and a\ is equal to 2A, for all i. Therefore we get 0 < (o-o + o-'0)/2 = (a, + a[)/2 = (a2 + a'2)/2 = (a 3 + a'3)/2.

(45)

By comparison of the equations which define a we see that the inequalities Ci >

hold for i = l,2,3, and k ^ i.

ff'k,

152

Let n = (CTO + 0o) /2- The entries of a have the following properties. (i) Since fi is non-negative then UQ G JO o r °~o €E J + . (ii) If (To € Jo then each entry of a belongs to Jo- This follows from (43) and \i > 0. (iii) From (46) we obtain that the entries of a given 8-tuple satisfy either the inequalities O'Q

< cr'i < a'j < a'k < fi < ak < o-j < ai <

CTO,

(46)

or ff0 <

CT

» ^

CT

j ^

CT

* ^ M < ^fc < ^

< cr< < °o-

(47)

As long as we are not interested in the value of the entries of any octect, but in their sign we will identify an octect with its corresponding array of signs (see Proposition 10); i.e., if the i-th entry of a belongs to Jk, we replace it by k in a, where k = - , 0 or +. Although the following remark is obvious, it is useful in order to identify all the cubic surfaces. Remark 11 Notice that the topological type of the cubic does not depend on the ordering of the 4-tuple (A,B,C,D). Hence, the octect generated by a per mutation of (A, B, C, D) corresponds to a component in the parameter space where the cubic has the same topological type as the component corresponding to (A, B, C, D). So, applying permutations to the parameters (A,B,C,D) in equations (42), we prove the following result Lemma 12 Suppose we have u,v,s,t 6 {+,0,—}. Then the octects of the form 1) (CTO,U,u,v,a' 0 ,s,s,t), 2) (ao,v,u,u,o-'Q,t,s,s) and 3) (CTO,U,v,u,o-' 0 ,s,t,s) are equivalent. Also the octects of the form 4) {o-0,u,u,v,a'0,t,s,s), 5) (o-0,v,u,u,a'0,s,t,s), 6) (o-0,u,v,u,o-'Q,t,s,s), 7) (o-0,u,v,u,cr'0,s,s,t), 8) (o-0,u,u,v,a'0,s,t,s) and 9) (a0,v,u,u,a'0, s,s,t) are equivalent. An 8-tuple <7 is trivial if it is equivalent to an octect generated by a set of parameters (A, B, C, D) which has al least one entry equal to 0. All these 4-tuples correspond to the connected component containing the (0,0,0,0) in R4\PBefore stating the conditions which imply that an octect a is a trivial one, we introduce some notation to simplify. We say that an octet
153 It is not hard to prove that an 8-tuple a is trivial if any of the following 7 groups of conditions is satisfied i)a'0cr3i ^ O ^

,a[a2,o'2ai,

,(ToO'l ,a'sa2,o'2(T3,

"K^i ,OQ02 ,a[(73,o' ai,

iii)a0a'2 iv)a'0(-,

v)a3a0,(T2(TU

vi)a2{-aQ), vii)o\o0,

3

$-(Tx),a2(-

-0-2), tf3("-0-3),

(48)

cr 1 CT 3 ,CT3(7 0 ,

(--°'s){-°$,°Wai(-- ^ 3 ) , &3<72, cr'2(J3.

Because of condition (44), we have that o"o = 0 or <JQ = +, for all a. To count all the possible octects, we will first consider the case OQ = 0, and then the case ao = +• To identify equivalent octects, we apply the set of constraints (43)-( 47), the conditions for trivial octects given by (48) and the Lemma 12. This way we get 20 equivalence classes of octects. It is important to remark that two non equivalent classes can repre sent surfaces of the same topological type. Let (A, B, C, D) be a point in a component represented by an octect of the form (si,S2, • • •,Sg) with s< in {+,0, —} , and P be a permutation in the set {A, B,C, D} . When we sub stitute the set (P(A), P(B), P{C), P{D)) into the Equations (42), an octect (s'j, s'2,- - • i s'g) with sj in {+, 0, - } is obtained. Notice that since (A, B, C, D) and (P(A), P(B), P(C), P(D)) determine surfaces of the same topological type, then we can consider that (s\, s2, ■ ■ ■, Sg,) and (s[, s 2 , • • -, s'&) are equiv alent. Using this method for the 20 octects identified above, they can be put together into 10 classes, as shown in the following table 0 1 2 3 4 5 6 7 8 9

[0,0,0,0,0,0,0,0], {+,+,+,+,0,0,+,+], [ + , + , + , + , - , - - - ] , [+,0,0,0,0,0,0,0], [+,0,0,+,-,0,0,0 , [+,+,+,0,0,0,0,0], [+,+,+,0,-0,0,0], [+,+,+,+,0,0,0,0 ) [+,+,+,0,-,0,0,+],[+,+,+,0,-,-,0,0 , [+,+,+,+,0,0,0,+], [+,+,+,0,-,-,-,+], [+,+,+,+,0,+,+,+], [+,+,+,+,-,0,0,0], [+,+,+,+,-,0,0,+], [+,+,+,+,-,-,0,0], [+,+,+,+,-,-,0,+] , +,+,+,+ -,-,-,0],

[+,+,+,+,-,+,+,+], [+,+,+,+,-,-,-,+]•

so, there are at most 10 different topological types.

(49)

154 5.2

Poincari Index Theorem and the topology of E°°

In this subsection we prove that E°° corresponding to regions 0, 1 and 6, which are found along the diagonal of the parameter space, is topologically a sphere, three disjoint spheres and a connected surface of genus 6, respectively. We begin by region 0. P r o p o s i t i o n 13 / / H3 = x3 + y3 + z3 + w3, then E°° = (H3 = 0) D S3 is a sphere S 2 . Proof: Indeed, the equations to be satisfied are x3 + y3 + z3 +w3 = 0, x2 + y2 + z2 + w2 = 1.

(50)

We will prove that to any point (x, y, z, 0) in the equator of 5 3 C R 4 , i.e. such that x2 + y2 + z2 = 1, there corresponds exactly one point of E°°, and conversely. In fact, this point generates the vector (x, y, z, — f/x3 + y3 + z3 1 satisfying Hz = 0. By homogeneity, the whole positive ray through that point belongs to {Hz = 0} . Intersection with S3 provides a unique point (x, y, z, -y/x3

+ y3 + z3) L - 1 1*1 2 2 2 3 3 3 2/3 (x + y + z + (x + y + z ) ) V

(51)

in E°°. The converse is clear. 3

■ 3

3

3

P r o p o s i t i o n 14 If H3 = x + y + z + w - {x+ y + z+ wf , then E°° = (Hz = 0) n S 3 is a connected surface Proof: We define the sets D+ = {{x,y, z, 1) | Hz(x,y,z,l) = 0} and D~ = {(x, y, z, — 1) | Hz (x, y, z, — 1) = 0} as the intersections of (Hz = 0) with the hyperplanes (w = 1) and (w = —1), respectively. The radial projec tion n + (x) : R 4 \ { 0 } -> S3 is defined by 11+ = x / ||x||, while we define II~ = —II + . Notice that II + projects D+ onto the north hemisphere (w > 0) while II~ (D+) is its projection onto the south hemisphere (w < 0). One can verify that

E°° = n + (D+)urir (D+). If we let 6 = y + z, c = y — z, we obtain D+ = (l + b)x2 + (l + b)2x + b(l + b) + ^b(b2-c2)

=0.

(52)

Since (1,0,0,0) G E°°, we can see that n + (D+) nUr (£>+) ^ 0. So to prove that E°° is connected it is enough to prove that n + (D+) is connected and this is true if D+ is connected. This fact is proved below.

155

Figure 2. Figure 5.1: The open regions A > 0.

If b 7^ — 1, we solve Equation (52) as

- (1 + 6)2 ± ^/(l + 6)4 - 46 (1 + 6)2- -( l + 6)6(6 2 -c 2 ) 2 (1 + b)

(53)

It gives real roots whenever the discriminant A = (1 + b) ((1 + 6) 3 - 46(1 + b) - b (62 - c 2 ))

(54)

is non negative. We need to locate the curves where A = 0 and then the regions where A > 0 in the plane 6-c. We first notice that A ^ 0 when 6 = 0. We can write A = 6(1 + 6) (c2 -{b)) for 6 ^ 0 , where 4>{b) = b + 1 — £. This function is always increasing and is undefined at 6 = 0 where it goes to ±oo. It is non negative in the intervals [—1/2 — \/5/2,0) and [—1/2 + N / 5 / 2 , OO) . Hence, the locus A = 0 consists of the line 6 = —1 and the two branches of the curve c? = is null, see Figure 5.1. The lower branch is asymptotic to the c axis on the 6 < 0 side.

156

By looking at the three factors 6,1+6 and c 2 - ^ (6) in the above expression for A, we verify that there are two open regions where A > 0, which are shaded in the Figure 5.1. These shaded regions are the projection of the set D+ along the x-axis.Then D+ is the union of three subsets A, B, C defined by A= Ux,y,z)\xeR,l + b = 0,b2 = c2}, B= \{x,y,z) | l + 6 > 0 , x = - ( l + 6 ) / 2 ± V S / 2 ( l + 6 ) } , C = {(x,y,z) | 1 + 6 < 0,s = - (1 + 6) /2 ± V S / 2 ( 1 + b)} .

(55)

We see that B projects on the upper shaded region, C projects on the lower shaded region, while A projects into the 2 points where they touch each other. For each point in the subset A > 0 there are exactly 2 points in D+ projecting into that one. Those 2 points are joined together as we approach the subset A = 0. There is an exception at the line b = — 1 because of the singularity in the expression for x. As we approach it from the shaded regions the solutions satisfy x —> ±oo. So, we conclude that D+ is connected. ■ According to Theorem 1 and considering the normal form (37) for if3, the surface E°° is invariant under the flow of the vector field given by the system of ordinary differential equations x = Z(z2 + ce2) + xX y = 3(w2 + de2) + yX z = 3 ( - x 2 - ae2) + zX w = 3 ( - j / 2 - be2) +wX

= Fj (x), = F2 (x), = F3 (x), = F4 (x),

, , ^°>

where e = ax + by + cz + dw and A = 3z (x 2 + ae 2 ) — 3x (z 2 + ce 2 ) + 3w (j/ 2 + 6e2) — 3y (w2 + de2), and x = (x, y, z, w). The equilibrium points of the vector field (56), are the solutions of the equations 3(z 2 + ce 2 ) + xA = 3(w2 + de2) + yX = - 3 ( x 2 + a e 2 ) + *A = - 3 ( y 2 + fee2) + wX =

0, 0, 0, 0,

.

. ^'>

with the extra condition that they must belong to the sphere S 3 . Lemma 15 The number of equilibria and total index of System (56) are as follows a) 6 equilibrium points with the global index 2 if all the parameters are null b) 10 equilibrium points with global index 6 if the parameters are -(1/2,1/2,1/2,1/2) c) 26 equilibrium points with global index -10 if the parameters are -(1,1,1,1).

157

We summarize the equilibrium points as follows. We first list the equilibrium points for the parameters a = b = c = d = — 1. They are separated in groups, acoording to the number of null coordinates. There are 4 points with 2 null coordinates and index 1 Pl:x

= 0 = z,y = ±,w=-^;P2 P 3 : y = 0 = w,x=±,z=-±;Pi

= -Pu = -P3.

^>

There are 16 points with one null coordinate and index - 1 . Instead of listing all of them, we give the list corresponding to the points with x — 0 or y = 0. The remaining points are obtained using the symmetries x <—► z, y <—* w and x <—> —x, which are made more precise below. P5:x

= 0,y = - ( ^ y \ z

P7:x

= 0,y = - ( ^ )

P9:x = - ( ^ ) Pu:x

1 / 2

=~ { ^ y

/

1 /

\ z

,y^0, \ y

={ ^ ) ( ^ y \ =e - ^ ) ( ^ y

2

/

\

W W

= -x, U , = ( l ^ ) ( ^ )

= 0,z = -x,W

=

= -l = -y-,

(59)

1 / 2

( ^ ) ( l ^ y

/ 2

The points Pg, Ps, Pio and P\i are the negatives of P5, P7, Pg and Pg, respectively. There are 6 points with no null coordinate. The first two points have index - 1 and the remaining four points have index 1 P21 : x = - 5 , y = - \ , z=\,w= i P22 : x = i , y = | , z = - i , w = - \ Pn:x

= -l/2(^)1/2;z

P25: x = - l / 2 ( ^ )

= l/2(^)1/*;x 1 / 2

;

2

=l/2(^)

= y,z = w 1

' ; x = y, z = w

P26 = - P 2 5

I f a = 6 = c = d = 0, then the equilibrium points are:

° i : (°'75' 0 '-^)' <>2 = -Ou 03:( ^ , 0 , - ^ , 0 ) , 04 = - 0 3 , O5 = (31 2' ~ 2 ' ""2)' ^

6

5

~ ~® '

The points O j , 0 2 , 0 3 and O4 have index 1, while 0 5 , 0 6 have index - 1 .

(61)

158

If a = b = c — d = —1/2, we get Qi = ( 0 , 1 / A 0 , - l / > / 2 ) ; Q 2 = - Q i , Q3 = (l/v/2, 0, - 1 / >/§, 0 ) ; Q 4 = - Q 3 > Q5= Q7

=

^

,

^

(

( f i ^ ! ,

^

,

ffi^!,

^

)

;

fi#l,

Q 9 = (1/2,1/2, - 1 / 2 , - 1 / 2 ) ; Q 1 0 -

Q

fi#!)

6

= -Q5,

(62)

, Q8 = _ Q 7 ,

-Q9.

All these points but Qg and Qio whose index is - 1 , have index 1. Notice that there two symmetries for the vector field (56). One of them is the symmetry <Si sending x <—► -x. The other one S2 ■ x <—► z, y *—► w, t 1—» —t is valid only when the parameters satisfy a = c and b = d. These two symmetries explain why the equilibrium points of the vector field appear in groups of 2 and 4 points. To compute the index for each equilibrium point, we give a chart of E°° about that point, which is equivalent to a choice of two independent variables, so that (56) becomes a system of two equations in the plane. Then one has to compute the eigenvalues of the linear approximation of the resultant planar system. We remark that g) a 3 j.( is a homogeneous function of degree 3, since it is a determinant where all the terms of the first row are homogeneous of degree 2, while the terms of the second row are linear. Let x*, xs denote the other 2 complementary variables. If $\x3'xl / O w e can take x^ and xs as independent variables, so that we can solve x< = Xi (x*, xa) and Xj = Xj (x*, x3). Computation of the partial derivatives J^-, | ^ , g^i and JEi show that they are quotient of homogeneous polynomials of degree three, and therefore they are even functions. On the other hand, the right hand sides Fi of system (56) are polynomials containing terms of degrees two and four. Hence, they are even functions. Then their partial derivatives of the first order are odd functions. In a chart about an equilibrium point where x^, xa can be taken as inde pendent variables, system (56) has the form ±k = Fk(xk,xs,Xi(xk,x3) X, = F, (xk,Xa,Xi

,Xj(xk,xa))

= $ f c (x f c ,x s ),

(Xfc,X,) ,Xj {xk,Xa))

= $a (Xfc,X s ) .

,g3.

The entries of the matrix of its linear approximation are clearly odd functions, since they have the form

^^dFk dxa dxa

+

dFkdx1 dxi dx,

+

dFkdx1 dxj dxa'

159

etc. The index of the equilibrium point is obtained from the sign of the determinant _ d$k d $ 3 dxk dxs

d$k d$3

dx, dxk '

(65)

which is an even function. When we apply symmetry <Si to an equilibrium point, we obtain a new equilibrium point with the same index, since V is invariant under S\. Let us apply symmetry 52 to one of the equilibrium points, from the cases that we are considering. By straightforward verification we see that V is invariant as well, so that, the new equilibrium point has the same index. Hence, for the component with 26 equilibrium points, is was enough to analyze only 9 of them. We conclude that the topology in the three components can be described as follows. T h e o r e m 16 Let H3 (x, y, z, w) = x 3 + y3 + z3 + w3 + (ax + by + cz + dw)3 a homogeneous cubic polynomial and W=(H$ — 0) n S3. Then a) W is diffeomorphic to a sphere if a=b=c=d=0. b) W is diffeomorphic to the disjoint union of three spheres if a=b=c=d= -1/2. c) W is a connected surface of genus 6 if a=b=c=d=-l. Proof: According to propositions 13 and 14, we have connected surfaces in cases a) and c). In case a) we have a sphere because the global index is 2, so that the genus is zero. In case c) the global index is -10, hence the genus is 6 as asserted. Finally, in case b) the global index is 6 and recalling that the genus must to be even (see Corollary 6), the only possibility is that the surface has three connected components of index two, i,e. spheres. ■

References 1. V. Arnold, A. Varchencko and S. Goussein-Zade\ Singularites des appli cations differentiates, MIR, (1986). 2. A. Cima and J. Llibre, Bounded polynomial vector fields, Trans. Amer. Math. Soc. 318, 557-579, (1990). 3. J. Delgado, E. Lacomba, J. Llibre and E. Perez, Poincare compactification of Hamiltonian polynomial vector field, in Hamiltonian Dynamical Systems, History and Applications The IMA vol. in Math, and its appli cations 63 , 99-115, Springer-Verlag, (1994). 4. A. Dold, Lectures on algebraic topology, Springer-Verlag, (1972).

160

5. M.Falconi, Estudio asintdtico de escapes en sistemas hamiltonianos polinomiales. Ph.D. Dissertation, UNAM, (1996). 6. M. Falconi, E.A. Lacomba and J. Llibre, Topological Classification of Real Cubic Surfaces in the 3-Dimensional Sphere. To appear. 7. S. Lopez de Medrano, Involutions on manifolds Springer-Verlag, (1971). 8. B. Segre, The non-singular cubic surfaces, Oxford Press, (1942).

RELATIVISTIC CORRECTIONS TO ELEMENTARY GALILEAN D Y N A M I C S A N D DEFORMATIONS OF POISSON BRACKETS RUBEN FLORES-ESPINOZA Departamento de Matemdticas, Universidad de Sonora, Rosales y Blvd. Luis Encinas, Hermosillo, Sonora 83000, Mexico. E-mail: [email protected] YU M. VOROBJEV Department of Applied Mathematics, Moscow Institute of Electronics and Mathematics, B. Vuzovsky per. 3/12 Moscow 109028, Russia. E-mail: [email protected] Deformations of Poisson brackets generated by the contraction of the Poincar6 algebra into the Galilei algebra are studied in the framework of the Poisson cohomology theory. We show that relativistic deformations of the Galilean Poisson structure are trivial, in particular, the cohomology class of the relativistic 2-cocycle is zero. This fact allows us to interprete an elementary relativistic Hamiltonian dynamical system as a Hamiltonian system on a homogeneous symplectic space of the Galilei group with a corrected Hamiltonian.

1

Introduction

The elementary relativistic (nonrelativistic) mechanics as the theory of dy namical systems on the homogeneous symplectic spaces of the Poincare group (the Galilei group) was developed in the fundamental work of Souriau 1 . Vari ous applications of this approach can be found, for example, in2-3>4>5>6. We are interested in relativistic effects coming from the contraction of the Poincare algebra into the Galilei algebra from the viewpoint of Poisson geometry. The Poincare group is generated by the Lorentz transformations and translations of the Minkowskii space. Its Lie algebra g is the semidirect sum of the Lie algebra of the Lorentz group and the 4-dimensional abelian algebra, j = o(3,l)jR4. So, the Poincare algebra g can be denned as a 10-dimensional vector space V consisting of all vectors of the form X = (A,v,x,t), where A € o(3), v,x, G R 3 , t € R, and the Lie bracket is given as 3,5 [Xi,X 2 ] = nAi,A 2 ] +vi ®v 2 -v2®vi,

Aiv2 -

A2v\,

Aix2 - A2xi + t2v\ - t\v2, vi ■ x2 - v2 • xA.

161

162

For each e > 0, define the vector space automorphism U€ : V —> V by (A, v, x, t) - - (J4, £ V V e 1 / 2 x, t).

(1)

Since C/£ is a vector space isomorphsim, we can defined a "new" Lie bracket on V as \X1,X2}e

U£-1[Ue(X1),Ue(X2)}.

=

The bracket [, ] e continuously depends on e including e = 0. For £ > 0, the Lie algebra ge = (V, [, }e) is isomorphic to the Poincare algebra, whereas for £ = 0, go is the Galilei algebra. This procedure is just the contraction of the Poincare^ algebra into the Galilei algebra 5 . Physically, the parameter e is equal to c - 2 , where c is the speed of light, and £ - > 0 corresponds to the nonrelativistic limit. Let V* be the dual of V. For each e > 0, consider the Lie-Poisson space g* = (V*,{, }£) equipped with the standard linear Poisson bracket {, }£ : C°°(V*) x C°°(V) — C°°(V) associated with the Lie algebra gE. We can modify the family of linear Poisson brackets { , } £ into a 2-parameter family of affine Poisson brackets { , }£<m on V in the following way. It is well known 1,3,5 that the antisymmteric 2-form on V \{Xl,X2)=vl-x2-v2-xl

(XUX2GV)

(2)

is a 2-cocycle of the Lie algebra g£ for each e > 0. Moreover, the second cohomology space H2(go) of the Galilei algebra go is one-dimensional and is generated by the cohomology class [A] of A. Thus the general second co homology class in H2(go) can be written as m[\], where, from the physical viewpoint, the parameter m is identified with the "mass" of the Galilei group (for details, see 3,5 ). For e > 0 (i.e., for the Poincare group), [A] = 0 in H2(ge). Now taking into account the general fact 5,8,10 : any 2-cocycle of a Lie algebra generates an affine Poisson bracket which is a deformation of the linear Poisson bracket, we obtain the family of affine Poisson brackets { , }CyTn on V* associated with the 2-cocycle m[A]. Using coordinates (A,x,v,t) on V in (1), we can identify the dual space V* with the Eclidean space R 1 0 = {£ = (J, K, P,H)}, where J,K,P G R 3 and H G R. In terms of these dynamical variables, the affine Poisson brackets { , } £ ] m are given by the relations 4 : {H,Ji)t,m

= {H,Pi)£<m

= 0,

=

\Ji,Jjfe,m GjjA: Jfe, {Ji, Pj}e,m = 6»jfc Pk, {#i, Pj}c,m = (eH + m)«y,

where Gij* is the Levi-Civita tensor.

{H, Ki}c<m = —Pi, \JiiK-i)e,m = €ijk "fc> {-^i, Kj}£<m = —£ G^fc Jfc, {Pi, Pj}c,m = 0,

(3)

163

Our goal is to investigate the family of Poisson brackets context of the deformation theory and Poisson cohomology7'8 applications 11 ' 12 ' 13 . For a fixed m > 0, we iterpret the Poisson brackets {, , } £ , m mation of the bracket {, , }o,m, where the parameter e plays the deformation parameter. In invariant terms, the Poisson bracket represented via the Poisson tensor r £ , m = r + mA + eTl.

(3) in the and their as a defor role of the (3) can be (4)

Here the bivector field T on V* represents the "limiting" bracket {, , }o,o> A corresponds to the cocycle A in (2), and the bivector field TZ defines the relativistic deformation of the Poisson structure ro, m . We have the following coordinate representations for 72. and A:

n

= -\^k]kwiAw-l + m>>wiAwl'

A=

<5ii »j

d Wi

d 8P/

(5) (6)

A

Since the brackets (3) satisfy the Jacobi identity for all £, the bivector field 72. satisfiew, the linearized Jacobi identity7 relative to ro,m- In the lan guage of the Poisson cohomology, this means that 71 is a 2-cocycle of the Poisson manifold (V'*,ro, m ). For convenience, the Poisson structure To,m and the bivector field 71 will be called the Galilean Poisson structure and the relativistic 2-cocycle, respectively. Our main result is that the Poisson cohomology class of the relativistic 2cocylce 71 is zero, or in other words, the infinitesimal relativistic deformation of the Galilean structure r 0 , m is trivial. For the case m = 0, this fact is not evident a priori by the following reasons: the Poisson structures T and r £ i o are not isomorphic and the second Poisson cohomology space of the Galilean structure T is not trivial (see §2). This result is applied to the investigation of elementary relativistic (nonrelativistic) dynamics1, that is, a dynamical system on a connected symplectic manifold, where the Poincare group (the Galilei group) acts transitively by symplectomorphisms. We consider the following family of Hamiltonian systems on V* associated with affine Poisson brackets (3):

j = JxVjf K = -$Lp

+ Kx VKf + Px + KxVjf-eJx

VPf, VJC/ - (eH + m)V P /,

(7)

164

P = PxV,f H = P-

+ (£H +

m)VKf,

VKf,

where the symbol x denotes the cross-product in R 3 and / = f£(0 G C°°(V*) is the Hamiltonian function smoothly depending on the parameters e. For e = 0, equations (7) represent a collection of the elementary Galilean (nonrelativistic) systems: the trajectories of (7) lie on the homogeneous symplectic spaces (symplectic leaves) O of the Galilean group. If m = 0, then each submanifold O C V* is a coadjoint orbit of the Galilei group. Under m > 0, O is an orbit of the affine representation of the Galilei group on V*. For e > 0, equations (7) describe an elementary relativistic dynamics whose spaces of motions coincide with the coadjoint orbits of the Poincare group. Following the general thesis: (deformtations of Poisson brackets) =>• (corrections to Hamiltonian functions) and, using the resutls 7 ' 8 ' 11 , in §2 we construct an infinitesimal transformation which transforms the relativistic system (7) into a nonrelativistic Hamiltonian system, up to 0 ( e 2 ) , with a corrected Hamiltonian. The generator of this infinitesimal transformation is determined by the primitive of the relativistic 2-cocycle TZ, for which we give an explicit formula. Moreover, in §3, for m > 0, from the Sudarshan-Makunda transforma tion 4 we deduce an isomorphism between relativistic dynamics and the ele mentary Galilean dynamics defined in sufficiently large domains of V*. This fact may be useful for a Hamiltonian description of a classical motion of a relativistic spinning particle in an external electromagnetic field ] . A Hamil tonian approach for systems of such kind based on the twistor technique was suggestied in 6 . 2 2.1

Deformations of the Galilean Poisson structure Basic notions

Here we recall some necessary facts from the Poisson geometry. A more ex tensive information can be found in 7 ' 8 ' 9,10 . Let M be a C°°-manifold. A skew-symmetric contravariant 2-tensor field (a bivector field) $ on M is called a Poisson tensor if the formula {f,g} =
(8)

defines a Lie bracket on the space C°°(M) of smooth funcitons on M. In this case, the bracket (8) is said to be a Poisson bracket. The pair (M, ty) is called a Poisson manifold urith the Poisson structure ^ .

165 Let (M, SP) be a Poisson manifold. The Hamiltonian vector field of / € C°°(M) is vj = -\P(d/). Dynamical systems generated by Hamiltonian vector fields on M are called Hamiltonian systems on the Poisson manifold. The symplectic leaf of a Poisson structure ^ through a point m € M is the set Cm of all points m! e M such that m and m' can be joinded by a piecewise smooth curve each segment of which is a trajectory of a Hamiltonian vector field. So, £ m is a connected submanifold of dimension d i m £ m = r a n k ^ ( m ) . Each symplectic leaf carries the symplectic structure u; with the property: w(w/,» 9 ) = {/,
I*,*1=0,

(9)

where [, ] : Vk(M) x V m (M) -» Vk+m~1(M) is the Schouten bracket7**10. This operation is a natural extension of the usual commutator between vector fields on M. In particular, for any f,g € C°°(M), [*, fj = vf, [[ [*, /],$]] = —{f,g}, and the Schouten bracket between # and a vector field w — wxd/d^ on M is given by the formula [*,wY> = * " 5 t t ^ - V'diW* -

w'ds*ij,

where tf = $<J'ft A d,- and ft = 3 / 9 ^ . It follows from (9) that a Poisson tensor ^ induces the coboundary op erator Z? = [*,•] : Vfc(M) -> V fc+1 (M), £>2 = 0, which is called the Lichnerowicz-Poisson cochain of the Poisson manifold (M, ty). The space of fc-cocycles Z f c (M,*) consists of all A e Vfc(M) such that DA = 0. A cocycle .A is a coboudary if DB = A for a certain i? € V f c _ 1 (M). The space of coboundaries is denoted by Bk(M, \t). Then W*(Af, * ) = Zk(M, *)/Bk(M,

*)

are called the Poisson cohomology spaces of (M, \I>). We have W°(M, ¥ ) = {Casimir functions}. A vector field z on M is called a Poisson vector field of its flow preserves the

166 Poisson bracket, that is, if Jtf,*! = 0. So, Hl(M,V) = {Poisson vector fields/Hamiltonian vector fields}. A bivector $ € V 2 (M) is called an infinitesimal deformstion of a Poisson structure \P if the bracket {f,9h

= {f,g} + et>(df,dg)

(eeflU])

2

satisfies the Jacobi identity up to 0{e ). An infinitesimal deformation $ is said to be trivial if there is an infinitesimal transformation £ >—> £+ew + 0(e2) generated by a vector field w on M, which transforms {, } into {, } £ up to 0(£2). L e m m a 1 A bivector field $ on M is an infinitesimal deformation of # if and only i / $ is a 2-cocycle, $ € Z2(M, # ) , and the triviality of$ is equivalent to the condition: $ is a coboundary, $ € B2{M,^), whose primitive is just the generator w of the above infinitesimal transformation. So, we have H2(M, ty) = {infinitesimal deformations/trivial infinitesimal deformations}. A submanifold N of (M, \P) is called a Poisson submanifold if any Hamiltonian vector field on M is tangent to N. In this case, the Poisson tensor on N is given by the restriction of ^ on N. We say that a Poisson submanifold N in M is regular if N is a union of regular symplectic leaves of (M, $ ) . A mapping 7 : M —» M between two Poisson manifolds (M,\P) and (M, * ) is called a Poisson mapping if ^*{^!{df,dg)) = ^{^'df^'dg) for all f,g € C°°(M). If / is also a diffeomotphism, then / is said to be a Poisson automorphism or equivalence. Two Poisson manifolds (M, ^ ) and (M, \P) are globally equivalent if there is^a Poisson automorphism between some regular Poisson submanifolds N C M and N C M. Note that if the topological struc ture of symplectic leaves of \& and # is different, then this is an obstruction to the global equivalence. A smooth deformation of a Poisson structure f on M is an e-family of Poisson tensors Hf£ on M smoothly depending on a parameter e G [0,1] and such that \I>J „ = * . In this case, $ = 4sH „ is an infinitesimal deformation of \P. 2.2

Symplectic leaves ofT£<m

Consider the Poisson tensor r £ ) m (4) of the affine Poisson bracket (3). There are two independent Casimiar functons of r e > m smoothly depending on pa rameters e > 0 and m > 0: CiJJ, = -p2

+ eH2 + 2mH,

(10)

167 Table 1. Topological structure of regular symplectic leaves

values of parameters

symplectic leaves

e > 0, m > 0; a > - m 2 / e , 0 > 0

e = 0, m > 0; a € R, 0>O

£ > 0, m = 0 ; a < 0 ,

0>O

e = 0, m = 0; a < 0, 0>O

Ci% = {(eH + m)J~Kx

topological type

o±

T*R 3 x S 2

Oo,m

T*R 3 x S 2

e,0

R 5 x S2 x S1

00,0

R 5 x 50(3)

u

P)2 - e(J • F ) 2 .

(11)

2

Here the terms such as P and J ■ P mean the norm and the inner product of vectors in the 3-dimensional Eucledian space. Funcitons (10) and (11) are related to the physical invariants of the Poincare group by the formulas Q,m = (M 2 — m2)/e and Ci,Jn = M2s2, where e~2M2 is the mass squared and s2 is the spin squared (see 1,2,5 ). The 8-dimensional regular symplectic leaves of the Poincare structuer Tem are given by the connected components of the regular level sets of the Casimir functions (10) and (11):

0 £ , m (a, 0) = {Ci'X = a, Ci% = 0>O}cV'^

R 10 .

(12)

For e — 0, submanifolds (12) are connected. For e > 0, each level set (12) has two connected components Ofm = Ofm(a,0): Otm = Oc,m n {sign {eH + m) = ± } .

(13)

The topological structure of the leaves depends on the values of parame ters e and m. We give the corresponding classification in Table 1. For 77i > 0, the topological type of the symplectic leaves does not change as e —♦ 0. Moreover, each Ofm or £>o,m is isomnorphic, as a symplectic manifold, to the direct product T*R 3 x S 2 , where T*R 3 is equipped with the canonical cotangent symplectic structure, and the symplectic structure on S 2 is defined by the area 2-form (see 2 ' 4 ' 5 ). The limit 0 —* 0 corresponds to

168

the singular points of the Poisson structure ro, m - If e > 0, then the singular points of r e i 7 n correspond to the levels: a = -m2/e (the massless case) or (3 = 0 (the spinless case). For m = 0, there is another picture: the symplectic leaves 0£ 0 and Oofi are not diffeomorphic. In particular, this implies that there is no global equivalence between the Poisson structures re>o and F. The singular points of T£io and T correspond to the critical values of the Casimir functions: a = 0 and (3 = 0. 2.3

Relativistic 2-cocycle

(i) Case m = 0. Let (V*, T) be the Poisson manifold, where T is the Galilean Poisson tensor in (4). Consider the regular Poisson submanifold (NQ, T) of V*, where N0 is the union of the 8-dimensional regular symplectic leaves (12),

N0=

(J

0o,o (a,/?).

(14)

aO

First, note that the vector fields

z

* = K-m+p-w>

Z2 = (Px(Px

K)) ■ A

(15) (16)

are linear independent Poisson vector fields of (iV~o,r) transversal to symplec tic leaves C?o,o- So, 7,13 n\N0,T)

w C°°(R 2 ) x C°°(R 2 ).

(17)

Using this fact and the general results on computing Poisson cohomology 7 ' 8 ' 10 ' 12 , we prove that the Poisson cohomology space Ti2(No,T) is not trivial, W 2 (iVo,r)wC 0 0 (R 2 ).

(18)

Theorem 1 The cohomology class of the relativistic 2-cocycle TZ (5) in H2(N0,r) is trivial: TZ is a coboundary, IT,W\=1Z, where the primitive W is the following vector field on N0: 2P2\

dK

DP)

(19)

169

P2\ +

P2

+iLf£^pnyPx{PxK)).».

p

P2{IfxP)2(-

V-K)-r(P-K)(P.J))(KxP).^.

(20)

Proof. We are seeking a solution of equation (19) in the form: W = a\Z\ + a2Z2 + b(K x P) • -J^, where Z\ and Z2 from (15) and (16), a\,a2,b G C°°(NQ) are some functions. Taking into account the relations

pi,p2} = ir,H2j, pi, (K x P)2fl = [r, 2HJ -(KxP)

+ (J. P)%

and evaluating equation (19) on the Casimir functions of T, we obtain a\ = H2/2P2 and a2 = -^{^ - 2\JKSy be found by direct computations.

- "(K^pp)-

Finally, the function b can

Corollary 1 The relativistic cocycle TZ defines a trivial infinitesimal defor mation of the Galilean Poisson structure T. The most general primitive of TZ has the form w = W + k\Z\ + k2Z2 + (Hamiltonian vector

field),

(21)

where k\ and k2 are arbitrary Casimiar functions oJT. Note that the primitive w (21) of T cannot be smoothly extended to the whole space V*. Indeed, if a point £ = (J, K, P, H) 6 NQ is moving to the set of singular points So = {K x P = 0}, then the vector field W becomes singular. The same effect takes place in several other examples 11 ' 12,13 . (ii) Case m > 0. Let To.m = T 4- mA be of the modified Galilean Poisson structure on V* defined in (4), (6). First, we observe that Z = 0/dH

(22)

is a Poisson vector field of Toim transversal to the symplectic leaves C?o,mProposition 1 For any m > 0, W1(^*,r0,m)«Coo(R2)

and n2(V*,T0,m)

Moreover, the cohomology class of 11 in H2(V*,ro,m) 11, where

« C°°(R 2 ). is trivial, [To,™, Wl =

V.-H«+ZK»+itPx(J+LPxK)).» m on

m

oK

2m V

p., V

m

//

oK

170

This fact can be derived from the resutls of §3. It is interesting to note that, by comparing with (20), formula (23) gives the primitive of r 0 ) m , which is well defined in the whole space V. But if m - » 0 , then W has no limit. So, adding the "mass" factor A to the Galilean structure T leads to qualitative changes in the Poisson structure. 2.4

Infinitesimal relativistic corrections

Now we are going to apply Theorem 1 for the investigation of the family of dynamical systems (7) as e —► 0 and m = 0. For a given function / = / £ (£) on V*, we consider system (7) as a perturbation of the limiting Hamiltonian system for e = 0. In general, the corresponding perturbation vector field is not Hamiltonian with respect to the Galilean Poisson structure T (see also 7,11 ). Denote by (V*, r £ ) o, / ) the dynamical system corresponding to system (7) with e > 0 and m = 0 and by (NQ,T, f) the dynamical system corresponding to (7) with e = 0 and m = 0, where the domain NQ is defined in (14). Proposition 2 Let fe € C°°(V*) be a function smoothly depending on e € [0,1]. Then the infinitesimal mapping £~Z

+ ew(t) + 0(e2)

(ZeN0),

(24)

where the vector field w is determined by (21), transforms the nonrelativistic Hamiltonian system (No, I\ fo+£f^), up to 0(e2), into the relativistic system (V, r ffi o, f c ) , where the "relativistic correction" to the Hamiltonian /o is given by the formula

/"> = Wo + f

(25) £=0

The proof of this fact follows from Corollary 1 and Lemma 1. The same fact is valid in the case m > 0. But, as we will see in §3, in this case there is a global equivalence between relativistic and nonrelativistic dynamics. 3

Equivalence between elementary relativistic and nonrelativistic Hamiltonian dynamics. Case m > 0

Let us consider the e-parameter family of affine Poisson brackets (3) for a fixed m > 0. Let (V*,ro,m) be the Poisson manifold associated with the "limiting" bracket (3) with e = 0, and (V*,TSim) be the Poisson manifold associated with the "deformed" bracket (3) with e > 0.

171

For any e > 0 and m > 0, using symplectic leaves (12) and (13), we introduce • the regular Poisson submanifolds G m and G £ , m of (V, To.m) as Gm=

U

O0,m(a,p)cV%

(26)

Q6R,/3>0

G£,m =

(J

O0,m(a,/9)cGm;

(27)

a>-m2/e,/3>0

• the regular Poisson submanifolds Vfm ^ m =

U

^

m

and V~m of (V*,r £ i m ) as

(a^),

P + m n P ~ m = 0.

(28)

a>-m2/e,/3>0

Note that G m , G £ , m , and Vfm are open 10-dimesional submanifolds in V*. In the limit as e —► 0 the submanifolds G £ , m and P ^ coincide with Gm. However, each point of Vfm goes to infinity as e —» 0. ^From the viewpoint of the dynamical system (7), these submanifolds have the following physi cal interpretation. The domain Gm (or P^m) can be viewed as a union of the symplectic phase spaces of nonrelativistic (or relativistic) systems corre sponding to various nonzero values of the spin (that is, for /3 > 0). For small e > 0, the invariant domains G£iTn and V+m approximate Gm and V~ is not essential. Now let us consider the map 7 £ : (J,K,P,H) i-» (J', K', P', H') defined by the formula H' = - ( - m + yj2m.eE + m 2 ), J' = J, K,

=

(29)

V2meH + m*K m

+

eP x (J - ±K x P) V-eP2 + 2meH + m2 + \f2meH + m2'

P ' = P. It is easy to see that 7 £ gives a C°°-map from G £ i m onto V+m. Theorem 2 For any e > 0 and m > 0, formula (29) defines a diffeomorphism le : G £ , m - V+m such that 70 = idGm and ^€ is a Poisson automorphism between the Poisson manifolds (G £ , m ,r 0 | m) and (P£ m ,r c>TO )>

(7e)»ro,m = r £>m . The transformation 7 £ symplectomorphically maps each symplectic leaf Oo,m(ot,(3) onto the leaf 0 £ > m (a, \/a2 + m2(3/m).

172

Formula (29) can be derived from the Sudarshan-Mukunda realization 4 . On the other hand, (29) is verified by direct computations. Note that as e —» 0 the infinitesimal part of the transformation (29) is just given by the vector field W in (23). Now Theorem 2 allows us to establish the following relation between the relativistic and nonrelativistic Hamiltonian dynamics associated to sys tem (7). For a given f£ e C°°(V*) smoothly depending on e S [0, oo), by (P+ ffl , T£ym, f£) we denote the restriction of the Hamiltonian system (7) with e > 0 to the invariant Poisson submanifold P + m C V*. The restriction of system (7) with e = 0 to the invariant Poisson submanifold G £ , m is denoted by (G e ,m,ro lT n,.F £ ), where Ft is a certain Hamiltonian function. Proposition 3 For every e > 0 and m > 0 the relativistic Hamiltonian sys tem ( P + ^ r ^ n n / e ) is isomorphic to the nonrelativistic Hamiltonian system (G£,m^o,rmFc), where the "corrected" Hamiltonian function Fe is given by

F£ = -y€of£ = f0 + e(Lwf0 + ^

)+0(e2).

(30)

Here 7 £ is defined by (29) and W is the vector field in (23). So, as £ —♦ 0 a nonrelativistic dynamical system can be interpreted as the Hamiltonian perturbation (30) of the nonrelativistic system ( G m , r o , m , / o ) This makes it possible to apply the pertubation theory of Hamiltonian sys tems, like the KAM theory, to the investigation of relativistic dynamical effects (see also 11 ). Acknowledgments The authors wish to thank R. de la Llave and H. Cendra for helpful conver sations and suggestions concerning this work. One of the authors (Yu. M.) was partially supported by the Consejo Nacional de Ciencia y Tecnologia de Mexico through the Catedra Patrimonial de Excelencia II, No. 940036. References 1. J.-M. Souriau, Structure of dynamical systems. A symplectic view of Physics, Progress in Mathematics, Vol. 149, Birkhauser, Boston, (1997); Structure des systemes dynamiques, Dunod, Paris, (1970). 2. N.M. J. Woodhouse, Geometric Quantization, Second Edition, Clarendon Press, Oxford, (1992).

173

3. V. Guillemin and S. Sternberg, Geometric Asymptotics, Mathematical Surveys, Vol. 14, Amer. Math. Soc, Providence, RI, (1977). 4. E.C.G. Sudarshan and N. Mukunda, Classical Dynamics: A modern per spective, R.E. Krieger Publishing Co., Malabar, Florida, (1983). 5. V. Guillemin and S. Sternberg, Symplectic Thecniques in Physics, Cambdrige Univ. Press., Cambdrige, (1984). 6. A. Bette, J. Math. Phys., 22 (6), 2158-2163, (1992). 7. Yu. M. Vorobjev and M. V. Karasev, Functional Anal. Appl., 22, 1-9 (1988). 8. V.P. Maslov and M.V. Karasev, Nonlinear Poisson Brackets. Geometry and Quantization, Translations of Mathematical Monographs, V. 110, Amer. Math. Soc, Providence, RI, (1993). 9. A. Weinstein, J. Differential Geometry, 18, 523-557, (1983). 10. I. Vaismann, Lectures on the geometry of Poisson manifolds, Progress in Mathematics, V. 118, Birkhauser, Berlin, (1994). 11. R. Flores-Espinoza and Yu.M. Vorobjev, in New Trends for Hamiltonian Systems and Celestial Mechanics, Eds. E. Lacomba and J. Llibre, World Scientific, London, 357-374, (1995). 12. R. Flores-Espinoza, Yu.M. Vorobjev, M.V. Karasev, in Quantization, Co herent States and Poisson structures, Eds. A. Strasburger et al., Polish Scientific Publishers PWN, Warsaw, 233-240, (1998). 13. R. Flores-Espinoza, Yu.M. Vorobjev, Russian Mathematical Surveys, 49, 6, 223-224, (1994).

HETEROCLINIC P H E N O M E N A IN T H E SITNIKOV PROBLEM ANTONIO GARCIA AND ERNESTO PEREZ-CHAVELA Departamento de Matemdticas, UAM-Iztapalapa, Apdo. Postal 55-5S4 Mexico DF. 09340, Mixico. E-mail: [email protected] and [email protected] We give the deduction of a Melnikov function for the Sitnikov problem. Using a perturbation method introduced by Melnikov and a thorough analysis of the geometry of certain auxiliary functions that we introduce, we prove analytically the existence of transverse heteroclinic orbits. As a consequence, we can embed a Bernoulli shift near these orbits. This shows that the Sitnikov problem possesses chaotic dynamics.

1

Introduction

In 1929, J. Chazy 2 predicted the existence of oscillatory motions in the threebody problem, that is, orbits whose behavior are unpredictable for long time intervals. K. A. Sitnikov11 was the first to answer positively this question giving a particular model. This kind of motions was studied widely by V. Alekseev1. However his work is very difficult to read. In 1973, J. K. Moser8 retook the Sitnikov model and was able to put the ideas of V. Alekseev in a more tractable way. The Sitnikov's model consists of two equal masses describing elliptic or bits in a fixed plane denoted by II; and a third massless particle moving on the orthogonal line to the plane through the center of mass of the former masses. J. K. Moser studied the Poincare map denned assigning the position of the primaries and the velocity of the third particle when it crosses II to the corresponding ones in the next crossing of the third particle with II. Taking the eccentricity of the ellipses as a parameter e, J. K. Moser showed that the problem is integrable for e = 0; and that for small e > 0 it is possible to embed a Bernoulli shift near the parabolic orbit of the massless particle, for the Poincare map associated to the flow. The basic fact used to prove this result is the existence of transversal homoclinic points. In this paper we prefer to use the time 27r-map as a Poincare map, and write the system modelling the Sitnikov problem as a perturbed Hamiltonian system, with the eccentricity e as perturbation parameter. For e = 0, the system is integrable and the level curves of the unperturbed Hamiltonian become the orbits of solutions. These curves are divided in two families, the

174

175 first one consists of bounded curves which correspond to the periodic orbits, and the second one consists of unbounded curves. Their mutual boundary consists of two special orbits, called the parabolic orbits, which escape to plus infinity and minus infinity with zero limiting velocity. Using the ideas of McGehee6, we prove that for e > 0, the points corre sponding to escape to ±00 can be seen as degenerate critical points for the flow. In order to clarify the panorama, and to distinguish the escapes above the plane with the escapes below, we have done our analysis in three different coordinate charts, one containing the origin in the phase space and the other two containing +00 and —00. In this framework we prove that the parabolic orbits still exist for e > 0 small enough, and that they act as heteroclinic orbits joining the two critical points at ±00. In fact McGehee proved, using a suitable change of coordinates, that the degenerate critical points are trans formed into periodic orbits, and that the stable and unstable sets of these periodic orbits are real analytic submanifolds. The aim of this paper is to prove, using the Melnikov techniques that the above analytic submanifolds intersect transversally along the heteroclinic or bits for e > 0 small enough. A similar result has been obtained by Dankowicz and Holmes3, however our approach is different. We never use elliptic func tions in order to compute the Melnikov function. But above all, the geometry of some auxiliary functions that we introduce and some basic computations allows us to prove analytically that the zeros of the Melnikov function are sim ple; this is our main result. We also give a detailed analysis of the deduction of the Melnikov function. 2

The Sitnikov Problem

In this paper we reanalize the classical problem studied by K. A. Sitnikov11 in 1960. We consider two non zero equal masses mi and TT12 moving around each other on ellipses of eccentricity e in a plane II, and a massless particle 7713 moving on the axis E perpendicular to the plane II passing by the center of mass. As in Moser8, we normalize the time so that the period T of the primaries is 27r and the gravitational constant G = 1. Then from the Kepler's third law, n2a2 = G(rn\ + m?) where n = 2it/T = 1. Hence if a = 1 then we must have mi = mi = 1/2. Now since the center of mass is at the origin, thus when the eccentricity e = 0, the primaries are moving in circles of radius 1/2 so that its mutual distance is one. With these hypotheses the motion of the first two particles is not afected by the motion of the massless particle which always remains on the axis E (see figure 1).

176

Figure 1. The Sitnikov Problem

Let x be the coordinate on the axis E of the massless particule, its motion satisfies the equation X

X

=

|*3

(2.1)

+ r2 , 3 / 2 '

where r (t) is the distance of each of the primaries to the center of mass located at the origin. Assuming that at time t = 0 the primaries are in the pericenter, we have r(t) = r(—t). The expansion of r(t) with respect to the eccentricity is given by (2.2)

r(t) = - (1 + ecost + h.o.t.) Taking x = y, the equation (2.1) becomes

GW-T

(2.3) 0

y ■ ( -

X

.

(.'+J)''

)

■

3x cos (

:-+»••» )

+

0 xB(x,i i.oy' («"

177

where B(x, t, e) is smooth and 27r-periodic with respect to t. In addition, (2.3) is a Hamiltonian system with Hamiltonian function given by H{x,y,t)

-y2-

=

1

(2.4)

2

(x + r 2 ) 1 / 2

= HQ(x,y) + eHi(x,y,t)

+ h.o.t.,

where H0{x,y) = -y2 Hi(x,
1

-

2

(* + i ) 1 / 2 >

cost 4(x ,

(2.5b)

1)3/2-

+

(2.5a)

Of course, if e — 0 the system (2.3) is integrable and therefore we may obtain explicitly the global flow. There are two possibilities for 7713 to escape: by above, when x —► +00; and by below, when x —> —00. To study the flow near these regions, we follow McGehee6 and Moser8 and make one of the following changes of coordinates: 1 X =

r

V = -P,

when x > 0,

y = p,

when x < 0.

1 X =

9 >

(2.6)

The equation (2.3) is transformed in: ■

q

1 3

~ 2q

P

'

n P

^

|l + rV! 3 / 2 '

(2.7)

The equation (2.7) can be extended to the points q = 0, these points can be thought as points at plus or minus infinity, each one associated with a different limit velocity. Once this extension is done, the phase space is covered with three different coordinate charts. In the first one containing the origin, we study the bounded motions governed by the equation (2.1). In the other two charts corresponding to the escapes to +oo and - c o respectively, the dynamics are given by the equation (2.7). We remark that the line x = 0 and the line y = 0 (or p = 0) are axes of symmetry; moreover, they are orthogonal to the flow except at the equilibrium solution. In (2.7), the point q = 0, p = 0 is a degenerate fixed point due to the factor q3 and it is contained in the line of fixed points q = 0.

178

Figure 2. Flow for e = 0 in the three local charts.

3

The case t = 0

In this case, H = H0 and the system (2.3) is integrable. The global flow is given by the level curves of (2.5a). See figure 2. There is one orbit that leaves +oo with zero velocity and reaches — oo with zero velocity and one orbit that does the opposite. Both are in the level set Ho = 0; we call them the heteroclinic orbits of the unperturbed problem, they are marked in bold lines in the figure (2). In order to give a parametrization of these orbits, we start by defining the following functions

/(*) =

V

J>

,

g(x) = J /(«) du.

(3.1)

H. Dankowicz and P. Holmes3 worked with these functions and obtained analytic expresions of the heteroclinic orbits using elliptic integrals. We prefer to use their geometric properties. The function g(x) is smooth, increasing and odd. It is convex if x > 0, and concave if x < 0. Let 0o(£) be the inverse of g(x), and MV = TtM*)- Then ^°W

=

77X77v>'

. " S 1 tf>o(0 = ±°o.

, l i m ^0(0 = 0.

/ (00 W )

t-»±oo

t->±oo

179 These equations show that the heteroclinic orbits are parametrized by

r 0 (t) = (*>(*), ifoW) and rs(t) = (-to(0,-iM*))-

(3-2)

We are going to deal with the first orbit, To(t). The analysis of the other orbit is similar. The functions 4>o(t) and ij>o(t) are analytic since they are the components of a solution of the analytic differential equation (2.3). 4

Melnikov function for the heteroclinic orbits

When e 7^ 0, the degenerate fixed point in the equation (2.7) corresponding to the escape to plus infinity with zero limit velocity is transformed into a 27r-periodic orbit 71. The parabolic orbits approach this periodic orbit when t —► +00. Let l{-) be the flow of (2.7), we use as a Poincare map the function cp^-ni') defined on a cross section z = ZQ. We denote it by Pt. Hence 71 becomes a fixed point of Pt, called +00. By the reversibility of the flow we also have a 27r-periodic orbit 72, corresponding to the escape by below, that becomes a fixed point, denoted by —00, of the Poincare map. These periodic orbits have as their asymptotic set the family of parabolic orbits. McGehee6 proved that the parabolic orbits form analytically inmersed submanifolds of the energy surface, and called them the stable and unstable manifolds of 71 and 72 respectively, denoted by W' (±00) and W" (±00). In a small neighborhood of the origin, we can apply KAM Theory to get periodic orbits for e ^ 0, these bounded orbits are called elliptic orbits of the perturbed problem. By adjusting the velocity of the massless particle, it is possible to get escape orbits (x —+ ±00) where the velocity of escape is positive (£(±00) > 0); these orbits are called hyperbolic orbits of the perturbed prob lem. This implies the existence of at least one parabolic orbit with x(0) = 0; that is, an orbit with x(0) = 0, limt_oo x(t) = 00 and lim ( _ 0 0 x(t) = 0. Using again the reversibility of the flow we get a heteroclinic parabolic orbit joining 71 with 72, we denote it by T e . This shows that W' (+00) and W" (—00) intersect along the parabolic orbit, the goal of this section is to show that this intersection is transversal along Fc using the Melnikov method. By Dankowicz and Holmes3, for a fixed to 6 R the orbits in the perturbed stable manifold can be approximated uniformly for all t > to by

* M,to) - {rAt)) - {Mt

+ to))

where the functions $i(i, to) and ^\(t,to) tions of the variational equation of (2,1).

+ e ^i{tto))

+ h.o.t.,

(4.1)

are smooth, in fact they are solu

180

Since to acts as a parameter, it is clear from our previous analysis that by varying to or ZQ = To (to) we get the same effect . So from here on, we will identify to with its image under T. Then for each z 0 6 To, let E*° be a cross section at ZQ in the phase space, and let zu(e,zo) and zs(e,zo) be the points in W" (—00) and W° (+00) respectively, that lie in the intersection with E Zo . In order to measure the splitting of W" (-00) and W° (+00) near the point zo = To(io) we define the function M* («, z0) = Ho (za(e, z0)) - H0 (z"(e, z0)),

(4.2)

for zo € To and 0 < e. We remark that using the equation (4.1) and appro priate E 2 °, the functions z"(e, zo) and zu(e,z0), M* are smooth. Using Taylor's expansion of the above function at the point ZQ, with re spect to the parameter t we get M* (e, zo) = cM(zo) + h. o. t., where the last function is the Melnikov function defined by u M(z0) = ±Ho(z°(e,z0))-^Ho(z (e,zo))

(4.3) t=o

Our next objective is to compute M{zo). First, define z3(e,t,to) as the solution of the equation (2.3), such that at time t = 0 it is at za(e, zo)- Thus z s (0,t,t o ) = r ( t + t o ), and pi

j t (z*(e, t, t0)) = JVH0 (za(e, t, t 0 )) + eJVtfi (za{e, t, t 0 )) + h.o.t. The following computation uses the previous equation and the fact that {Ho, Ho} = 0, where {Ho, Ho} = VH0JVH0 is the Poisson bracket and J = (_?i 0) is the canonical symplectic matrix.

H0(z°(e,A,t0))

- ^H0(z'(e,zo))

=J

-£-HQ(z°(e,t,t0))dt

= JQ §-e (vffo (zs(e, t, to)) ^ {z\c, t, to))) dt =J

§; (eVtfo (za(e, t, t0)) (JVff, (z'(e, t, to))) + h.o.t.) dt

= 1

^(t{H0,H1}{za(e,t,to))

= f Jo

+ h.o.t.) dt

{Ho,Hl}{za(e,t,to))+er1(e,t,to)dt,

181 where r)(e,t,to) is a smooth function. Therefore at e = 0 we get (4.4)

= -^(a,«o))

-lH0(z'(e,zo)) £=0

£=0

{#o,ffi} {T{t +10)) dt.

/ + Jto

Taking the limits when A —> oo we get --H0(z°(e,z0))

= -

lim

A->OO

e=0

at

(4.5)

—H0(z'(€,A,to)) e=0

/»O0

+ /

{ff 0 ,ffi}(r(t + to))
Ahead we shall prove the existence of the former limit. Using the same arguments, we also get (4.6)

-H0(z"(e,z0)) £=0

/

J-oo

{H0,Hl}(T{t

+ tQ))dt+

lim

A-+-00 d e

^-H0(zs(t,A,t0)) £=0

Let us observe that, in the local chart containing +oo ^H0(z'(e,A,t0))

VH0(z°(e,A,t0))^(za(t,A,t0)).

=

(4.7)

Then by (4.1), J^(z'(e, A, to)) is bounded in a compact neighborhood of +oo, and linu^+oo VH0(zs(e, A, to)) = 0. Therefore rA

M(z0)=

lim

/

A-+ooJ_A -A +oo

/

{ff 0 ,ffi}(r°(t + io))dt

{H0,H1}(r0(t + t0))dt.

■OO

Remember the identification of to with T(io) = -Zo a n d let us observe that M(ZQ) does not depend on the particular choice of the functions za(e, ZQ) and zu(c, zo), hence the dependence of M* (e, zo) on the particular choice of those functions occurs only in the terms of bigger order in e. With the previous analysis we have proved the following result.

182

Theorem 4.1. The Melnikov function M(-) for the Sitnikov problem is given by r+oo

{ff0,ffi}(r°(t + to)))dt ■oo

r+

-£ 5

°° 3<j>o{t + to)Mt

+ to) cos{t) dt.

(4.8)

Transversality for e > 0

In this section we study a perturbation method developed by Melnikov7 in 1963, in order to prove the existence of transverse heteroclinic orbits. As a consequence, using the ideas of J. K. Moser 8 , we can embed a Bernoulli shift near the parabolic orbits in the Sitnikov problem, concluding that the system possesses chaotic dynamics. We start stating the Melnikov theorem in this case. Theorem 5.1 (Melnikov). Let ZQ = r°(io) o. point on the heteroclinic orbit for the unperturbed system (3.2) such that M{z0,to)

=0

and

dtQ

* 0.

(5.1)

(20,to)

Then W* (+00) and W? (—00) intersect transversally at (zo,to) on the cross section E 2 0 . Now we have to check that the Melnikov function given in the theorem (4.1) satisfies the hypothesis of the theorem (5.1) for to = kit, k £ Z. First, we need to simplify the form of M(ZQ) given by the equation (4.8). Lemma 5.2. The Melnikov function is given by: M(z0) = Asm(t0),

3 r A^irmM^^du. 2 Jo

where

Proof. Let u =■ t + to in (4.8), thus the Melnikov function becomes:

M(zo)=J

+

°° 3<j>o(u)ip0{u) cos(u - to) du

*(<%(«) +\)S/2

3 ,, , [+°° A>(uM(u)co8(ti) =-cos(to) j ,^IS , l v , / 9 du 00 (*§(«)+ * ) 6 ' 2 +00

3 -i«w* )^ /f+°° + T sin(t i ( *0 o ) /

, . „Mu)Mu)s\n{u) , . . ,^r,5/a du. n

4 - ^ y - c o "wg(«) + D

(5.2)

183 The first integral is zero because the integrand is an odd function, and since the function in the second integral is even we get the result. □ L e m m a 5.3. The integral /0°° ^fflff+1 °.°[u) du is positive. Proof. We integrate by parts, using that rpo(t) = -ji4>o(t), [°° 4>o(u)ip0(u)sin(u)

oo

sin(i)

(5.3) H o(u) + \)V* The first term on the right hand sideJois zero. In order to compute the second term we study the function:

Jo

5 2

dU

($(«) +l) '

~

+

f

1

F(u) =

3 ($(«) +

(5.4)

l\3/24)

By the Implicit Function Theorem and the definition of F'(u) =

-v/20o(u)

($(«) +i)"'

4

and

1

F"(u) =

4>Q(U)

we have

- (Wo(«) -

"($(«) + i) 4

-!)•

Using that <j>o(u) is increasing and that its limit when t —► oo is inifinity we have that F(u) is even, decreasing if u > 0, lim„_oo F(u) = 0 and it has a unique inflection point on the interval [0, +oo) at uo = ff("7ff )• That is, F(u) is bell shaped. Two consequences of theses facts are that if 0 < u < UQ then F(u) is above the line L\{u) joining the points (0,F(0)) and {UQ,F[UQ))\ and if 7r/2 < u < 37r/2 then F(u) is below the line L,2{u) joining the points (■7r/2,ir(7r/2)) with (37r/2,F(37r/2)). Using Mathematica we compute the following values, where we use four significant digits: u0 = 0.1199, <j>Q{uQ) = 0.2357, F(u0) = 1.974, <j>0(0) = 0, F(0) = 2.667, c£o(7r/2) = 2.079, F ( J T / 2 ) = 0.341, 0O(3TT/2) = 4.524, F(37r/2) = 3.536 x 10" 3 . Using this values we obtain: Li(u) = -5.779u + 2.667

L2(u) = -9.724 x 10 _ 3 u + 4.936 x 10" 2 (5.5)

Let A(u) = F(u)cos(u), thus the second term in the equation (5.3) can be written as /•OO

/•7T/2

/•37T/2

/•OO

/ A(u) du = / A(u) du -1- / A(u)du4- / A(u)du4■ / \{u)du. Jo JZ-IT/2 Jo Juo J-x/2

184

We denote the above integrals as A\, A2, A3 and A4 respectively. First, /•5w/2

A4 — I

,.7w/2

\{u) du +

J3n/2

/-9w/2

X(u) du+

/.llir/2

X(u) du+

J5x/2

J7n/2

X(u) du+ ■ ■■ . J9TT/2

Using the fact that F(u) is decreasing, and the symmetries of the cosine function, it is easy to check that this series is alternating; the terms of the series are decreasing, converging to zero and the first one is positive. These facts imply that the series converges to a positive number, proving that A4 is positive. Clearly A2 is positive. Finally, using the relation between L\(u) and L2{u) with F{u) we have: /•«o

Ay. = / Jo

/"«o

F(u) cos(u) du >

L1{u)cos{u)du

= 0.2776

(5.6)

Jo 3TT/2

\As\

j F(u)cos(u)du T/2 Jn/2

< /

L2(u) |cos(u)| du

■K/2 J-K/2

/•3TT/2 ron/1

< - / K/2

L2(u)cos{u)du

= 0.03762

(5.7)

+ A3 + A4> 0.2776 + A2 - 0.03762 + A4 > 0.

(5.8)

Summarizing, we have that Ai+A2

□ Using the above lemmas we get our main result. Theorem 5.4. Let to = kn, where k 6 Z. Thus the Melnikov function M(to) of the Sitnikov problem satisfies M(to) = 0 and dt' "' ^ 0. Moreover, these are the only points with the above properties. The fact that W* (+00) and Wu (—00) intersect transversally give us im portant consequences. The results of Smale and Birkhoff for homoclinic points as given in J. K. Moser8 and restated by Dankowicz and Holmes 3 , for degener ate fixed points allows us to guarantee the existence of a hyperbolic invariant set where the Poincare" map is topologically equivalent to a Bernoulli shift. That is, in a neighborhood of the heteroclinic points we can introduce symbolic dynamics and with this, we assure the existence of infinitely many periodic orbits with large period and infinitely many oscillatory orbits. Recalling that an orbit is said to be oscillatory if limsupx(t) = 00 and liminf x(t) < 00 as t —♦ ±00 (See Z. Xia 12 ). This also shows that the Sitnikov problem is very sensitive with respect to initial conditions and therefore chaotic and nonintegrable.

185

Acknowledgments The authors are partially supported by by Conacyt-Mexico, projects 4002005-1406 PE and 400200-1 Exp. 980108. References 1. V. Alekseev, Quasirandom dynamical systems I, II, III, Math. USSR-Sb. 5, 73-128 (1968); 6, 505-560, (1968); 7, 1-43, (1969). 2. J. Chazy, Sur l'allure finale du mouvement dans le probleme des trois corps I, II, III, Ann. Sci. Ecole Norm. Sup. 3 e Ser. 39, (1922); J. Math. Pures Appl. 8, (1929); Bull. Astron. 8, (1932). 3. H. Dankowicz and P. Holmes, The existence of transversal homoclinic points in the Sitnikov problem, J. Diff. Equations 116, 468-483, (1995). 4. A. Garcia and E. Perez-Chavela, Melnikov techniques in the Sitnikov problem. To appear. 5. J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Sys tems, and Bifurcations of Vector Fields, Springer-Verlag, New York, (1983). 6. R. McGehee, A stable manifold theorem for degenerate fixed points with applications to celestial mechanics, J. Diff. Equations 14, 70-88, (1973). 7. V. K. Melnikov, On the stability of the center for time periodic pertur bation, Trans. Moscow. Math. Soc. 12, 1-57, (1963). 8. J. K. Moser, Stable And Random Motion In Dynamical Systems, Annals Math. Studies 77, Princeton University Press, (1973). 9. T. J. Morrisey, A degenerate Hartman theorem, Israel Journal of Math ematics 95, 157-167, (1996). 10. C. Robinson, Dynamical systems; stability, symbolic dynamics and chaos, CRC Press, (1994). 11. K. A. Sitnikov, Existence of oscillatory motions in the three-body prob lem, Dokl. Akad. Nauk. USSR 133, 303-306, (1960). 12. Z. Xia, Melnikov method and transversal homoclinic orbits in the re stricted three body problem, J. Diff. Equations 96, 170-184, (1992).

D O U B L Y - S Y M M E T R I C P E R I O D I C SOLUTIONS OF HILL'S L U N A R P R O B L E M R. CLARISSA HOWISON 619 Washington Blvd. Baltimore, MD 21230, USA. E-mail: [email protected] KENNETH R. MEYER Department of Mathematical Sciences, University of Cincinnati Cincinnati, Ohio 45221-0025, USA E-mail: [email protected] The existence of a new family of periodic solutions to the spatial Hill's lunar problem is established. These solutions have large inclinations and are symmetric with respect to two coordinate planes. In this family the infinitesimal particle is very close to the primary.

1

Introduction

This paper establishes the existence of a new family of periodic solutions to the spatial Hill's lunar problem. The periodic solutions of this family have large inclinations and are symmetric with respect to two coordinate planes — hence the name doubly-symmetric periodic solutions. In this family the infinitesimal particle (the moon) is very close to the primary (the earth). These periodic solutions are perturbations of circular solutions of the Kepler problem. By the Kepler problem we mean the spatial central force problem with the inverse square law of attraction. A related paper of the authors 7 established the existence of two new families of periodic solutions to the spatial restricted three-body problem by Poincare's continuation method. These families exist for all values of the mass ratio parameter \i and have large inclinations. In one of the family the infinitesimal particle is far from the primaries and in the other case the infinitesimal is very close to a primary. In this note we will indicate that the latter family exists in Hill's lunar problem also. This is reasonable since Hill's lunar problem is a limit of restricted problem developed to study the motion of the moon 5 . The small parameter e will be introduced as a scale parameter in such a way that s small means the infinitesimal is close to the primary. The per turbation problem is very degenerate. First of all, even to the second ap proximation the characteristic multipliers are all + 1 , and second, the periodic solutions that we establish are undefined when e = 0. These difficulties are

186

187 overcome by exploiting the symmetries of the problem and using the implicit function theorem of Arenstorfx. In 1965, Jeffreys8 showed that there exist doubly symmetric, periodic so lutions to the three dimensional restricted three-body problem. His method of the proof depends heavily on a symmetry argument, together with a standard perturbation method applied to the mass ratio n of the restricted problem There is no natural parameter in Hill's lunar problem corresponding to (J. and so a scaling parameter is introduced. The problem becomes considerable more difficult. 2

Hill's lunar p r o b l e m

One of Hill's major contributions to celestial mechanics was his reformula tion of the main problem of lunar theory: he gave a new definition for the equations of the first approximation for the motion of the moon 5 . Since his equations of the first approximation contained more terms than the older first approximations, the perturbations were smaller and he was able to obtain series representations for the position of the moon that converge more rapidly than the previously obtained series. Indeed, for many years lunar ephemerides were computed from the series developed by Brown, who used the main prob lem as defined by Hill. Even today, most of the searchers for more accurate series solutions for the motion of the moon use Hill's definition of the main problem 4 . Before Hill, the main problem consisted of two Kepler problems — one describing the motion of the earth and moon about their center of mass, and the other describing the motion of the sun and the center of mass of the earth-moon system. The coupling terms between the two Kepler problems are neglected at the first approximation. Delaunay used this definition of the main problem for his solution of the lunar problem, but after twenty years of computation was unable to meet the observational accuracy of his time. In Hill's definition of the main problem, the sun and the center of mass of the earth-moon system still satisfy a Kepler problem, but the motion of the moon is described by a different system of equations known as Hill's lunar equations. Using heuristic arguments about the relative sizes of various physical constants, he concluded that certain other terms were sufficiently large that they should be incorporated into the main problem. In a popular description of Hill's lunar equations, one is asked to consider the motion of an infinitesimal body (the moon) which is attracted to a body (the earth) fixed at the origin. The infinitesimal body moves in a coordinate system rotating so that the positive x axis points to an infinite body (the sun)

188

infinitely far away. The ratio of the two infinite quantities is taken so that the gravitational attraction of the sun on the moon is finite. The Hamiltonian of the three-dimensional Hill's lunar problem is H = o (l/i + vl + »I) - x iy2 + Z23/1 - = (x? - x\ - x\) -

,

=^ (i)

9

(see Llibre, Meyer and Soler ). 3

a

Symmetries and Special Coordinates

The Hamiltonian (1) is invariant under the two anti-symplectic reflections: 111 : (xj,22,2:3,2/1,2/2,2/3) —*

{xi,~x2,-x3,-yi,y2,y3), (2)

TZ2 : (xi,x2,x3,yuy2,y3)

—► (xi, - x 2 , x 3 , -yuy2,

-y3).

These are time-reversing symmetries, so if (xi(t),x2(t),x3(t), 2/1 (*),2/2(*), 2/3(*)) ^ a solution, then so are (xi(—t), — x2(—t),±x3(— t), —2/1 ( — 0,2/2( — 0,-F2/3(—t)). The fixed set of these two symmetries are Lagrangian subplanes, i.e. £1 = {(si, 0,0,0,1/2,1/3)},

£2 = {(xi,0,x 3 ,0,1/2,0)},

are fixed by the symmetries TZi,TZ2. If a solution starts in one of these La grangian planes at time t = 0 and hits the other at a later time t = T then the solution is 4T-periodic and the orbit of this solution is carried into itself by both symmetries. We shall call such a periodic solution doubly-symmetric. Geometrically, an orbit intersects £1 if it hits the Xi-axis perpendicularly and it intersects £2 if it hits the xi,X3-plane perpendicularly. To be more specific, let (Xi(t,a,/3,7),X 2 (<,a,/9,7),X3(t,Q,i9,7).Vi(*,a,i9,7),V2(*,a l /3,7),l'3(*,a,/3, (3) be a solution which starts at (a, 0,0,0,/?,7) 6 £1 when t = 0, i.e. X 1 (0,a,/3,7) = a,

X 2 (0,a,/3, 7 ) = 0,

X 3 ( 0 , a , / ? , 7 ) = 0,

yi(0,a,/?,7) = 0,

y2(O,a,/J,7) = 0,

Y3(0,a,M)

(4)

=T

"These is a typographical error in the Hamiltonian of Hill's lunar problem found in Meyer and Hall10.

189

The solution with a = Q 0 , 0 = (3Q, 7 = 70 will be doubly-symmetric periodic with period AT if it hits the £2 plane after a time T, i.e. *2(T,ao,/3o,7o)=0,

^ (T,a 0 , A), 7o) = 0,

Y3(T,ao,0ano)

= 0.

(5)

This solution will be a nondegenerate doubly-symmetric periodic solution if the Jacobian d(X2,Yi,Y3) -£— ^ - { T , a 0 , 0 o , 70) (6) d(t,a,0,-r) has rank three. It follows from the Implicit Function Theorem that nondegenerate doublysymmetric periodic solutions can be continued under a small conservative perturbation which preserves the symmetries. In general, a nondegenerate doubly-symmetric periodic solution may not be nondegenerate in the classical sense, i.e. a nondegenerate doubly-symmetric periodic solution may have all its multipliers equal to one. Jefferys8 proved the existence of nondegenerate doubly-symmetric peri odic solutions of the spatial restricted three-body problem by first setting the mass ratio parameter \i equal to zero to get the Kepler problem in rotating coordinates. He then showed that some of the circular solutions of the Kepler problem where nondegenerate doubly symmetric periodic solutions. Thus, by the above remarks these solutions can be continued into the restricted prob lem for small \i. Since there is no natural parameter like [i we will introduce a scale parameter e. This makes the analysis much more delicate. We follow Jefferys by using a variation of the Poincare-Delaunay elements. First, the Delaunay elements (£, g, k, L, G, K) are a coordinates on the elliptic domain of the Kepler problem. The elliptic domain is the open set in K6 which is filled with the elliptic solutions of the Kepler problem. The elements are: £ the mean anomaly measured from perigee, g the argument of the perigee measured from the ascending node, k the longitude of the ascending node measured from the x\ axis, L semi-major axis of the ellipse, G total angular momentum, K the component of angular momentum about the x 3 -axis. £,g, and k are angular variables defined modulo 2ir, and L,G and K are radial variables. If i is the inclination of the orbital plane to the X\,X2 reference plane, then K = ±Gcosi, and so an orbit is in the xi,X2-plane when K = G. (Often, k and K are denoted by h and H, but we are Hamiltonophiles.) An orbit hits £1 at time t = 0 if it is perpendicular to the Xi-axis. So its orbital plane must be through the ii-axis or k = 0 mod n, its perigee must be on the xi-axis or g = 0 mod TT, and it must be at perigee (apogee) or £ = 0 mod TT. Thus, C\ in Delaunay elements is defined hy £ = g = k = 0 mod -K.

190

An orbit hits £2 at time t = T if it is perpendicular to the xi,X3-plane. So its orbital plane must be perpendicular to the xi,X3-plane or k = n/2 mod 7r, its perigee must be in the Xi,X3-plane or g = n/2 mod ir, and it must be at perigee (apogee) or t = 0 mod 7r. Thus, £2 in Delaunay elements is defined by t = 0, g = k = TT/2 mod TT. Since these coordinates are not valid in a neighborhood of the circular orbits of the Kepler problem, we change to Poincar£ elements as follows: first make the symplectic linear change of variables 9i = I + 9 + k, q2 = -k -g, Q3 = l + 9,

p! = L - G + K, p2 = L-G, p3=G-K,

and now apply the symplectic change of variables defined by the generating function P2 W(q, P)=qiPx + -2- tanq 2 + P3q3 so that P2 = v^2p2 cos q2 and Q2 = y/2p2sinq2. changes gives the new variables:

This combination of variable

Q\ = q\ = I + 9 + k,

Pi=Pi=L-G

+ K,

Q2 = - V 2 ( L - G ) s i n ( f c + g),

P2 = y/2(L - G) cos(k + g),

(7)

Q3 = q3 = l + g, P3=p3 = G-K. These variables are valid on circular orbits which occur at L = G (see Howison6 or Szebhely 12 ). The circular orbits with L = G correspond to Q2 = P2 = 0. Thus, C\ in Poincare elements is defined by Q2 = 0, Q\ = Q3 = 0 mod 7r, and £ 2 in Poincar6 elements is defined by Q2 = 0, Q\ = 0 mod n, Q3 = TT/2 mod 7r. 4

Approximate Solutions

Move the infinitesimal mass close to the origin by scaling the variables: x —» e2x, y —> e~1y, which is symplectic with multiplier e - 1 . Letting H —> e~lH, expanding the potential in e, and by dropping the constant terms, the Hamiltonian becomes

H = £_3

M "Rf

{xiV2

~ X2Vl) + £3i/t(x'y'e)'

(8)

191 where H^ is analytic and order 1 in e. The solutions that will be establish will have the new x, y coordinates of order 1 in £ and so the original x will be order £ 2 and the original y will be order £ _ 1 . We will not scale time and the solutions we establish will have periods which are order 1 in s. Note that as £ —» 0 the Hamiltonian tends to infinity, thus we can not just set £ = 0. We will need approximate solutions to the equations and good estimates. (Scaling time does not remove the difficulties of the problem. If we scale time so that the Hamiltonian becomes order 1 in £ then the new periods will tend to infinity.) In Delaunay elements, the Hamiltonian becomes H = ^

r

- K

+ e3HHe,9,k,L,G,K,e).

(9)

Since these coordinates are not valid in a neighborhood of the circular orbits, we change to Poincare" elements (7) and the Hamiltonian becomes H

= , w " £ „v2 ~Pl + \ {P2 +Ql) 2 (Pi + Pz) *

+S3Ht(QUQ2,Qz,Pl,P2,P3,£)(10)

Thus the equations of motion are Q = Hp, P = —HQ or £~3

^1==77^ ^ 3 \P\ + P3)

1 + £3

/i>

- P i = 0 + £3/4,

Q 2 = Pi + £ 3 /2,

P2 = - O a + £ 3 /s,

^3

-^3=0 + £ 3 / 6 ,

=

. p l p ^ 3 + £ 3 /3,

(11)

(Pi + P3) where the /j are the appropriate partials of W. First let us consider the approximate equations in order to find the correct approximate periodic solutions. Consider the approximate equations Wl

Qi

(Pi

+ P3)3

= P2,

(Pi

Pi = 0, P2 = - Q 2 ,

e- 3

Qz-

1,

+ P3) 3 '

P 3 = 0.

(12)

192

These are of course, the equations of motion for the Kepler problem in scaled, rotating Poincare elements. The solution of equation (12) are O lt\ Ql(i)_

/ e-3 -l)t V(Pl+P3)3 1

+ qu

Q2(t) ~-= qi cos t + P2 sin t,

Q3(t) = ■

(

£

~

W) = —q2sint

\ t 4- /jo

3

Pi(t)=pu

V(Pl+P3> 3 1 t-l-93.

P*{t) -=

+p2Cost,

(13)

P3,

for initial conditions {qi,q2,Q3,Pi,P2,P3) at t = 0. The periodicity conditions are the same as those in the Section 3. That is, at t = 0; Qi = in, Q2 = 0, Q3 = jn and at t = T; Qx = {i + k)n, Q2 = 0, Qs — 0' + m + l/2)7r where i and j are 0 or 1, and k, and m are arbitrary integers. To satisfy these symmetry condition at t = 0 and at t = T we have so solve the equations

«^-(ra^- 1 ) T + i , r = ( i + i f c ) '' Q 2 ( r ) = p 2 s i n T = 0,

Q3(T) =

(14)

( ( ^ F ) T + J7r = {j + m + 1/2)7r>

The second equation is solved by taking p? = 0, thus selecting a circular orbit of the Kepler problem. The difference between the first and third equa tion has a solution with T = (m — k + l/2)w. It remains to solve the third equation. With this choice of T it becomes .

,,

£ - 3 (m - k + i )

Recall that Pi + P3 = L which is the semi-major axis in the Kepler problem and that we want solutions which are order 1 in L. Let n be a fixed small integer. To solve (15) set m+l/2 = £ - 3 , k = m — n, and (pi +P3) 3 = (n + 1 / 2 ) . With this choice the period becomes T = n +1/2 We shall indicate in Section 5 that these solutions of the approximate equations (12) are actually approximations of actual doubly-symmetric peri odic solutions of the true equations (11). Thus, our main theorem is

193 Theorem 4 . 1 . There exist doubly-symmetric periodic solutions of the spatial Hill's lunar problems with large inclination which are arbitrarily close to the primary.

5

Outline of the proof

Here we will outline the proof and refer the reader to Howison and Meyer7 for a more detailed account of a similar result. Consider the equations (11). The periodicity conditions remain: at t = 0; Q\ = in, Q
Qi(*)= ( (

£ +

^ - M t + gi+g^i.

Q2(t) = 92 cos t + pi sin t + e3g2, Q&)

=

^i(*)=Pi+e394, ^M*) = p? cos t - q
it 1 \a) f + 93 + £3$3, \(Pl +P3) J

Pz(t) = P3 + £396,

for initial conditions (qi,q2,q3,Pi,P2,P3) and for time ie[0,7], where ft = 3i(*,9l.92,93,Pl,P2,P3)To satisfy the symmetry condition at t = 0 we have the solution

Q i ( ' ) = \T^ - s - l l i + Mr + e ^ i , \(Pl+P3) J Q2W = p 2 s i n ( + £ 3 g 2 ,

Q*W = I 7-^

^ 1 * + i* + £333-

\(Pi+Ps) / Next we need to solve this for the symmetry condition at t — T which are now:

194

GiW = I T~^ <& - 1 I l + i7r + £39i = (* + *) T, \(Pl+P3) / Q2{t) = p2sint

+ e3g2 = 0,

<&(*) = (7—^ ^ ) * + ^ + e3Sa = ( i + m + i ) 7 r , V(Pi + Ps) / or =• - 1 I t - kit + e3g! = 0,

(P1+P3)3

i

p2 sin t + eJp2 = 0, 3 3 - 1 h - ( m + 2i)7r + £ 5 3 = 0.

V(A+ft)

)

This is done by applying the Arenstorf implicit function theorem twice. Roughly speaking Arenstorf's theorem applies to situations where the prob lem is undefined when e = 0, all that is required is that the perturbation and the derivatives of the perturbation are sufficiently small. See Arenstorf1 or Howison and Myer7 for details. First we consider the difference of the first and third equation, together with the second equation. This is the system of equations t + (k - m - i ) 7r + e3g3 - £35i = 0, p2 sin t + e3g2 = 0. This has solution t = (m + 1/2 — k)n, p2 = 0 at e = 0. Along this solution, the determinant of the derivative of the system with respect to t and P2 is given by 1 0 0 sin(m + 1/2 - k)ir)

±1^0.

Thus by the Arensdorf implicit function theorem, there exists a neighbor hood N of 0 and functions T(pi,p3,e) near (m + 1/2 — k)n and p2(p\,P3,e) near 0 for e G M and (pi,P3) arbitrary such that

195 T(puP3,e)

~ (m + \ - fc)* + e 3 5 3 - £39i = 0,

P2(pi,P3,£)sinT(p 1 ,p 3 ) e)+£ 3 p2

=0.

Putting this solution for T into the third equation, we need to solve Qz{T) (m + \)ir — 0, or £-3

(m+--fcW-£353+£35i

(pi +P3) 3

- (m + - j w + e3g3 = 0,

which is equivalent to solving m + - - fc J TT - £ 3 5 3 + e 3 3i - f £3 f m + - J TT - e 6 5 3 J ( P l + p 3 ) 3 = 0 for (pi +P3) 3 whenever e &M — {0}. Now the solution for T left both m and fc arbitrary, so for the moment regard m and k as free variables. Then letting m + k = e~3 and letting fc = m - n for n a small integer, we need to solve R =

v

+

2) * ~£3fl3+£3si ~(*~e6g^ ^Pl+P3^3 = °'

By this choice of m and A;, T becomes T(pi, p 3 , e) = (n + 1/2)TT + £ 3 g 3 - e3g\ which is uniformly bounded as e approaches zero. dT/dpi at e = 0 is given by -Ad{e3gs—£392)/dp\ where the partialsof the & are evaluated along solutions; t = (n + \)ir, P2 = 0, £ = 0, (pi,p 3 ) arbitrary. By the another Grownwall argument the partials of the
196

References 1. R. F. Arenstorf, A new method of perturbation theory and its application to the satellite problem of celestial mechanics, J. Reine Angew. Math., 221, 113-145, (1966). 2. E.A. Belbruno, A new regularization of the restricted three-body problem and an application, Celestial Mech., 25, 397-415, (1981). 3. P. Guillaume, New periodic solutions of the three dimensional restricted problem, Celestial Mech., 10, 475-495, (1974). 4. M. C. Gutzwiller and D. S. Schmidt, The motion of the moon as computed by the method of Hill, Brown, and Eckert, Astronomical Papers XXIII(I), U.S. Naval Observatory, Washington, DC, (1986). 5. G. W. Hill, Researches in the lunar theory, Chapter I, Amer. J. Math., 1, 5-26, (1878). 6. R.C. Howison, Doubly-symmetric Periodic Solutions to the Three Di mensional Restricted Problem, Ph.D. Dissertation, University of Cincin nati, (1997). 7. R.C. Howison and K.R. Meyer, Doubly-symmetric periodic solutions of the spatial restricted three-body problem, to appear in J. Diff. Equations. 8. W. H. Jefferys, A new class of periodic solutions of the three-dimensional restricted problem, Astron. J., 7 1 , 99-102, (1965). 9. J. Llibre, K. R. Meyer, and J. Soler, Bridges Between the Generalized Sitnikov Family and the Lynpunov Family of Periodic Orbits, to appear in J. Diff. Equations. 10. K.R. Meyer and G.R. Hall, An Introduction to Hamiltonian Dynamical Systems, Springer-Verlag, New York, (1991). 11. D.S. Schmidt, Families of Periodic Orbits in the Restricted Problem of Three Bodies, Ph.D. Dissertation, University of Minnesota, (1969). 12. V. Szebehely, Theory of Orbits, Academic Press, New York, (1967).

O N P R A C T I C A L STABILITY REGIONS FOR T H E M O T I O N OF A SMALL PARTICLE CLOSE TO THE EQUILATERAL P O I N T S O F T H E REAL E A R T H - M O O N S Y S T E M ANGEL JORBA Departament de Matematica Aplicada i Analisi, Universitat de Barcelona, Gran Via 585, 08007 Barcelona, Spain. E-mail: [email protected] This is a summary of a preliminary study on the existence of stability regions near the equilateral libration points of the real Earth-Moon system (here, by "real" Solar system we refer to the model defined by the ephemeris DE406 from JPL). We start from a model that takes into account the main effects coming from Earth, Moon and Sun to locate regions where trajectories are confined for long time intervals. Then, these regions are checked against a numerical integration that takes into account the real motion of Earth, Sun and the planets. This shows the existence of regions such that any initial condition inside them produces a trajectory that remains there for at least 1000 years.

1

Introduction

Let us start first with the simplest model for this problem, the well known Restricted Three Body Problem (from now on, RTBP). The RTBP models the motion of a particle under the gravitational attraction of the Earth and Moon (also called primaries), under the following assumptions: i) the particle is so small that it does not affect the motion of Earth and Moon; and ii) Earth and Moon are point masses that revolve in circular orbits around their common centre of mass. It is usual to take a rotating frame with the origin at the centre of mass, and such that Earth and Moon are fixed on the X axis, the (X, Y) plane is the plane of motion of the primaries, and the Z is orthogonal to the (X, Y) plane. The units of distance, mass and time are defined such that the Earth-Moon distance, the total mass of the system (Earth+Moon) and the gravitational constant are all one. With these units, the period of the motion of Earth and Moon around the origin equals 2n. Defining momenta as p^ = x — y, py = y + x and pz = z, the equations for the motion of the particle can be written as an autonomous three degrees of freedom Hamiltonian system. The corresponding Hamiltonian function is HRTBP = M

+ Pi + Pz) + VPx ~ XPv ~ ~T~~ ~ ~T~,

(1)

being rpE = (x - fi)2 + y2 + z2 and r2PM = (x - /i + l ) 2 + y2 + z2. It is well known that this Hamiltonian has five equilibrium points, that

197

198 l

1

1

1

i

-

0.8 0.6

_

0.4 0.2

-

L,

U

X,

0

-0.2

-

.

M

E

-

-0.4 -0.6 -0.8

-

-1 -1.5

'

-

XL« i

-0.5

•

0.5

1.5

Figure 1. The five equilibrium points of the RTBP.

are displayed in Figure 1: three of them lay on the X axis (they are called collinear points, Eulerian points, or simply 1-1,2,3), and the other two form an equilateral triangle (in the (X, Y) plane) with Earth and Moon (they are called triangular points, Lagrangian points or simply £-4,5). For more details see, for instance, [1] or [2]. It is also well known that, for the value of the masses of Earth and Moon, the triangular points are linearly stable. It is known that, under very general conditions, there exists a neighbourhood of these points such that any initial condition inside this neighbourhood produces a trajectory that remains close to the point for huge intervals of time (see [3], [4], [5], [6] and references therein). On the other hand, let us consider the motion of a particle in the real Solar system, close to the triangular points of Earth and Moon. Here, by real system we refer to the one obtained from the well known JPL ephemeris [7]: the ephemeris contain the position of the main bodies of the Solar system (Earth, Moon, Sun and planets) so we can easily write the vector field acting on the particle. This is a numerically defined vector field, that we can integrate for the time span for which the ephemeris are available. Note that, for the real system, the triangular points are not longer equilibrium points since the hypotheses used to derive the RTBP are not satisfied. It is know that arbitrary trajectories starting near the (geometrically defined) triangular points go away after a short time (see [8] and references therein) and hence, the RTBP is not a good model to study this problem because it displays a quite different qualitative behavior. In this talk we will first use the so called Bicircular Problem (from now

199 on, BCP). This model can be seen as a time dependent periodic perturbation of the KTBP, that includes the main effect coming from the Sun. As far as we know, this model is first introduced in [1]. As we will see, the BCP model has some of the main features of the real model (for instance, the neighbourhood of the equilateral points is unstable). Our purpose is to use the BCP model to describe some properties of the real model. In particular, we will compute, for the BCP, families of 2-D in variant tori near the equilateral points, as well as their normal behaviour. We will show (by computing the normal modes of the tori) that some of them are hyperbolic, while some others are elliptic. Those elliptic tori are found at some distance from the equilateral points. It is known that lower-dimensional normally elliptic tori give rise, under suitable hypotheses, to regions of effec tive stability around them (see [9], [10]). Finally, we will show, by means of numerical simulations, that some of these stability regions found in the BCP model subsist in the real system, at least for time spans of 1000 years. This talk is based on results contained in [11], [12] and [13]. They can be downloaded from the web server http:Wwww.maia.ub.es/dsg, in the p r e p r i n t s section. 2

The Bicircular Problem

Let us assume that Earth and Moon are point masses that revolve in circular orbits around their common centre of mass (as in the RTBP), and let us also assume that this centre of mass revolves in a circular orbit around the Sun (that is also considered as a point mass) that is fixed at the origin. We are also assuming that the motion of the three bodies is coplanar. The purpose of the Bicircular Problem is the study of the motion of an infinitesimal particle under the attraction of these three bodies. It is usual to take the same reference frame and normalized units as in the RTBP. So, in this system of reference, Earth and Moon are fixed on the x axis while the Sun is turning around the origin. Defining momenta as px = x - y, py — y 4- x and pz = z, then the motion of the infinitesimal particle can be described by a Hamiltonian system that depends on time in a periodic way: HBCP

= 2 (Px + P\ + Pi) + yPx ~ xpv

-

2" (ysmO — xcosw), (2) as where rPE = (x - /x)2 + y2 + z2, r2PM = (x - fj. + l ) 2 + y2 + z2, rPS = (x - xs)2 + (y - ys)2 + z2, *s = as cos6, ys = -as sin0, and 6 = u>st. The TPE

TPM

Tps

200 /i LJS

= =

0.012150581623433623 0.925195985518289646

ms as

= =

328900.54999999906 388.81114302335106

Table 1. Floating point values used for the different constants of the BCP. Not all the digits are significative (they appear because the definition of these quantities involves quotients).

values used for these constants are listed in Table 1. We stress that the BCP can be written as a periodic time dependent perturbation of the RTBP:

ire "BCP

=

ZJ fi "RTBP + £nBCP,

G I L "BCP = -"*S I \rpS

1

ysmu -xcoso o as

\ . ) ' \°) )

and it is clear that HB^9p = HRTBP, and that H££QP — HBCPNow, let us focus on the L5 point (due to the symmetries of the RTBP and the BCP, the same results will be valid for L 4 ; of course,J;his is not true for the real system). As the vector field of the perturbation EHBCP does not vanish on L5, this point is not longer an equilibrium solution. An straightforward application of the Implicit Function Theorem shows that, under a generic non-resonance condition (satisfied in this case) and assuming e small enough, L 5 is replaced by a periodic orbit with the same period as the perturbation. This orbit tends to L5 when e tends to zero. Let us consider now the Poincare section defined by t = 0 mod Ts, where Ts = 2n/u)s is the period of the perturbation. Let us call PE the corresponding Poincar6 map. Note that the fixed points of this map correspond to periodic orbits (of period Ts) for the flow, and vice versa. Now, by means of a continuation process, let us compute the fixed points of PE for e ranging between 0 and 1. The results are displayed in Figure 2 (left), where the horizontal axis shows the x coordinate of the fixed point and the y coordinate refers to the value of e. Note that, for e = 1, there are three fixed points of Pe that are close to L$. The (x,y) projection of the corresponding periodic orbits for the flow are displayed in Figure 2 (right). By computing the eigenvalues of the differential of Pc at the fixed points, one can see that orbit P O l is unstable, and that orbits P 0 2 and P 0 3 are linearly stable (see Figure 2). Moreover, the three periodic orbits have an elliptical mode very close to the {z,pz) plane. In what follows, we will refer to this mode as the vertical mode of the corresponding periodic orbit. For more details about the BCP, see, for instance, [14] or [15].

201

Figure 2. Left: Continuation of the periodic orbit that replaces i 5 in the BCP. Right: The three periodic orbits that appear in the left plot for e = 1.

2.1

The vertical families of 2-D tori of the BCP

Let us now consider the vertical mode of one of the periodic orbits. It is known that, under generic conditions (see [9], [10]), there exists a Cantor family of 2-D invariant tori that extend this linear mode into the complete (i.e., nonlinear) system. This is very similar to the well-known Lyapunov centre theorem: instead of obtaining a (smooth) family of periodic orbits, one obtains a (Cantor) family of 2-D tori. Next, we will focus on the computation of these vertical families of twodimensional tori. Their interest will be clear later on. 3

Computation of invariant curves

In this section we will briefly explain a numerical method for computing fam ilies of invariant curves of high dimensional maps. The method does not require the knowledge of an algebraic expression for the map, it only requires a procedure to numerically evaluate the map (and its differential) at a given point. So, it is a suitable method to be applied to the Poincar£ map of an ODE. More details can be found in [11]. 3.1

A Newton method

Let us consider a map / from an open set of R n into itself, with an invariant curve with (irrational) rotation number u\ this is, we assume that there exist a map x : T 1 —♦ R n such that f(x{6)) = x{6 + w),

for all 0 € T 1 .

(4)

202

To simplify the discussion, let us assume that we know the value u , so we only need to determine the Fourier coefficients of the function x(8), x(0) = OQ + ^2 ak cos(fc0) + bk sin(fc0),

k € N.

k>0

Let us now fix a truncation value N (the selection of N will be discussed in Section 3.3), and let us try to compute an approximation to the 2N + 1 unknown coefficients UQ, ak and bkj 0 < k < N. First, let us define a mesh of 2iV + l points on T 1 , 0J = » y . . 0<j<2N. 3 2N + 1 -J Imposing condition (4) on the points of this mesh we obtain the system of 2N + 1 equations, f(x{0j)) - x{9j + u) = 0,

0 < j < 2N,

(5)

that we will solve, by means of the Newton method, to determine the 2N + 1 unknown Fourier coefficients. The differential of this equation can be easily obtained by using the chain rule: Let F be the function that maps each sequence of 2N + 1 Fourier coefficients (ao, a\, b\, 02,62,..., aw, 6AT) on the vector of 2N + 1 components given by

(f(x(0j))-x(0j+uj))oiJS2N. Hence, denoting by Df(p) the Jacobian of / at a given point p G K n , we have

dF -DfWOi)) dak Note that Df(x(9j))

^l(0j)

^M)

^

(9,).

can be easily computed, and that

= Incos(k9J),

dX

^

+

U})

(9j)-Incos(k9J+kcj),

(6)

where /„ denotes the n x n identity matrix. Similar formulas hold for the derivatives with respect to bk- From these formulas one can easily assemble the Jacobian matrix of F: it consist of (2N +1) x (2N + 1) boxes, being each box a n x n matrix (we recall that ak and bk are arrays of n components).

203

3.2

On the uniqueness

If x(9) denotes a Fourier series corresponding to an invariant curve then, for any

Error estimation

To select the truncation value N used in the discretization (see Section 3.1), we will use the following estimation for the error: E(x,u)

= sup \f{x{6)) - x{9 + w)|. 0ST>

Clearly, x is an invariant curve with rotation number u iff E(X,LJ) is zero. For each curve, we have estimated the quantity E(x,w) by simply tab ulating its value on a mesh of points on T 1 . Of course, this mesh must be finer (typically, we have used a 100 times finer mesh) than the one used to obtain (5) from (4). When this estimate is bigger than a prescribed threshold (here we have used 10 - 1 2 ), the program automatically increases the number of Fourier coefficients and the number of discretizing points (i.e., it increases the truncation value N) and recomputes the invariant curve. This process is repeated until the estimated error E(X,UJ) is lower than 1 0 - 1 2 . 3.4

Results

By means of a standard continuation procedure (see details in [11]), we have computed the vertical families of 2-D tori for the three periodic orbits POl,

204 Ep6ilon«0-69

EptlkxM).91

Figure 3. Vertical families of invariant tori for several values of e. The horizontal axis shows the value of the coordinate z of the invariant curve when z = 0, and the vertical axis shows the corresponding rotation number. The gaps that conform the Cantor structure are too narrow to be seen at this scale.

P 0 2 and P 0 3 , for several values of t (see Figure 3). For definiteness, let us call Fj the family that is born in P O j , j = 1,2,3. Note that the families F l and F2 are connected, as it is suggested by the diagram in Figure 2 (left). As P O l is hyperbolic, we expect that the tori in F l are also hyperbolic, at least for moderate values of the vertical amplitude. P 0 2 and P 0 3 are elliptic orbits so, under generic conditions, it is expected that most of the tori in families F2 and F3 with a small vertical amplitude are also normally elliptic. 0 The study of the normal stability of these families is the topic of Section 5.

"Although most of them are going to be elliptic tori, a small measure set of them is ex pected to be normally hyperbolic, due to some resonances involving internal and normal frequencies.

205 4

Stability regions

It is also known that, under generic conditions of analyticity, non-degeneracy and non-resonance, a lower dimensional and normally elliptic invariant tori is surrounded by a region of effective stability. Here, by "region of effective sta bility" we mean that trajectories starting inside this region need a extremely long time to escape of a neighbourhood of the torus [10]. Here we will try to estimate this region by means of purely numerical methods (see [16] for a semi-analytical approach in the case of a periodic orbit). So, given a 2-D torus on one of these families, we will first select the unique point a^ on it satisfying t = 0 (i.e., we are selecting the point on the Poincare section), z = 0 and z = ZQ > 0 (io will be used as a parameter). Then, we will take a 2-D mesh of points around the (x, y) coordinates of ao (and inside the (x,y) plane), keeping fixed all the remaining coordinates. Then, we will use each point on the mesh as initial condition for a nu merical integration of the vector field. If the trajectory does not cross the y = -0.5 plane, and it does not approach collision with Earth and Moon dur ing the integration time (see Section 4.1), we admit that the initial condition belongs to the stability region. Next, in Section 4.1, we will summarize the results from the simulations for the BCP and for the real system. 4-1

Some numerical simulations

Let us first select the 2-D torus of the family F3 such that, in the Poincare section t = 0, z is zero and z is 0.58 (the reasons for this selection will be clear later). Let us first make a numerical simulation around this torus, as explained above. Figure 4 (left) displays the set of initial conditions (x, y) such that the corresponding trajectory does not escape for a time span of 5000 periods of the Moon. Then, let us take this mesh to make a numerical simulation for the real system, by using the JPL ephemeris (file DE406). Here, the initial integration time is the first full Moon of year 2000 (this is to have a relative position of Earth, Moon and Sun similar to the one of the BCP when t = 0). Figure 4 (centre) shows the initial conditions that do not escape after 500 years, and Figure 4 (right) contains the ones that do not escape after 1000 years. The unit of distance in the scales of the plots is the Earth-Moon distance. Note that the size of these regions is quite big. Let us now consider the torus (of the family F3) corresponding to z = 0 and i = 0.83. Doing the same simulations as before, one obtains the results displayed in Figure 5. As before, the left graphic refers to the BCP (the

206

Figure 4. Left: stability region for the BCP, for i = 0.58. Centre: stability region for the real model (500 years). Right: stability region for the real model (1000 years). The box corresponds to the explored region. The position of Ls is marked with a big "+" sign.

-1

«f

<0«

«.4

-0.2

0

92

-I

-0J

-Ol

-0.4

42

9

02

-1

410

-Of

-0.4

«2

0

02

Figure 5. As Figure 4, but for the case z = 0.83.

integration time is again 5000 revolutions of the Moon), and the centre and right graphics refer to the real system (for 500 and 1000 years respectively). This kind of stability was already observed in [14], but with smaller regions and much shorter time intervals (around 60 years). The main improvements in this work come from the computation of the vertical families of 2-D tori for the BCP and to use initial conditions around them. A natural question is whether this is the typical behaviour around the vertical family. To this end, we have repeated the above simulations around several tori in family F3. The results are summarized in Figure 6 (more com plete results and explanations can be found in [12]). The horizontal axis refers to the z coordinate of the torus in the Poincare section, and the vertical axis contains the number of points of the initial mesh that do not escape for 1000 years (as the meshes used have the same density, the number of surviving points can be considered an estimate of the size of the stable region). Appar ently, the stability region has two big components, centered (approximately) around the values z = 0.58 and z — 0.83. Simulations for longer time intervals on the BCP model (as well as a discussion on the relevance of the total simulation time) can be found in [15] and [12]. For longer simulations on the JPL model, see [12].

207

0.2

0.3

0.4

0.S

0.6

0.7

0.8

0.9

Figure 6. Horizontal axis: value of the i coordinate. Vertical axis: an estimate of the size of the stability region (see text).

5

Normal behaviour of lower dimensional tori

Note that the numerical simulation is not a good method to detect if a given object is normally stable. For instance, an hyperbolic object can have almost coincident stable and unstable manifolds (i.e., with a very small splitting), such that the trajectories look like confined in a neighbourhood of moderate size (in fact, they keep escaping and coming back). To detect this situation, it is very convenient to have methods to compute the normal behaviour of invariant tori. Note that the linear approximation to the normal behaviour of a torus is given by a linear differential equation that depends on time in a quasi-periodic way. The study of linear systems that depend on time in a quasi-periodic way is an old topic of the theory of ODE. Here we will focus on the numerical computation of the normal behaviour of these systems. This problem has been addressed in [17] and [18] (see also references therein). Here we will explain a method for numerically compute the normal modes of an invariant curve. This method can also be applied to tori of higher dimensions (see [13] for the details). Let x(6) be an invariant curve of a n-dimensional map / , and let UJ be the corresponding rotation number. Hence, they satisfy equation (4). Then, it is not difficult to see that the linear behaviour around this invariant curve

208

is given by the dynamical system xi+i = A(6i)xi, 1

,?s

where Xi € R n . To keep the notation simple, we have used again the letter x to denote an infinitesimal displacement w.r.t. the invariant curve. 5.1

On the reducibiUty of linear quasi-periodic systems

The system (7) is said to be reducible if there exists a linear change of vari ables, Xi = C{0i)yi, such that the transformed system, Vi+i =

!+!=of+%'}

m=c l{e+w

~

w*)cW'

satisfies that the matrix B does not depend on the angles 9, i.e., that the matrix B is constant. Note that the normal behaviour of a reducible system like (7) is completely determined by the eigenvalues of the reduced matrix B. It is known that not all the linear quasi-periodic systems like (7) are reducible, but is also known that reducibiUty appears in many cases. In the concrete case of elliptic lower dimensional tori of Hamiltonian systems, it turns out that the usual proofs of existence produce reducible tori. Hence, the possible non-reducibiUty must be confined into a set of very small measure. As far as we know, the only paper that proves the existence of lower dimensional tori without using reducibiUty conditions is [19], although it does not show the existence of non-reducible tori. Let us start by considering reducible linear systems like (7). This is, we are assuming that there exists a matrix C{6), 9 G T 1 , such that the matrix B = C~1(0 + ui)A(9)C(9) does not depend on 9. Let V be the vector space {V>: T 1 i-» R n } , and let us define the translation 7^ as Tux{9) = x(9 + u>). Let us consider the following generalized eigenvalue problem: to look for couples (A, V) € C x V such that A(0)1>{0) = XT^{9).

(8)

It is not difficult to show that (see [13] for more details): 1. If A is an eigenvalue of B, then A is an eigenvalue of (8). 2. If A is an eigenvalue of (8), then there exists k € Z such that Aexp(v / -Tfcw) is an eigenvalue of B.

209 The numerical method we will use to compute the normal behavior of the invariant curves is based on computing the spectrum of a discretization of (8), as explained in Section 5.2. It is interesting to note that the reducibility hypothesis introduces strong restrictions on the spectrum of (8). Hence, it seems possible to detect nonreducibility by computing the spectrum of a discretized version of (8) and showing that it is not compatible with the reducibility condition (see [13]). It is also worth noting that all the examples included in this paper have a discretized spectrum compatible with reducibility. 5.2

Methodology

The numerical method for computing the normal behaviour of the invariant curves is then the following: once the invariant curve has been obtained (by means of the Newton method explained in Section 3), we can easily obtain the matrix corresponding to the discretization of the operator that maps ip(8) to A(6)ip(6): note that this matrix is precisely the first term that appear in the differential of F in (6), Df ■ $*-, but composed with the linear operator that, given a set of points, produces the Fourier coefficients. Let us denote this first part of the differential by A. Then, the initial eigenvalue problem is equivalent to the (standard) eigenvalue problem T^Aip

= Xrp.

(9)

Note that, as the matrix of 71„ is block diagonal (in the usual basis of sinus and cosinus, these blocks are 2 x 2 ) and very easy to write, we obtain the discretized matrix of T-UA by simply multiplying the matrices of these two discretizations. Then, it is not difficult to obtain the corresponding eigenval ues. Once these eigenvalues have been obtained, the next problem is to classify them. We recall that, if the initial system is reducible, these eigenvalues can be expressed in the form exp(i/--!kw)X, where A is an eigenvalue of the reduced matrix. Note that u is known here, and we are only interested in identifying the values A (note that, in the elliptic case, these values are not uniquely defined, see [13]). 5.3

Some results

Figure 7 displays, in the complex plane, the eigenvalues of (7) for some tori in the families F l and F2. More concretely, we display the transition from F l to F2 (see also the bottom-right plot in Figure 3). In the left plot, we observe three circles, two of them correspond to hyperbolic behaviour (those

210 ■h0.S34217767* dz-0.1915664043

»-OS350333394 dbO.OMSOMMS ' 1

. J . ■ • - »■*» a ° .

•*•■ -v -v. •tf ■*.

0.5

*

„ 0

-05

-1

1

■

-

$

fc

->v. ......*: a "

■««><> Q a o

B

a

/

«y-#,, ■

05 ■

°

*

^

w-0 5370035099 tlz-0 1810663949

■

0

■05

<*>

• **»»

' ' , «ll-»* x '

•1 -1

-0.5

0

0.5

1

Figure 7. Eigenvalues in the transition from family Fl to F2. The corresponding rotation number (w) and the value of i (dz) are included at the top of each plot.

are the ones of radius different from 1), while the circle of radius 1 correspond to neutral and elliptic behaviour. Looking at the (numerical values of) the eigenvalues of modulus one, it is easily observed that there are only two of them that are real: 0.915799019153 and 1.091942641438, while all the others can be found as a rotation (of an angle multiple of u>) of them. In the set of eigenvalues of modulus 1, one finds a couple of eigenvalues equal to one (they come from the symplectic structure of this problem), their rotations by a multiple of u, the normal frequencies and their corresponding rotations. They are studied with more detail in [13]. The sequence of the three plots in Figure 7 shows how the change of stability takes place. The study of the bifurcations that appear here is left for a future work. Figure 8 (left) shows the eigenvalues for a torus in the family F3. In the right plot, we display the modulus of the eigenvalues against an index labelling them. It is clearly seen the mild instability of this torus. In this case, there are no eigenvalues of modulus different than one that are real. This corresponds to a complex saddle, where the modulus of the eigenvalues of the saddle are 0.999897464349 and 1.000102546166. Due to the errors introduced in the truncation of the linear operator (9), we expect that high order modes will have a rather big error. Those modes are easily detected since there are no other eigenvalues obtained from them by rotations of angle leu. Moreover, they do not persist if the truncation order of the operator is increased.

211 w.O 5055149066 Oz-0 8958352596

w-0.5055149066 dz>0.8S58352S96

0. 0.99965

1 ,;.-.. 1.00005 1.0001 1.00015 ■ 1.0002 ■1

-0.5

0

0.5

1

0

100

200

300

400

500

Figure 8. A mild instability in family F3.

6

Conclusions

The main purpose of this work has been to show the existence of regions of practical stability close to the equilateral points of the real Earth-Moon system. We have focused the discussion on the L5 point, and similar results hold for Li ( [12]). Although this result is based on a numerical integration of the real system, it has been shown how some standard techniques from dynamical systems theory allow to guess, approximately, where these regions should be. So, the final numerical exploration has only been done in a small part of phase space. An interesting point is the persistence of these regions for longer time intervals because, in this case, they could contain some kind of dust or even some asteroids. Note that an asteroid inside this region must oscillate up and down across the ecliptic plane. They can reach a distance between 0.5 and 0.9 (we recall that the unit of distance of the RTBP is the Earth-Moon distance) from the ecliptic, and it seems reasonable (we recall Figure 6) that they either reach a distance close to 0.6 or to 0.8 adimensional units. In any case, note that they must spend most of their time away from the ecliptic plane and, when they cross that plane, they do it with their highest speed. Hence, we should not expect to find them very close to the triangular points. Finally, let us mention that these regions could also be interesting for some space missions, since no control is necessary to keep an spacecraft there.

212

Acknowledgements The author wants to thank interesting discussions and remarks by C. Simo. This research has been supported by the Spanish grant DGICYT PB94-0215 and the Catalan grant CIRIT 1998S0GR-00042. References [1] V. Szebehely, Theory of Orbits, Academic Press, (1967). [2] K.R. Meyer and G.R. Hall, Introduction to Hamiltonian Dynamical Sys tems and the N-Body Problem, Springer-Verlag, (1992). [3] A. Giorgilli, A. Delshams, E. Fontich, L. Galgani and C. Simo, Effective stability for a Hamiltonian system near an elliptic equilibrium point, with an application to the Restricted Three Body Problem, J. Diff. Equations 77, 167-198, (1989). [4] C. Simo, Estabilitat de sistemes hamiltonians, Mem. de la Real Acad. de Ciencias y Artes de Barcelona, Vol. XLVIII, no. 7, (1989). [5] A. Celletti and A. Giorgilli, On the stability of the Lagrangian points in the spatial Restricted Three Body Problem, Cel. Mech. 50,31-58,(1991). [6] G. Benettin, F. Fasso and M. Guzzo, Nekhoroshev-stability of L4 and L5 in the spatial Restricted Three-Body Problem, Regular and Chaotic Dynamics 3, 56-72, (1998). [7] h t t p : / / s s d . j p l . n a s a . g o v / h o r i z o n s . h t m l [8] C. Diez, A. Jorba and C. Sim6, A dynamical equivalent to the equilateral libration points of the Earth-Moon system, Cel. Mech. 50, 13-29, (1991). [9] A. Jorba and J. Villanueva, On the Persistence of Lower Dimensional Invariant Tori under Quasi-Periodic Perturbations, J. of Nonlinear Science 7, 427-473, (1997). [10] A. Jorba and J. Villanueva, On the Normal Behaviour of Partially Elliptic Lower Dimensional Tori of Hamiltonian Systems, Nonlinearity 10, 783822, (1997). [11] E. Castella and A. Jorba, Numerical computation of a family of twodimensional invariant tori in the Bicircular problem, preprint, (1999). [12] A. Jorba, A preliminary numerical study on the existence of stable motions near the triangular points of the real Earth-Moon system, preprint, (1999). [13] A. Jorba, Numerical computation of normal frequencies of invariant curves of higher dimensional maps, in progress. [14] G. Gomez, A. Jorba, J. Masdemont and C. Simo, Study of Poincar6 Maps for Orbits near Lagrangian Points, ESOC Contract 9711/91/D/IM(SC), Final Report, (1993).

213

[15] C. Sim6, G. G6mez, A. Jorba and J. Masdemont, The bicircular model near the triangular libration points of the RTBP, in From Newton to Chaos, ed. A.E. Roy and B.A. Steves, 343-370, Plenum Press, (1995). [16] A. Jorba and J. Villanueva, Numerical computation of normal forms around some periodic orbits of the Restricted Three Body Problem, Phys. D 114, 197-229, (1998). [17] A. Jorba, R. Ramirez-Ros and J. Villanueva, Effective Reducibility of Quasiperiodic Linear Equations close to Constant Coefficients, SIAM J. on Math. Anal. 28, 178-188, (1997). [18] H. Broer and C. Simo, Hill's equation with quasi-periodic forcing: reso nance tongues, instability pockets and global phenomena. Bol. Soc. Bras. Mat. 29, 253-293, (1998). [19] J. Bourgain, On Melnikov's persistency problem. Math. Res. Lett. 4, 445-458, (1997).

VARIATIONAL M E T H O D S FOR Q U A S I - P E R I O D I C SOLUTIONS OF PARTIAL D I F F E R E N T I A L E Q U A T I O N S RAFAEL DE LA LLAVE Mathematics Department, The University of Texas at Austin Austin, Texas 78712-1082, USA. E-mail: [email protected] We show how to use variational methods to prove two different results: Existence of periodic solutions with irrational periods of some hyperbolic equations in one dimension and existence of spatially quasi-periodic solutions of some elliptic equa tions.

1

Introduction

In this note, we will present two results. The first is a remark about a question of ( ! ) on whether one can use variational methods to produce solutions with irrational period of

&

du a ? + V(u) == 0 «(0,t) = «(!,*)

We will show that for some particular numbers there is an extremely simple answer. The second result is to show how one can use variational methods to produce quasiperiodic solutions to some difference equations as well as par tial differential equations and to equations involving some pseudo-differential operators. The method of proof is related to some of the results of AubryMather theory in dynamical systems and we recover the corresponding results in dynamical systems as particular results. Of course, the Aubry-Mather the ory for dynamical systems is much more developed and it includes not only results about existence of minimizing periodic orbits, but also construction of connecting orbits, etc. Many of these results do not have an analogue yet for PDE's. The connection of Aubry-Mather theory with PDE's was pointed out in ( 2 ). Results for difference equations were obtained in ( 3 ). Here we will present two proofs for partial difference equations that are based respectively in ( 4 ) and ( 5 ). These strategies can be adapted to the more complicated case of partial differential equations. As it will appear in the proof, the arguments rely just on translation invariance, comparison principles, periodicity, and a variational structure (for differential and pseudodifferential equations we will also need some compactness properties, which amount to regularity results) and, hence they work in great generality. For

214

215

the purposes of these notes we will just present the simplest cases. We point out that an important element in our proof is the consideration of the heat flow as a tool. The relevance of the heat flow for Aubry-Mather theory seems to have originated in (6) and was used in ( 7 ). 2 2.1

Periodic solutions of irrational periods for wave equations Statement of results

We recall that a number T € K is called of constant type if \Tk — l\> C|fc| _1 V k, I £ Z. It is well known (Liouville's theorem) that all irrational numbers which solve quadratic equations with rational coefficients are of constant type. We sketch a proof of the delightful classical argument. In effect, if £ is an irrational number and P(f) = a£ 2 + b£ + c = 0 with a,b,c e Z, we have P'(£) ^ 0 because, otherwise P(x) = a(x - £) 2 = ax2 - 2£ax + a£ 2 , and, then £ = —b/2a would be rational. If n, m G Z and \m/n — £| is sufficiently small, by the mean value theorem \P{n/m)\

= \P(n/m) - P(()\ < 2\P'{0\ \n/m - f|

On the other hand 2 = —Trlan + brim + cm2\ > —zi m2 m because the numerator is an integer and different from zero by hypothesis. Putting together the two inequalities above, we have:

\P(n/m)\

n/m -- £ l >

l

2|P'(0|

1 m2

Quadratic irrationals are not the only constant type numbers. It is easy to show that a number is constant type if and only if it has a bounded con tinued fraction expansion. In particular, the set of constant type numbers is uncountable (but it has measure zero). As usual, given a function / : R x R —> R satisfying f(x, t) = f(x + l,t) = f(x,t + T). We will write

= £/*,

ie2*i[kx+l(t/T)}

f(x ,t) =

fc,/€Z

and denote

H3

= {/HI/112 == E i/wif(i fc.iez

+ k2 + i2y < oo> .

216

We will also denote by D the operator ^ j — gfrTheorem 2.1 Let T be a number of constant type. Let V : R —► R satisfy i) 0 < a < V < 0, where a is any number bigger than zero and 0 depends on T in a way that will be made eocplicit ii) \V"{x)\ < K Let f : R x R -> R satisfy Hi) f{x + l,t)= f(x,t-rT) iv)

= f{x,t)

feL2 Then, there exist a u € L 2 such that:

«;(&-

-&)u + V(u) = f

b) u{x + l,t) =u{x,t

+ T)

=u{x,t)

As we will see later, the smallness hypothesis on 0 are optimal. Of course, u satisfies a) only in the weak sense at this stage (as it is typical of variational results). For rational T, ( 1,s ) contain a beautiful argument that shows that weak solutions are strong solutions. This argument does not seem to be available in our case. Nevertheless, we can obtain smooth solutions for constant type periods imposing smoothness in V and / and smallness in / . Theorem 2.2 Let T be an irrational constant type number as before. Assume that f e Hr and V € Cr+2 and that its Cr+2 norm is sufficiently small. Then, there is one u e HT which satisfies Du + V(u) = f The solution is unique in a ball in Hr around the origin. Note that, in particular, when V(0) = 0, / = 0, the only solution with period T is u = 0 contained in a sufficiently small ball around the origin. This should be compared with the results of ( 9 ) which construct solutions of D u 4- V(u) for many frequencies. 2.2

Proof of Theorems 2.1, 2.2

The main observation is that, under the hypothesis that T is a constant type number D has closed range in H'^. In effect, the range is just the closed subspace of functions / such that /o,o = 0.

217 Note that D is diagonal on trigonometric functions and that if / is such that /o,o = 0 if we denote by

□ _ 1 / = E fk,i(27ri)2(_fc2

+

p / T 2 ) Ekll{x,t)

where Ek,i{x,t) = exp[2ni(kx + l(t/T)], we have that on functions / with zero average

nn" 1 /-n _1 a/ = /. (Even if the first can only be defined for functions with zero average, we can define the second for any / and have D D / = / - /o,o-) The crucial remark is that when T is constant type we have |A;2 - (Z/T) 2 | > C> 0 V(fc, 0 € Z 2 - (0,0) Since k

2

-^

= T~2(Tk-l)(Tk

+ l)

we see that when k,l are of the same sign, we can bound \Tk — l\ > C|fc| -1 and \Tk + l\ > T\k\, therefore the product can be bounded from below by CT. When k, I are of opposite sign, \Tk -l\> T\k\ and \Tk + l\ > C]^-1. I This shows clearly that D can be defined as a bounded operator on the space of functions with zero average. Since it is clear that the average of a function in the range is zero, R a n D is precisely the set of functions in Hr with zero average. Clearly, it is a closed subspace. Note also that Ran^)- 1 - = {constant functions}. The rest of the proof of Theorem 2.1 is very similar to parts of the cor responding result in ( 1 , s ), which we reproduce for the sake of completeness. Unfortunately, the bootstrap of regularity in these references, does not go through in our case, indeed, we suspect that the conclusions of the bootstrap may not be true in our case. Note that D u + V{u) = f implies that V(u) - / G Ran(D) and that, since 0 < a < V < 0 we can find V~l and that / H < ( V - 1 ) ' < a - 1 . If we write V(u) - f = w,u = V~l(w - f) the original problem becomes

v-^w-f)-^ fv-i(w-f)+nr*w = o

218

where w € Ran(D), that is V-1(io-/)+D

i-i

toGRanO"1

This problem has a variational structure. If V is a primitive of V - 1 , then (2.2) is the equation for the critical points of

+ (l/2)(w,n _ 1 w)

S(w) = fv(w~f)

considered as a functional on the space Y — Ran(D). Since Y is closed, it is a Banach space. Now we turn to computing the second derivative of the functional. We leave to the reader the easy verification that the functional admits second Gateaux derivatives and that they are what one would expect through a formal calculation. The only non-trivial part is to check that w —► JV(w — f) is differentiable, which follows easily from the Moser estimates for Sobolev spaces (See ( 10 ) 13 §3.) We have that:

D2S{wh = fv"{w - /) 7 2 + ( 7 ,D _1 7 ) Since the second derivative of V is the first derivative of V - 1 we see that V > - / ) 7 2 > / r 'Thw2^

I

-i-i

Since ( 7 , D

7) < T - ^ h l l ^ if fTxT

>

T-1C4TT2

, the second derivative of the functional S is strictly positive definite and, therefore the functional is strictly convex. The condition above is the condition on /? that we alluded to in the statement of the theorem. Note that it involves T and its number theoretic properties. If this condition holds S is a convex functional, moreover, we have S(w)>a\\w\\l0-b\\w\\Ho-c +

(1)

for some a, 6, c € R . The functional is bounded from below and we can find a sequence wn such that S(wn) —► inf r i 5 ( w ) . The bounds (1) show that ||w n ||H 0 is bounded, hence we can extract a weakly convergent subsequence by BanachAlaoglu theorem.

219

As it is well known, convex differentiable functionate in Hilbert spaces are lower semi-continuous in the weak topology. Hence, the infimum is reached by the limit point and this limit point satisfies the variational equations. This finishes the proof of Theorem 2.1. ■ Note that the conditions in 0 we obtained are optimal for theorems of the form indicated. If we consider V(u) = Xu, and A is an eigenvalue of D , we do not expect any solutions. The eigenvalues of D get as close as desired to the value we claimed. The proof of Theorem 2.2 is quite elementary. If we write u = u\ + 6, where u G Ran(D), 6 G R = Ran(D) x , the equation D u + V(u) = / is equivalent to

Ul =

-n~y(Ul+s)-/)

6

=±JlV(ul+6)-f}

We only have to check that the right hand side of the equation is a contra diction if we consider it as an operator acting on (u\, S) with a contradiction constant that can be controlled by ||V||c"-+a. This follows from the Moser estimates standard in Sobolev spaces. See e.g. (10) 13 §3) I

3

Aubry-Mather theory for configurations on lattices

The classical Aubry-Mather theory is concerned with variational problems for functions defined on the integers. A prototype example is the variational principle

L(u) = 5Z( 1 /2)(«. - "i+i) 2 - 5(Ui) i€Z

w i t h S ( u + l) = S(u). The Euler equations become - u i + i - t i i _ i +2ui-S'(ui)

= 0

(2)

which is the well known "standard map" of dynamical systems. Notice that, besides being a variational principle for the standard map, we could consider L(u) as the energy of a chain of penduli coupled by springs. The Ui would be the angle between the iilx pendulum and the rest position, S(ui) the potential energy due to the position of the pendulum. Another alternative physical interpretation (which appeared in solid state physics) is a chain of atoms in a one dimensional periodic potential coupled by a harmonic

220

interaction. With the latter physical interpretation, this is called the FrenkelKontorova model, which is a qualitative model of deposition on the surface of (one-dimensional) crystals. The goal in this lecture is to investigate the existence of quasi-periodic solutions of similar equations when the independent variables are in Z d not just in Z. This could be interpreted physically as arrays of penduli coupled to their nearest neighbors. That is, we want to consider the variational principle for maps u : Zd -» R

L(u)= £ (l/2)( Ui - Uj ) 2 -£S(u<) and its variational equation (-At*)* - S'(Ui) = 0

(3)

where A is the discrete Laplacian (Au)* = £)ij-ii=i Uj — 2dui and, again S(u + 1) = S(u) which we will assume is a C2 function corresponding to the position and (1/2)(UJ — Ui+i) 2 the elastic energy of the spring between two consecutive penduli. One of the first results of Aubry-Mather theory is that, for every u € R, (2) has a solution Uj such that supi |UJ — wi\ < oo. This result is somewhat surprising since for the apparently very similar equation Ui+\ +Uj_j — 2u< = a with o ^ O , the solutions grow parabolically. We refer to ( n ) for an up-to-date review of Aubry-Mather theory that, nowadays, includes not only the existence of solutions as indicated above, but much deeper geometric characterizations of the quasi-periodic invariant sets, their stable manifolds and many other geometric and dynamical properties of these sets. We will present two proofs of the following result. The proofs are based on ( 4 ' 5 ). There, the reader will find more details as well as generalizations. Another proof can be found in ( 12 ). T h e o r e m 3.1 With the notations above, for every u; G Rd there exists a solution of (3) such that sup|uj - (w,i)| < oo t

The proofs we will present make use of the heat flow, whose importance was noted in ( 6 - 7 ). Our goal here is to present the argument in its simplest form so that it is clear that it is a consequence of only:

221

• Variational structure • The variational structure is somewhat local • Periodicity in space of the variational problem • Periodicity in the u's of the variational problem • Twist conditions (comparison theorems for the gradient flow) • Some mild regularity assumptions that guarantee the existence of a flow for all times as well as differentiability of functional As detailed in (4'5) there are several other contexts where we have the properties above and, hence we have an analogue of Theorem 3.1. As for the locality of the variational principle, we point out that it is very mild. It will be apparent in the details of the proof. The locality requirements in the proof in (4) are slightly stronger than those in ( 5 ). We again note that the variational structure of the problem with period icity conditions is crucial since the equation —Au + a = 0 leads to quadratic growth at oo. From the point of view of variational theory these are some what non-standard problems since the spaces of trial functions are modeled in the spaces £°°(Zd) and, the functional are only formal — the sums are not meant to converge. If we try to regularize the problem by well defined ones, we see that we have to get some control at oo to guarantee the linear growth. This is conveniently achieved by introducing the customarily called Birkhoff configurations. Definition 3.1 We say that u is a Birkhoff configuration if for every k € Zd, l e Z i o e have either ui+k + / > in V i S Zd

or

ui+k +1 < Ui V i £ Zd

That is, if we translate the graph horizontally and vertically by integers, the graph does not cross itself. Given u) 6 Rd we denote Bu, = {u | u is Birkhoff Ui - (u>, i) € t°°] . We note that Bu is not empty since (a», i) belongs to it. We also note that if u € Bu, then Ufc - UQ is restricted to lie in an interval of length 1. In effect, if Ufc — uo + ' > 0 then because of the Birkhoff property unk+k — unk + I > 0 and adding we obtain U(„+1)fc - UQ + nl > 0 (similarly for <). Since "(n+i)fc — (wi (ft + l)k) is to remain bounded independently of n we see that

222

the inequality we have to pick in Definition 3.1 is the same as that when we compare (w, k) with I. For k e Z d , I € R we introduce (Cku)i = ui+k; 1llUi = u* -f1. Note that k j C C = Ck+>, nlTln = Kl+n and Ck1ll = ft'C*. A more compact notation can be obtained as follows: We say that u > v <=> Ui > Vi for all i (similarly for <) [note that this is not a total order, there are many non-comparable elements] Using this notation, we can say that a configuration u is Birkhoff if and only if, for every k € Z d , I € Z CkTZlu >u

or

CkTllu < u .

This makes it clear that CkBu = KlBM = S w We also note that Bu is closed under pointwise limits. It will be important to note that the fact that a configuration is Birkhoff implies a priori bounds for the components. If u G Bw, then Uk+i + I — u< has to have the same sign as (w, k) + I. (Otherwise, applying the Birkhoff property, we would get a contradiction with u^ — (u, k) G £°° ) Therefore, evaluating for i = 0 we obtain that when u & Bu, Uk — UQ has to lie on an interval of length two (which we can write explicitly as a function of*). Hence, if we consider the subset of u e Bu with UQ € [a, b], applying Tychonov theorem, we conclude it is compact under the pointwise limit topology and, since it is closed, we obtain it is compact. Relatedly, if we consider two configurations as equivalent if they differ by an integer — this is natural since this does not affect the variational principle — we can choose a representation with UQ € T 1 = R/Z. Hence, BU/TZ is compact. The next ingredient in the proofs is the heat flow. This is the solution of the equation

du

= Au-- S'(u) ~dt~~ Note that, formally, this is the gradient flow for the (strictly speaking, mean ingless) variational principle. There is no problem in showing that the heat flow is defined for all time since the R.H.S. of the equation is Lipschitz on C°°(Zd). We denote by $t(u) the solution at time t of initial condition u. An important result for the proof is the comparison principle

223

Lemma 3.1 Ifu > v then $ t ( u ) > $t(*>). The proof of this lemma is not difficult. The crucial observation is that the twist condition implies that the off-diagonal terms in the variational equations of the flow are non-negative. A simple calculation shows that the elements of the matrix of the varia tional equation are: — gu.g„\ • The non-negativity of this quantity is the twist hypothesis. Now we observe that comparison is true for linear equations with diago nal coefficients (they become exponentials) and for those who have positive entries — use the Taylor expansion of the exponential. Using Trotter product formula, it is true for sums of diagonal and positive entries, hence for the variational equations of heat flow. Once the result is true for the variational equations it is true for the full equations by interpolation. We refer to ( 4 ) for more details if needed. I We also note that nlCk$T = $TCk1ll. Hence, the heat flow of a Birkhoff configuration is Birkhoff. Note also that if u € (w,t) +l°° then A(u) + 5(u) € *°°, therefore $ T B W C Bu and, moreover, the heat flow projects to BwfR,. As remarked before, the heat flow is the gradient flow of the variational principle. But recall that since L(u) = L(lZlu) the variational principle is "defined" on the compact set B^/H. Hence, the gradient should vanish at one point in the set. Unfortunately, the previous argument does not work since the variational principle L is not "defined" in anything except a formal sense. So we need to regularize somehow and study the convergence of the regularizations. The first solution that we present is that in ( 4 ). For a cube A we define:

LA(«)=

(i/2)k-^| 2 -£ 5 ( u ')

£ i€A

i€A

We note that L\(u) is a well defined function in B^/71 continuous when BU/TZ is given the pointwise convergence topology. Moreover we have: |LA(*CU) = - ] T ( - A - S ' ( U ) ) 2 + t6A

where the 7i are "gradients."

Y, t€9A

J

>(U)

(4)

224

The intuition is that for large enough cubes the boundary terms should be ignorable. A moment's reflection shows that this should be the case, except when the bulk terms EigA are very small, but in this case, we should have a critical point! More precisely, Lemma 3.2 Assume that there is no solution of (3) in Bu then, we can find e > 0, N such that ifutBu |A| > N then £ i 6 A ( - A u + S'(u)) 2 > e. We show that the negation of the conclusion implies the negation of the assumptions. The negation of the conclusions is: • 3{uW}~ = 0 ,{A n }~ = 0 such that u<»> e Bu, |A n | -> oo,

• rAn(«(»)) = E
r ck „A„(c fc "uWHr A >W) we see that the sequence Cknu^ — consisting of configurations in Bw — makes arbitrarily small. Hence TA (u n ) —► 0 where A„ C A n + i . By passing to a subsequence (the Tychonov topology is metrizable in this case) we can assume that u^ -* u*. Since TA_ (u) > TA (u) for all u, we have r A „(ii ( n + m ) ) —» o and therefore T An (u*) = 0 for all n. Hence u* satisfies (3)

I

Corollary 3.1 In the assumptions of Lemma 3.2 we can find C > 0 such that r A (tt) > C(|A| - 1) V u € Bu Given a large enough cube, divide it in cubes of size N and apply to each of the cubes Lemma 3.2. ■ It is very easy to show that the terms J* are uniformly bounded, hence if there was no critical point, applying Corollary 1 to estimate the right hand side of (4) we have: ^A«T(«)

< - C [ ( | A | - 1) + lAI 1 -' 1 /")] V u € Bu

Hence, for big enough |A|, the right hand side would be negative. But this is impossible since L A is a continuous function in the compact set B^/TZ. We, therefore conclude that there is a critical point.

225

A second proof of the result of Theorem 3.1 can be obtained by observing that if w = (1/N)(ni,..., n^) with AT G N, r i i , . . . , n j € Z we can define LN(U)=

(l/2)|«i-«j|a- £S(u«)

Y,

(5)

\i\oo
In this case, the Birkhoff orbits are periodic and satisfy: U t i , . . . , t j + N , . . . , i d = W.,,...,»j,...,t d +

1j

This implies that

jtLN{u) = - J2 (-Au + ^tt))? Hence, we can conclude that the heat flow leads to a solution of 3 when

w eQ d . To prove the result for any u we observe that if we find a sequence uA") —♦ u with u^ € Bu* solving 3, (we can assume without loss of generality \UQ \ < 1). By the fact that u' n ) in in Bu for every j e Z d we know that Uj lies in an interval of size \j\ max n (w' n ',u) that is, it remains bounded. Hence, by passing to a subsequence, we can obtain that the u^ —► u pointwise. Since the u( n )'s satisfy (3) so does u. Remark 3.2 The proofs above do not need that the interaction is nearest neighbor. They go just through without changes assuming 1.

L(u) = £

HB{u)

(6)

BCZd

where HB(U) is a C 2 function which depends only on u\B and the sum is formal. 2. The twist condition: ^ " g ^ > 0 when i ^ j . 3. Translation invariance: H-RkB(1lku) 4. Periodicity: HB{du)

= HB(u)

= HB(u)

Vu,VB C Zd,fc € Z d .

Vu,VS C Z d ,/ € Z.

5. Some conditions of decay for HB when B is large.

226

Note that (6 ) is quite common in Statistical Mechanics. HB describes the interaction between the bodies in B. The conditions of translation invariance appear very frequently in Statistical Mechanics since it is a reflection that the interactions do not depend on the translations. The twist condition appears in statistical mechanics as ferromagnetism. As for the condition on decay, we will just say that it is easy to check that the conditions are satisfied when the interaction is finite range (i.e. HB = 0 when the diameter of B is large enough. ) Once we prove the result for LA(U) = 5Zscz d diamB
227 the equilibrium configurations are equicontinuous in small enough scales. (The equicontinuity in the large is still obtained from considering Birkhoff config urations as before.) For PDE's this can be obtained using elliptic regularity theory. Some equations to which the the above conditions apply are

Au + V'(x,u) = 0 where V(x + e,u + l) = V(x,u) Vi € R d ,u € R,e € Z d , / € Z k

Y,L2i+V'(x,u)=0 t=i d

where Li are Z periodic vector fields satisfying Hormander's hypoellipticity conditions and V is in the previous case. (~A)1/2u

+ V'(x,u)

(7)

with V as above. We note that the last equation is not a PDE, since (— A) 1 ^ 2 is a non-local operator. Nevertheless all the argument can be carried though. The comparison principle for the heat flow follows because e'^ - A ^ is convolution with the Poisson kernel, as it is well known in harmonic analysis, which is positive. Then, one can use Trotter product formula as indicated in the proof. (An alternative, more general, argument, suggested by C. Gutierrez is to use the subordination identity.) Problems involving (— A) 1 / 2 similar to (7) appear in several applications. Let us just note that they appear in the study of geostrophic flows and in Levy diffusions. One can also combine this treatment of PDE's with the previous remarks and obtain quasiperiodic solutions in other homogeneous spaces besides the torus. We need to assume that the fundamental group satisfies the condition of being residually finite.

■

4

Acknowledgements

The research of the author has been supported by NSF grants. Section 3 is based on joint work with A. Candel, H. Koch and C. Radin. Both D. Bambusi and an anonymous referee made suggestions which improved the exposition.

228

References 1. H. Brezis, Periodic solutions of nonlinear vibrating strings and duality principles, Bull. Amer. Math. Soc. (N.S.), 8, 409-426, (1983). 2. J. Moser, Minimal solutions of variational problems on a torus, Ann. Inst. H. Poincare Anal. Non Liniaire, 3, 229-272, (1986). 3. M. L. Blank, Metric properties of minimal solutions of discrete periodical variational problems, Nonlinearity, 2, 1-22, (1989). 4. H. Koch, R. de la Llave, and C. Radin, Aubry-Mather theory for functions on lattices, Discrete Contin. Dynam. Systems, 3, 135-151, (1997). 5. A. Candel and R. de la Llave, On the Aubry-Mather theory in statistical mechanics, Comm. Math. Phys., 192, 649-669, (1998). 6. S. B. Angenent, Monotone recurrence relations, their Birkhoff orbits and topological entropy, Ergodic Theory Dynamical Systems, 10, 15-41, (1990). 7. C. Gole, A new proof of the Aubry-Mather's theorem, Math. Z., 210, 441-448, (1992). 8. H. Brezis and L. Nirenberg, Forced vibrations for a nonlinear wave equa tion, Comm. Pure Appl. Math., 3 1 , 1-30, (1978). 9. W. Craig and C. E. Wayne, Newton's method and periodic solutions of nonlinear wave equations, Comm. Pure Appl. Math., 46, 1409-1498, (1993). 10. M. E. Taylor, Partial differential equations. Ill, Springer-Verlag, New York, (1997), Nonlinear equations, Corrected reprint of the 1996 original. 11. J. N. Mather and G. Forni, Action minimizing orbits in Hamiltonian systems, In Transition to chaos in classical and quantum mechanics (Montecatini Terme, 1991), pp 92-186. Springer, Berlin, (1994). 12. M. L. Blank, Chaos and order in the multidimensional FrenkelKontorova model, Teoret. Mat. Fiz., 85, 349-367, (1990).

T H E SPLITTING OF INVARIANT L A G R A N G I A N SUBMANIFOLDS: GEOMETRY A N D D Y N A M I C S JEAN-PIERRE MARCO Institut de Mathematiques de Jussieu, Universiti Paris 6, 175 rue du Chevaleret-75013 Paris, Prance. B-mail: [email protected] We give precise definitions for the splitting of invariant manifolds, in an abstract setting, both from the point of view of Euclidean and symplectic geometry. We then discuss the relevance of these notions in the problem of heteroclinic connections between nearby tori, and in the estimation of transition times along chains of hyperbolic fixed points.

1

Introduction

This paper is intended to clarify the basic definitions and properties, in higher dimensions, of the so-called "splitting of separatrices" of a hyperbolic fixed point for a symplectic map in the plane. It may be viewed as an introduction to the paper 4 , where the properties of the splitting of stable and unstable manifolds of a hyperbolic invariant torus are investigated both by symplectic and analytical methods, and then applied to the problem of estimating the transition times along a chain of such tori. The content of the present paper is rather elementary but the geometric features of the splitting do not seem to have been yet investigated in any detailed way, and we find it necessary to establish firmly the grounds of this theory before getting into the quite involved and unavoidable calculations. We give in the two first paragraphs the definitions and basic properties of the "angles" between two Lagrangian submanifolds of a symplectic manifold. Paragraph 1 is devoted to the linear case: we examine first the natural and primitive notion of symplectic angle of two Lagrangian subspaces in a sym plectic vector space: being quite easily computable it proves to be the fun damental notion for our purposes, in spite of its weak geometric significance. We introduce then the refined notion of Euclidean angle, which necessitates a scalar product in addition to the initial symplectic form and encapsulates the entire geometrical information. We refer to 4 for a presentation of the transformation formulas for these two angles. The globalization of these angular notions is given in the second para graph, we study first the case of submanifolds of general symplectic manifolds (with an almost-complex structure for the Euclidean notions), and then limit the discussion to the case of cotangent bundles, the most important one being

229

230

of course the cotangent bundle T*Te of the ^-dimensional torus. In the third paragraph we specialize the preceding notions to the case of the stable and unstable manifolds of a hyperbolic torus invariant under a Hamiltonian flow. We need first some definitions borrowed from the hy perbolic theory of dynamical systems: oddly enough, the classical definitions of (partially) hyperbolic tori usually involve special normalizations in their neighborhoods; we find it unsatisfactory and take here the opportunity to fill this gap and define the hyperbolicity of an invariant torus by means of invariant decompositions of its tangent space, adapted to the hyperbolic fea tures of the linearized system. The existence of invariant stable and unstable manifolds for hyperbolic tori can then be deduced from the general theory of hyperbolic systems. One can prove furthermore that, under the natural assumption that the hyperbolic torus be isotropic, these invariant manifolds are Lagrangian (at least if the necessary condition on their dimension is met); their splitting is thus naturally defined following our preceding general study. The last paragraph illustrates two dynamical interpretations of the split ting quantities, mainly in the problem of transition chains (see 1). The first question is to pass from the homoclinic data to heteroclinic ones, the easiest problem being to find the maximal distance between two heteroclinically con nected nearby tori along a simple resonance surface. We show that, due to the existence of a natural horizontal and vertical decomposition of the space, the relevant quantity in this problem is the symplectic angle of the invariant man ifolds. We address finally the question of transition times along a given chain of hyperbolic objects. In the case of chains of hyperbolic tori in Hamiltonian systems, this problem is closely related with the question of the optimality of the so-called Nekhoroshev exponents, we refer to 2 and to forthcoming work for a general study. Here we limit ourselves to the problem of transfer around a fixed point for a symplectic map, i.e. the number of iterates needed for a given submanifold, transverse to the stable manifold of the fixed point, in or der to intersect a given ball centered on the unstable manifold, with controlled C 1 distance to the unstable manifold. This is thus a quantitative version of the usual A-lemma, which proves to be very useful for the computation of transition times along chains of hyperbolic objects.

2

Angles of Lagrangian planes

Let (W, B) be a vector space endowed with a nondegenerate bilinear form, and denote by 0(W, B) the orthogonal group relative to the form B, i.e. the

231

space of all linear isomorphisms satisfying B(cj>(u), 0(i/)) = B(u, v),

V(u, v) E W2.

The d-Grassmannian G<j = Gd{W) is the set of all d-dimensional linear subspaces of W. Given a pair (Vi, V2) of elements of G<j, we define the angle AB{V\, V2) of (Vi, Vb) relative to B as the orbit of the pair under the diagonal action of 0(W, B) on G j x G
= {(4(Vi),*(K 2 )) | S

0(W,B)}.

Suppose now that (W, fi) is a symplectic vector space of dimension 2£, and recall that a subspace V of W is said to be : - isotropic when the induced two-form iyQ vanishes, or equivalently when V is contained in its symplectic orthogonal, - Lagrangian when it is isotropic and ^-dimensional, or equivalently when V is equal to its symplectic orthogonal. We examine in this paragraph the properties of the angles of Lagrangian subspaces of W, relative first to the symplectic form fi and then to an addi tional Euclidean form on W, satisfying some compatibility conditions w.r.t. the symplectic form ft. To begin with, we recall the basic definitions con cerning the variety of all Lagrangian subspaces, together with the classical construction of coordinates over particular open sets. 2.1

The Lagrangian- Grassmannian.

It is well-known that one can identify W with R2*, with coordinates (m,Vj), in such a way that the symplectic form fi reads ft = J2i=i dviAdui. We shall explicitely do this identification below and denote by Ho (horizontal) and Vo (vertical) the Lagrangian subspaces R' x {0} and {0} x R* respectively. The set of all Lagrangian subspaces L of R2e will be denoted by A(R2*, ft), or simply A. The symplectic group Sp(2£, R) acts transitively on A, and the stabilizer of Ho is the subgroup C C Sp(2£, R) of all symplectic lifts of linear isomorphisms of H 0 . So C ~ Gl{£,M.) and the Lagrangian-Grassmannian A is thus an algebraic variety of dimension £(£ + l ) / 2 which is isomorphic to the quotient Sp(2£,R)/Gl(£,R); we will see another identification below. The construction of local coordinates on the manifold A is classical. For getting first about the Lagrangian character of the spaces, consider a pair of ^-dimensional transverse spaces H and V of R 2 ' and denote by pn and py the projections on H and V associated with the direct sum R 2 / = H © V. Let then L be an £-dimensional subspace transverse to V. The restriction to L of

232

the projection pa is an isomorphism of L onto H, by transversality of L and V; denote its inverse by q : H —> L. Then L is the graph of the linear map JL = Pv °Q from H to V (identifying H© V with H x V); we say that the linear map JL is the equation o/L relative to H and V. Since L = {a+jt(a) \ a G H}, where a and jh(o) are considered as elements of R2t, one may also consider the map lL : H —♦ R 2 ' defined by li,(a) = a+ji,(a), giving a linear embedding of H onto L which we call the natural embedding for L, relative to H and V. Assume now that H and V are Lagrangian subspaces of R2e. Note that the dual space H* of H can be naturally identified with V, by means of the symplectic form: for v G V, simply define v*(u) = Q(u,v) for u G H, and use the fact that V is Lagrangian to prove that this gives rise to an isomorphism. Given an ^-dimensional subspace L (not necessarily Lagrangian), one can now define a bilinear form on H through the relation 7 L (a,6)

= n(a, jL(&)),

a G H, 6 G H;

the form TL is the cooordinate of L relative to H and V. Using the identification of H* with V, this is the classical identification of a linear map from H to H* to a bilinear form on H; so the correspondence L — i > TL is one-to-one. The crucial property is that 7L is symmetric if and only i/L is Lagrangian. Indeed note that n{a + jL{a),b + jL(b)) = n(a,j L (&)) + n(j L (a),&) since H and V are Lagrangian, and the symmetry of 7L is thus equivalent to the vanishing of fi on L. Let us denote by Ay the set of all Lagrangian spaces which are transverse to the space V; clearly Av is an open subset of A (classically refered to as a "Schubert cell"). The correspondence L t-> TL allows us to identify Ay with the space 5(H) of symmetric bilinear forms on H. Since «S(H) is isomorphic to R'( / + 1 )/ 2 , this identification can be considered as a coordinate map over the open set Av- Varying V and H, we obtain in this way an atlas for the Lagrangian-Grassmanian A. The following remark will be useful in the sequel. We have seen that V is naturally identified with the dual space H*; on the other hand one can consider the quadratic form Q L on H associated with TL (i.e. QL{O) = 571,(0,0)), and its derivative dQ^ from H to H* = V. Then one finds that

dQh(a)(b) = T L M ) = 7 L ( M - 0(6, jh(a)),

233

so that dQ^a) is identified with JL{O)- In other words the Lagrangian space L is also the graph of the derivative of the quadratic form QL- The form Q L is said to be a generating function for L. 2.2

The symplectic viewpoint.

The first remark concerning the possibility of denning symplectic angles is disappointing: the action of the symplectic group on A2, which would be the most natural thing to consider in order to define symplectic angles, does not lead to an interesting notion. Indeed, given two pairs (Li,L2) and (Lj,L 2 ) of Lagrangian spaces, a necessary and sufficient condition for the existence of a symplectic isomorphism ip of R2e such that ip(L\) = L'x and ^(L2) = L 2 is simply that the intersections Li n L2 and L\ n L'2 have the same dimension. In this respect this is no different from the action of the full linear group Gl(2£, R): the symplectic angle of two Lagrangian spaces is just the dimension of their intersection. One thus has to refine the primitive notion, which we do by considering linear Lagrangian embeddings, instead of abstract Lagrangian subspaces. This leads to the Definition 1 Consider two linear Lagrangian embeddings li and I2 of two (.-dimensional vector spaces E\,E2 into R2*. The symplectic angular form of the pair (li,h) is the bilinear form A, on E\ x E2, defined by At(a,b) = n(li(a),l2(b)),

aeEu

b e E2.

The geometric significance of the angular form A3 is quite weak and does not go much beyond the following transversality result, whose proof is straight forward. Proposition 2 Let Lt = li(Ei), for i = 1,2. Then KenA.

:= {a € Ei | A.{a,b) = 0,V6 € E2) = ff'fL, n L 2 )

(with an analogue for Ket2At). The spaces L\ and L2 are thus transverse in R2< if and only if the symplectic angular form is non-degenerate. The form As cannot be defined for a pair of Lagrangian subspaces (work ing with embeddings amounts to fixing bases for the two spaces), but one encounters frequently situations where such embeddings are "natural". The most important one is given by the previous construction of coordinates on A, when a decomposition R 2 ' = H © V is given from the start. The corre spondence L H-» ZL identifies Ay with the set of linear Lagrangian embeddings of H into R 2 ' whose ranges are transverse to V. As a consequence, if a pair

234

(Li,L2) in Ay is given, one can define as above the symplectic angular form of the associated pair of embeddings (l^ ,IL3)Definition 3 Let (Li,L/2) be a pair of elements of Ay. The splitting form of the pair (Li,!^) relative to the subspaces H and V is by defi nition the symplectic angular form As of the two natural embeddings l^ and ZL3 relative to H and V (here of course E\ = E2 = H^. Given a basis B H of H, the matrix S in B H of the bilinear form CTHV(LI, L2) is the splitting matrix of the pair (Li,L2), relative to H, V and B H . CHV(LI,L2)

As one would expect, the splitting form is directly related to the coordi nates relative to H and V: P r o p o s i t i o n 4 Let (Li,L2) be a pair of elements of Ay. The splitting form of the pair (Li,L2) is given by the difference of the coordinates ofh\ and L2 relative to H and V: CTHV(LI,L2)

=7L3 - 7 L l -

It is thus a symmetric form on H. The proof is a simple calculation. One can also consider the splitting form as a quadratic form on H and the preceding proposition shows that it is also (twice) the difference QL 3 — Qhi of the generating functions of Lx and L 2 relative to H and V. <7HV(LI,L2)

One would like now to interpret the splitting form in a more quantitative way. As far as transversality properties alone are concerned, one finds as before that Li and L2 are transverse in R2e if and only if the splitting form is non-degenerate, with dim(Li n L2) = dimKer ( O " H V ( L I , L 2 ) ) - But even in R 2 , where each line is a Lagrangian subspace, there is no relation between the splitting form of two lines and (for instance) their usual angle. An element L € Ay0 is given by an equation J'L(°) = 7fli ^vith 7 6 R and a € Ho = 1 (so its usual equation reads y = 7 1 ) , and the form 7L associated with L is simply given by its matrix [7] relative to the basis e\ = 1 of R. The splitting matrix of a pair (Lj, L 2 ) of A Vo , with equations j ^ k (a) = 7* a (k — 1,2), is thus just the difference [72—71]- The angle is 6 = t a n - 1 (72)—tan - '(71) € ] — n/2,w/2], (mod 7r) and it is for instance possible to make 6 go to infinity while keeping the matrix [72 - 71] constant. By contrast, the knowledge of both coordinates TLJ and TL 2 completely determines the relative positions of the two subspaces. The best way to see this will be to introduce first the notion of Euclidean angles for Lagrangian

235

subspaces, and then to show how to compute their determinations from these data. 2.3

The Euclidean viewpoint.

We now assume that a compatible complex structure is given on the ambient symplectic vector space (W, Q), meaning that W is identified with Ce, with coordinates IU, = Uj +ivj. The usual Hermitian form 2 is given by i

2(tu, w') — > J uij w'j = n(to, to') + i £l(w, w') i=i

for w and w' in C*, where II is the usual scalar product of R 2 '. The identifi cation W s; R 2 ' ~ C* is explicitely used below. The three groups relative to the forms 2, II and fi are of interest in the study: they are respectively the unitary group U(£,C), the Euclidean orthogonal group 0(2£, R) and the symplectic group Sp(2*,R). We recall the following useful relations: U(£, C) = 0(2*, R) n Sp(2£, R) = Gl(£, C) n 0(2*, R) = Gl(i, C) n Sp(2£, R). Using the expression of 2, one sees that an ^-dimensional subspace L is Lagrangian if and only if every R-basis of L, orthonormal for II, is a unitary C-basis of C ' for the hermitian complex structure. In this case, the union B U iB is a symplectic II-orthonormal basis of R2*. This is also equivalent to the equality i L = Orthn(L), where Orthn(L) is the ^-dimensional subspace II-orthogonal to L. As a consequence, the unitary group U(£, C) acts transi tively on A, and the stabilizer of the subspace Ho is 0(£, R). The manifold A is thus also isomorphic to the quotient U(£,C)/0(£, R). Let Gf denote the Grassmannian of all ^-dimensional subspaces of R 2 / . Given a pair (Li, L2) in G 2 , one defines the Euclidean angle of (Li, L2) as the orbit of (Li,L 2 ) under the diagonal action of the orthogonal group 0(2*, R) on G 2 , so the space of Euclidean angles is the quotient space G 2 / 0 ( 2 * , R ) . This notion is valid in particular when (Li, L2) is a pair of Lagrangian spaces, but in that case the intersection of the orbit of the pair with A2 is exactly the orbit of the pair under the diagonal action of the unitary group U(£, C) on A2: if ip € 0(2*, R) transforms a Lagrangian space into another one, tp is indeed unitary. As we are only concerned with the study of Lagrangian objects, the following definition is thus natural: Definition 5 Let (Li,Lj) be a pair of Lagrangian subspaces ofCe. The Eu clidean angle A(Li,L2) o/(Li,L2) is the orbit of the pair under the diagonal

236

action ofU(6,C) on A 2 . The space of Euclidean angles of Lagrangian spaces in R2t is thus the quotient A2/U(t,C). The central result of this paragraph is the following Proposition 6 Let (Li, L2) be a pair of Lagrangian subspaces of R 2< . There exists a unique unitary isomorphism VL1L3 G U(C, C), diagonalizable relatively to a unitary basis contained in Li, with eigenvalues of the form ete, 9 € ] — 7r/2,7r/2], such that ¥>LIL 2 (LI) = L2. If is a unitary isomorphism with L; = <£(L<) for i = 1,2, then tph>ih>2 =<j>o tpuu ° 4>~XThe proof is based on the following result, useful in itself. Proposition 7 Let L be in A, transverse to VQ. There exists a unique unitary isomorphism tp such that tp(H0) = L, diagonalizable relatively to a unitary basis contained in Ho, with eigenvalues of the form e10, 6 e ] — n/2,ir/2]. If 7 w the coordinate o/L relative to Ho and Vo, tp and 7 are diagonalizable in the same unitary basis contained in Ho, and if a is an eigenvalue 0 / 7 , the corresponding eigenvalue for tp is given by p = (1 + ia)/ \l + ia\. Let us say some words about the proof. It is useful to note first that given a Lagrangian space L, the equation JL relative to Ho and Vo extends uniquely to C ' as a complex linear map, i.e. there exists a unique complex endomorphism / of Ce such that f(a) — ji,(a) e Vo, for all a G Ho. One easily sees that / has the form i JL> where J L is a self-adjoint endomorphism of C* sending Ho into HQ. This simply means that the matrix of Ji in a unitary basis BH0 of C ' contained in Ho is real and symmetric: indeed this is just the matrix of 71 relative to J5H0 • The map I + iJi, is thus clearly the unique extension to C ' of the embedding l^ as a complex isomorphism, sending Ho onto L. To prove the lemma, consider the self-adjoint endomorphism J = J L , let f?H0 = (0.1,...,at) be a unitary basis of eigenvectors of J, contained in Ho, and let a^ be the (real) eigenvalue corresponding to a*. Then the a^ are also eigenvectors for I + iJ and I — iJ, with eigenvalues 1 + i ctk and 1 — i a* respectively. The product P = (7 + iJ) o (/ — iJ) is thus symmetric and positive definite, with eigenvalues 1 + a\. Consider the square root R of P denned by the diagonal matrix diag(^/l + a\,..., ^/l + af) in the basis BH 0 I and define tp = (I+i J ) o . R - 1 . Since R maps Ho onto itself tp sends Ho onto L, and tp is clearly unitary, with eigenvalues (1 + i a*)/11 + i otk\ of the required form (i.e. with Ok e ] — 7r/2,7r/2[). This proves the existence of tp when L is transverse to Vo. Since the matrix of 7 in BH0 is the same as the one of J , the second assertion is also proved.

237

As for uniqueness assume (p is a unitary map as in the proposition. Then consider a unitary basis JE?H0 = (a\,...,ae) of eigenvectors of ip, contained in Ho, and let pk be the (complex) eigenvalue corresponding to a*. One sees that pk ak is in the intersection of the complex line C a* with L2, which is thus at least one-dimensional, and exactly one-dimensional since L is Lagrangian. This condition determines pk in the required form and this proves the unique ness of f. The lemma is proved. One can easily extend the proof to the case L € A and non transverse to Vo (see 4 ) . Finally to prove the proposition, note that if <po is a unitary isomorphism mapping Ho onto Li, the set U(Li, L2) of unitary isomorphisms sending L\ onto L2 is given by U(U, L 2 ) = {ifo o ip o v~l

I

= 9 + v,

(45)

and the angular velocity <\> can be found, if necessary, through the angular momentum G = <^>r2. Neither the classical orbital elements nor Delaunay elements are well de fined on circular orbits because the argument of the pericenter is not denned. If we give initial conditions (in rectangular coordinates) for a circular orbit, then there exist arbitrarily small variations in these initial values which will cause the argument of the pericenter g to vary from — n to 7r. The same applies to any angular variable having the pericenter of the orbit as origin, i.e. any of the anomalies. In contrast to that, the magnitudes e sin v and e cos v are well denned and depend smoothly on the variations of the initial conditions. The same can be said about e sin £, e cos I, e sin E, e cos E. In the transformation from polar coordinates and momenta to Poincare elements, direct use of the angular variables £, v, E will be avoided, and pairs such as (e sin E, e cos E) will be used instead. Notice that in each one of the pairs (esin^,ecos^), (e sin u,e cost;), (esinE,ecos.E), each component can be expanded as a power series in the components of any other pair. The differences v — E, E — £ and v — £ can be expanded in the same way as well. We will quote a few of these expansions that will prove useful in what follows. A standard reference for the subject is 1. ^From Kepler's equation (42) the following expansions can be derived in a variety of ways esin^ = esini? — esini? ecos.E + 03(esinE,

ecosE),

e cos £ = e cos E + (e sin E)2 + 03(e sin E, e cos E),

(46) (47)

342

e s i n £ = esin^ + esin^ ecos^ + ( ^ ( e s i n ^ e c o s ^ ) , 2

e c o s £ = e c o s £ - (esin£) + 0 3 (esin^,ecos£).

(48) (49)

The difference v — E can be expanded from (41) as v — E = -esinE

+ - e s i n £ ecosi? + C>3(esmE,ecosE),

(50)

and the difference E — i is just given by Kepler's equation (42). We will give explicitly the change from Poincare variables to polar co ordinates and momenta. Poincare variables A, A, £, 77 are defined in terms of Delaunay elements by the following set of formulas (see 6 ) A = L,

(51)

A = * +
(52)

i = ^2(L-G)cosg,

(53)

r) = -y/2(L-G)sing.

(54)

We look now for the formulas that change from polar coordinates and momenta to Poincare variables. We have in mind a particular case: that of a circular orbit of radius unity and angular momentum unity; the trajectory is given by r = 1, = t, R = 0, $ = 1 in polar variables and by A = 1, A = t, £ = 0,77 = 0 in Poincare variables. As we want to study a neighbourhood of this orbit, we will use the variables 6r,6R,8$,6A, defined by r = R = $ = A=

1 + fir, 8R, 1 + <5$, 1 + <5A,

and work out the formulas for the transformation. i,From the relation of the energy 2A2

2 V

rf)

r'

we readily obtain A as A=(-J?2+2/r-$2/r2r1/2. i,From equation (43), we obtain the following expression for ecosE ecosE = - l + r.R2 + $ 2 / r ,

343

and from (44) esinE = ifrA - 1 so that we can write e sin E and e cos E as power series in 6r, 6R, 6$. Now, according to (52) and (45), we have A = (P + £-E

+

E-v

whence we obtain, using (42) and (50), \ = +

Si{6r,6R,6$),

where S\ is a series in the indicated variables with no constant term. Finally we will find expressions for £ and T) as functions of the polar coordinates and momenta. Notice that, according to the definitions (53) and (54), those variables are not essentially different from e cos g and —e sin g. As g — A — £, we can now write e cos g = cos A e cos £ + sin A e sin £

(55)

e sin g = sin A e cos I — cos A e sin I

(56)

and using (55), (56), (46), (47), we can expand the right hand terms as power series in Sr, 6R, 6$ whose coefficients are trigonometric expressions in . Now from (53) and (54) $ = v^A(A + $ ) _ 1 / 2 e c o s p , 77 = -\/2A(A + $)~ 1 / 2 esinp, and we finally get series expansions for £ and 77. In all the expansions found, <j> appears as the argument of trigonometric expressions which are the coefficients of a power series in 6r, 6R, 6$. In the final expression, we will take <j> + 8

and expand with respect to 6

e+r?=l{L-G), whence * = A-\(S2 + V2).

(57)

344

From the same expressions we can write ecos<7 =

-LsA-\2A-\(e + W, l

2

2

1 2

esing = --L 7? A- (2A-i(C + ?3 )) / , and the right hand sides can be expanded as power series in 6 A, £, 77. Invert ing (55) and (56) we get esin^ and ecos^ as power series in the mentioned variables and coefficients trigonometric polynomials in A, and from (48) and (49) we eventually find esini? and ecosE, again as series of the same type. From (43) and (44) and what we already know, we get the needed expansions for r and R. The expansion of <j> follows a similar reasoning using 4> = X + v-E

+

E-£,

and expanding v — E and E — I as before. The above procedures will yield the series up to any order if one takes care to expand to the required order in each step. The result for polar coordinates and momenta as functions of Poincare variables is, up to second order and expanding with respect to 6X, r = 1 + 2<5A — cos A £ + sin A 77 3 3 + 6A 2 - - cos A 6A£ + - sin A 6A77 + sin A 6A£ + cos A 6Xr) + sin 2A £77 + sin 2 A f2 4- cos2 A T72, 0 = A + <5A + 2sinAf + 2cosA77 — sin A <5Af - cos A 6Arj + 2 cos A 6X£ 5 5 5 — 2 sin A <5A?7--- sin2A 772 + - cos2A £77 + - sin2A £ 2 , 4 2 4

3

3

R = sin A £ + cos A 77 — - sin A 8A£ - -- cos A 0 A77 + cos A 6X£ ~ sin A 6XT] + 2 cos 2A f 77 + sin 2A(f2 - 772)

$ = 1 + 6 A - U2 Poincare variables in terms of polar coordinates and momenta are given by A = 1 + 6$ + ~6r2 -- 2<5r6$ + i<5i?2 + 25$ 2 , X = (p + 64>-- 26R - hrSR + 6R6$, £ = cos 4>(-6r + 2<5$) + sin^> 6R

345

+ sin 4>{8r8(j)--26cf>8$ +UR6*) + cos <j>(6r2 + 26$2 --6r6^ -

+ 6(f>6R),

r} = sin4>(6r - 2<5$) + cos

+ cos - -2(506$

+ pR6$)

- sin (6r2 + 2<5$2-hr6$ -

+ 8
We finally give some of the terms z^k'^(6z,t) in the expansion of the flow near a circular orbit as a power series in the small variations of the initial conditions. Let V be the transformation from polar coordinates and momenta to Poincare variables as computed above, that is, (A,A,C,r?)=P(r,^fl,$), and let .^(Ao, A0,fo,77o) be the flow of the Kepler problem in Poincare vari ables, given by A(t) = Ao, \(t) = Ao +

AQ

3

t,

Z(t) = &, v(t) = %• We are interested in the flow in a neighbourhood of the circular orbit r = l,<j> = t,R = Q,$ = 1. If we take initial values 1 + Sr, 6, 6R, 1 + 6$, then the flow in polar coordinates is given by x{V(6x,t)

= V-1 TtV{\ + 6r,6cp,6R, 1 + «*).

The zeroth and first order terms in the £'s of this flow are r(o,o) = 1 ;

0
tf

^(0,0)

= Qt $(o,o) =

1(

and r (o,i)

=6rcost

^(o.i) = -26rsmt RW 0

= -6rsmt

& 'V = 6$.

+ 6Rsint + 6$(2-2cost), + 6

(58) + 4smt),

(59) (60) (61)

346 We will use a subscript star to indicate that the particular initial values 6(f> = SR = (5$ = 0 are taken, so that only 8r ^ 0. In this case we have ri°' 2 ) = 6r2(l - cos* + sin 2 t),

(62)

4 ° ' 2 ) = 6 r 2 ( - - t + 2 sin t + - sint cost),

(63)

R[°'2)

$i°'

2)

= Sr2 {sin t + 2 sin t cost),

(64)

= 0.

(65)

The flow j/°)(2/o,t) which describes the unperturbed motion of P^ with initial values j/o — (^A &9, L + 6L, G + 5G) is t(t) = 6e + (L +

6L)-%

9(t) = 69, L{t) = L + 6L, G(t) = G + SG The only term in /x which will be relevant to the problem is R^'°\ matter of fact, the terms x^1'0^ are given by

x^{t)

= M*(o)(i) j T

As a

X^irWiiz^ir)), 1 0

and from this formula we get the following expression for itf - ) i? (1 ' 0) (t) = iisint f + fxcostf

(-VP(T)<°>

sinr - 2vj,0) (r) cosT) dr

(-VW(r)cosr

+ 2Vj, 0 ) (r)sinr) dr,

where Vr (T) and Vi (r) stand for the partial derivatives of V evaluated on the unperturbed solution z^0,0^(t). In order to solve the equations (27)-(29) we write the above expansions with initial values 8<j> = 0, 6R = 0, 6$ = 0, 81. = 0, 8g = 0, 8G = 0, set t = Tk+8T and expand in 8T. We indicate these operations by a superscript star and quote the results, using generic constants a* throughout to stand for quantities whose value is not relevant to Theorem 2. The expansion for the angular variable

0*(6T,6L,6r,M) = kit + ST + aui - — Sr2 + ST 0(6T, 8L, Sr) + /10{8T, 6L, Sr, /x),

(66)

347

where Ok stands for a series in the specified variables beginning with terms of degree at least k. We remark that in equation (66) there are no linear terms in either SL or Sr and there are no quadratic terms in either SL2 or 6L6r. Maybe a few comments on the nature of these terms will be helpful to get some insight into the physics of the problem. Terms in 6T are natural to appear just because an expansion around Tk has been made. The absence of terms in SL (both linear and quadratic) is clear because any change in the orbit of P2 could possibly affect the motion of Pi only through the gravitational interaction, that is, they must appear multiplied by p.. The absence of a linear term in Sr is due to the fact that the expansion of the energy (the value of Hamiltonian (5) when c\ = a2 = 1 and \i = 0) just happens to have no term linear in Sr and so neither has the period. The expansion for g, the other term in equation (27), is g*(ST, SL, Sr, p) = a2p + p. 0(ST, SL, Sr, p).

(67)

As for equation (28), we can write £*{6T, SL, Sr, p) = L^ST

- ZL^TkSL

(68)

+ a3p + 02(ST, SL) + p 0(6T, SL, Sr, n) Finally, we give the expansion of equation (29) R*{6T, SL, Sr, n)=C/i-

SrST + nO(ST, SL, Sr, ft) + 04(Sr)

(69)

where C is given by the following expression C = (-1)* /

(-VJ:0){T)cosrdr

+ 2V(J?)(T) s i n r j dr.

(70)

It is easily seen, even though they haven't been calculated, that there are no terms in Sr3 in the expansion of R*. First and second order terms are known from (60) and (64) so that, if Sr ^ 0, we can write Ri°\t)

= -Srs\nt

+ 6r2{sint + sm2t) +Sr3p(t) + 04(Sr),

where p{t) is an analytic function; now if we take t = kn we get R(Q)*(kn) - Sr3p{kw) + 04(Sr) but on the other hand we know that fi(°)*(fc7r) = 0 for any value of Sr because at time t = k-n the planet P2 is at an apsis of its orbit and so its radial velocity R is 0, which implies p(kn) = 0 and justifies the absence of 0$(8r) in equation (69).

348

Acknowledgments The author is partially supported by a DGES grant number PB96-1153. References 1. Brouwer and Clemence, Methods of Celestial Mechanics (Academic Press, New York, 1961). 2. P. Guillaume, Families of Symmetric Periodic Orbits of the Restricted Three Body Problem, when the Perturbing Mass is Small, Astron. Astrophys. 3, 3-57 (1969). 3. K. R. Meyer and G. R. Hall, An Introduction to Hamiltonian Dynamical Systems (Springer-Verlag, New York, 1991). 4. H. Poincare, Les Methodes Nouvelles de la Micanique Celeste, 3 Vols. (Gauthier-Villars, Paris, 1892-1899, reprinted by Dover, New York, 1957). 5. D. Schmidt, The family of direct periodic orbits of the first kind in the restricted problem of three bodies, Ordinary Differential Equations (Aca demic Press, 1972). 6. V. Szebehely, Theory of orbits (Academic Press, New York, 1967).

T H E P H A S E SPACE OF F I N I T E S Y S T E M S

KURT BERNARDO WOLF Centro de Ciencias Fisicas Universidad National Autdnoma de Mexico Apartado Postal 48-3, 62251 Cuernavaca, Morelos E-mail: [email protected] NATIG M. ATAKISHIYEV Institute de Matemdticas Universidad National Autdnoma de Mexico Av. Universidad s/n, Col. Chamilpa, 62191 Cuernavaca, Morelos E-mail: [email protected] SERGEY M. CHUMAKOV Departamento de Fisica, Universidad de Guadalajara, Mexico Czda. M. Garcia Barragdn y Olimpica, Sector Reforma, 44281 Guadalajara, Jalisco E-mail: [email protected] The Newton equation of a shallow optical waveguide (a harmonic oscillator) is compatible with the algebraic structure of SU(2). Wavefunctions satisfy orthog onality relations with respect to a discrete measure of 2( + 1 points, of position, momentum, or mode number. The Wigner function for SU(2) in the representation t devolves the classical 'c-number' variables of position, momentum, and energy, picturing the state —a (21 + l)-dimensional vector of sensor data— on the surface of a sphere of radius t+ \- The three coordinates are canonically conjugate in this SU(2) sense. Linear SU(2)-optical systems rotate the sphere rigidly, while others deform it nonlinearly.

1

I n t r o d u c t i o n : discrete a n d finite s y s t e m s

Parallel processing of finite data sets by optical means can achieved doping a strip on a transparent substratum with channels that are planar multimodal waveguides. If a finite set of coherent light-emitting devices modulate a com mon field by factors {/m}m=o, propagation along the waveguide to an array of field sensors will correspond to the fractional Fourier transformation on this data set. This is because the waveguide is the (first-order) optical analogue of the quantum harmonic oscillator, which is well known to rotate phase space. When T is the axial oscillation wavelength of the guide, the Fourier transform occurs at | r , inverted imaging occurs at | r , and the inverse Fourier transform for \T = —\T. These considerations led us in Ref. l to propose a Fourier-

349

350

Kravchuk transform based on an oscillator model 2 with an su(2) Lie algebra of difference operators; the Kravchuk 3 functions were further investigated in Ref. 4 . Since the SU(2) group is compact, the system is discrete and finite. The quantum harmonic oscillator is a very well-known Hamiltonian sys tem (indeed!); the phase space representation of its wavefunctions can be clearly depicted with the Wigner quasiprobability distribution function 5 . One of us had proposed a generalized definition of the Wigner function on the full Heisenberg-Weyl group for polychromatic optics 6 , and the group SU(2) became a natural testing ground for the proposed generalization 7 . It has also provided a phase space picture for discrete, finite systems. In Sec tion 2, we base our construction on the oscillator Newton equation; perhaps surprisingly, several algebraic structures are consistent with it 8 , among them the finite oscillator model of the shallow waveguide. In Section 3 we turn the finite oscillator into a Hamiltonian system through identifying the dynamical variables with the generators of SU(2) in any fixed (and well-known) finite-dimensional irreducible representation space, and in Section 4 we characterize the eigenfunction basis of the Hamil tonian through Kravchuk functions. In Section 5 we present a digest of the generalized Wigner function for SU(2) introduced in Ref. 7 , which gives opera tional meaning to a so-defined 5ft3 'meta-phase' space, two of whose 'c-number' coordinates are those of 'ordinary' phase space, and the third is the mode number of the oscillator. Section 6 recounts the reduction of this 5R3 function to a function on the sphere; this sphere we regard as a proper phase space for finite systems. On it we can plot the face of a finite signal and follow its evolution along a waveguide, and through prisms and lenses of various kinds. The concluding Section 7 condenses the authors' present lines of research on SU(2) optical systems. 2

Newton's equation for the finite waveguide model

In the classical one-dimensional harmonic oscillator potential V(q) = \q2, a particle of unit mass obeys Newton's equation q = —q (the dots are time derivatives); geometric optics uses the same equation for the transverse ray co ordinate across a waveguide, and where the evolution parameter is the length along the optical axis. As does quantum mechanics, paraxial wave optics assumes that there exists an operator Q whose eigenvalues q represent position, and replaces time derivatives by Lie commutators with a Hamiltonian evolution operator H (times i). With these assumptions, Newton's equation becomes the Lie-

351 Newton equation [H,[H,Q]) = Q.

(1)

The commutator of the Hamiltonian and position operators is by definition (—i times) the momentum operator (whose eigenvalues provide the momen tum coordinate), and so the Lie-Newton equation (1) becomes the simultane ous pair of Hamilton equations [H,Q] = -iP,

[H,P} = iQ.

(2)

So far, we have not specified what [P, Q] is. And indeed, more than one algebraic structure is consistent with the Newton equation (1). To close into a Lie algebra, the three operators Q, P and H must satisfy the Jacobi identity, from which it follows that [H,[P,Q]] = 0

=»

[P,Q] = if(H),

(3)

allowing any function f of H (including constants and/or central operators). The requirements that [P, Q] be within the algebra and also preserved under Hamiltonian evolution, are satisfied by the following two distinct Lie algebras: oscillator algebra [P, Q) = iA, SU(2) algebra [P, Q] = iH,

Eq. (2) and [A, Q) = [A, P] = [A, H] = 0, [Q, H] = iP, [H, P] = iQ.

Also, two distinct q-algebras are consistent with the oscillator Newton equa tion 8 , but we shall not address these here. 3

The finite oscillator on the SU(2) group

In the familiar oscillator algebra, Q and P generate the Heisenberg-Weyl subalgebra, with A; their spectra are 3? and serve as classical phase space coordinates. The eigenvalue of A labels irreducible representations: ft for quantum mechanics, X/2n =/= 0 for (polychromatic) paraxial optics. In the new finite oscillator, the algebra is su(2). The spectra of Q, P and H are discrete (the integers and half-integers) and, within an irreducible representation of spin £, they are finite: m = —£,—£+ 1,...,£. This we interpret as the (normalized) positions qm of 2£ + 1 equidistant sensors in the wavefield 7 . The set of measured values is f = {fm \ m — -£, -£ + 1 , . . . ,£}, fm = f(qm) & C, and form a complex vector in the $t2e+1 space of signals. We are in effect representing the generators of SU(2) by means of difference operators 2 ' 7 ; denoting by q € 3? the (continuous) coordinate and using the

352

unit-shift operators e±a"f(q) = f(q ± 1), we can write position Ji = Q = q-, momentum J2 — -P = -i^[ae(-q)

(4) dq

e

dq

- a((q) e~ ],

atti) = ^/{e + q){£-q + l), mode J3 = H-(l+$)l = -\[ctt(q)e-d<

+at(-q)e9*].

(5) (6) (7)

These operators satisfy the commutation relations of the Lie algebra su(2): [Ji, JI|(/|J3l/)|2. 4

(8)

Kravchuk's 'discrete' wavefunctions

The solutions to the eigenvalue equation of the finite oscillator or waveguide (7) with 2 ^ + 1 points and levels, H(q)4>n{q,2l) = {n + \)4>n(q,2Z),

n = 0 , 1 , . . . ,2*.

(9)

are the normal modes of the wavefield in the finite waveguide. The wavefunc tions satisfy the discrete orthogonality and completeness relations on this set of 2£ + 1 points qj, £M«*>2*)4>n{qj,2t) = «„.,„,

qj =j-e,

(10)

j = 0,1,...,21

(11)

j=o it

2 > „ t e , 2 « ) 0 „ t o ' , 2 O = 6j'j,

They are given in terms of the symmetric Kravchuk polynomials kn(q, 2£) of degrees n in q 3 , and as usual normalized and multiplied by the root of the

353

binomial distribution:

*.(..«) - fm+1, v) Jnt+ffg - ;j| g + I .
W h a t is phase space? — T h e generalized W i g n e r function

Given a function ip{q), q € Sft, its (traditional 5 ) Wigner function is a sesquilinear functional of ip, and a function of phase space (q,p) G 3?2, of the form W+(q,p) = ^jdxxP{q-\xYe-ixi,xl>{q+\x).

(13)

The plots of this function show intuitively the one-dimensional objects of the theory (particles, coherent states, light beams) as bumps and oscillations of a two-dimensional surface; its level curves draw the partitura of the signal on a pentagram of canonically conjugate variables. The Wigner function is not strictly a probability distribution because in small regions it can have negative values; but there is agreement that its properties merit the place it has as one of the foundations of quantum mechanics, on par with the Heisenberg and Schrodinger interpretations. Equation (13) can be seen to define phase space as the domain of the Wigner function, which here is associated to the Heisenberg-Weyl group of translations of phase space in position and momentum (multiplication by linear phases). To define a 'phase space' associated to the finite oscillator built on SU(2), we have generalized the definition of the Wigner function (13) to d-parameter

354

Lie groups Q with generators J = {«/fc}fc=n whose elements (in polar coordi nates) are g[y] = exp(—iy ■ J), where y ■ J := X2jt=i VkJk, and square brackets indicate that arguments are in this coordinate set. The generalized Wigner function between two wavefunctions , i/» in a Hilbert space of inner product (<£ | ip) is the matrix element 6*7 W(
(14)

of the Wigner operator given by W ( x ) = / dg[y} exp[iy ■ (f - / ) ] Jg = f dg[y] exp(ix-y)g[y} Jg

(15)

~ (2n)dY[6(xk

- Jk),

(16)

k

which is a function of £ 6 SR^. The last expression involving Dirac 6's is meant to suggest that, were the Jk some c-numbers, the arguments Xjt of the Wigner operator would have support on that point only; but since the Jk are operators, the xk in (16) serve as the continuous classical coordinates for the d generally non-commuting observables Jk, associated to a set of physical quantities. When the polar parametrization is well known 7 ' n : 9[y\ = 9[nv(6,4>)} = exp(-iy • J) = exp[-ij]{vi Jx + V2J2 + V3J3)}, dg[y] = I sine \r)dy'= \ sin 2 ^77 dr] dv, dv(iD, y?) = sin 1? dfi dip,

(17) (18)

where dy = dyidy^dy^ is the Cartesian measure in 3R3, sincx = x _ 1 s i n x , and dg is the invariant Haar measure on SU(2). The angle TJ = \y \ is counted modulo 47r and distinguishes between the conjugation classes of the group. The integral over the group is over the manifold of a 3-sphere. Observe that while the range of the polar parameters y is the compact SU(2) manifold, the space x in the argument of the Wigner operator is SR3. The Wigner operator (16) acts linearly on column vectors f G 9J 2 ' +1 (the wavefield values), and can be thus represented by a (2£+ 1) x (2^+ 1) matrix We(x): W(x) f = f

dg\y] exp(ix • y)T3?\jf] f = We(x)

f.

(19)

JsU(2)

This we call the Wigner matrix of spin £, and is essentially the Fourier trans form of the unitary irreducible ('Wigner-D') matrix representations D f [y ] of

355

SU(2) in polar coordinates, We(x)=

[

±sinc±|j7|dyexp(ix.y)D<[y],

(20)

./SU(3)

&W]=

, - 1 i,~, / dxexp{-ix-y)W'(x). 7i-> sine 51 j/1 7*3

(21)

In the 5R2<+1 space of vectors f and g, the SU(2) Wigner function (14) is the sesquilinear form We(f,g\x)

e £

= fW(x)g=

[fm}mW^m.(S)gm.,

(22)

m,m'=—(

The elements of the Wigner matrix W ( x ) — {W^m,(x)}m,m' = —V have been computed in analytic and numerical form. In particular, from the unitarity of the Wigner D-matrices, D < ([y]) t = D*[—y] (and choosing the in tegration range to be invariant under inversions), follows that the Wigner W^-matrices are self-adjoint: W £ ( f )* = We(x). 6

The Wigner function in 3-space

The asignments (4)-(7) establish the correspondence between the 'c-number' arguments of the Wigner function Wt(f\x) and the observables of a wavefield in a shallow waveguide; explicitly,

(

q —p

\

/ position \ = I -momentum I ,

(23)

E - £ - \ ) \ mode / where E is the energy. Because (x\,xi) = (q, —p) are coordinates of ordinary phase space, but there is now one more, X3 = mode number, we may call x e SR3 the meta-phase space of the SU(2) model. For large £ and small £3, the sphere \x\2 = q2+p2 + (E-£-^)2 « £2 can be approximated by its osculating paraboloid, the classical energy of the (infinite) oscillator, E « (p2 + q2)/2£. We expect that, within this 5R3 meta-phase space for 2^ + 1 points, the values of the Wigner function should peak around radii x = 1^1 w ^- Indeed, when we integrate over the sphere angles, we compute the radial marginal distribution: MLia.(fIX) = I du W<(f |*u) = ||f||2 Rl{X), Js2

(24)

356

where ||f||2 = (f,f) = X)l/il 2 ' s t n e squared norm of the signal, and the dependence on the radius is contained entirely in the factor Re(x)> which is expressible as x _ 1 times a sum of four sine-integral functions, which peaks strongly i n ^ < x < ^ + l - Picturing the Wigner function in 5ft3, we find it concentrated on a sphere, and we can choose between plotting spherical slices of it, or integrating over the radius x to obtain a function over the unit sphere. For the latter we want yet an analytic expression, but the former has been be handled numerically and plotted through level curves on a sphere 7>12>13. The plots for different slices of x (arithmetic or geometric mean, or elsewhere between I and (. + 1) are not too different and stand out starkest for £ + ~. The Wigner function (14)-(15) defined over the group SU(2) has all the desirable properties of the classical Wigner function (13), such as being real for <}> = x, being SU(2)-covariant: We(Be[y) f, T>e[y ] g| x) = We(f, g| R [ - y ] x ) ,

(25)

with the 3 x 3 'rotation' matrices R[y] € SU(2), and with total marginal distribution M'(f,g)=/

d £ W ' ( f , g | x ) = 7r 3 (f,g).

(26)

Thus the Hilbert-Schmidt norm of the SU(2) Wigner operators is w3. Among the many formulae we would like to derive analytically (which are at present computed only numerically) are the marginal distributions on arbitrary planes and axes (through one and two integrations). 7

Outlook

In more mathematical terms, the domain of a Wigner function are HilbertSchmidt operators —above we considered only those operators that are sepa rable into a ket-bra pair of wavefunctions p = \ip)(4>\— and on the co-adjoint orbits of the group (classified by irreducible representations). The multipli cation of the Haar measure (18) by a radial function of the classes r\ = \y\ leaves its covariance property (25) invariant. This freedom allowed in Ref. 14 the definition of the Wigner function on Lie groups whose right- and leftinvariant Haar measures are distinct, as they are in the two-parameter affine group. There, the Wigner function and the wavelet transform are essentially the same. The Wigner functions for the compact and noncompact one-parameter groups (and all abelian groups) are given in Ref. 15 , and the non-abelian two-parameter group in Ref. 14 . Among the three-parameter Lie groups, the

357

Heisenberg Weyl 6 , SU(2) 7 , and the Euclidean IS0(2) 15 groups have been visited. In the latter, the Hilbert spaces of wavefunctions can be the oscilla tory Helmholtz wavefields in two dimensions, or the discrete, infinite sets of data in a discrete version of two-dimensional optics. A challenge is contained in Sp(2,3?) = SU(1,1) = S1(2,5R), because the group is not of exponential type and the structure of its irreducible representations is arduous; yet it is the pro totypical dynamical algebra for several one-dimensional mechanical systems and models of half-infinite discrete data 16 . Also the Poincare and conformal groups are of interest. The partitura of a signal on phase space is its Wigner function. When we place in the waveguide a flat prism of angle a, 'optical rays' are deflected by the group element exp(iuaQ) {v is a fixed coefficient). With free flight and prisms we can generate all SU(2) transformations of data sets that rotate rigidly the two-sphere of phase space. Data sets derived from an original signal by SU(2)-optical elements can be recognized for having the same, albeit rotated, Wigner function. On the other hand lenses, which act on the wavefield by exp(ry<22), are in the enveloping algebra of SU(2); their action results in deformations —aberrations— of the Wigner function on the sphere. In Ref. 12 we studied in this way the quadratic nonlinearity of the optical Kerr medium Hamiltonian HK„t = H0,c + 0H^c. The Wigner function, being sesquilinear in two wavefields (one is the standard beam and the other is the object beam) serves for holographic encoding and decoding 17 of finite signals. 8

Acknowledgements

This work was performed under the support of Project DGAPA-UNAM 104198 Optica Matemdtica. At various stages in the program we have ben efited from the collaboration, computer and numerical support of Guillermo Krotzseh, Ana Leonor Rivera and Luis Edgar Vicent. References 1. N.M. Atakishiyev and K.B. Wolf, Fractional Fourier-Kravchuk transform. J. Opt. Soc. Am. 14, 1467-1477 (1997). 2. N.M. Atakishiyev and S.K. Suslov, Difference analogs of the harmonic oscillator. Theor. Math. Phys. 85, 1055-1062 (1991). 3. M. Krawtchouk, Sur une generalization des polinomes d'Hermite. C. R. Acad. Sci. Paris 189, 620-622 (1929); A. Erdelyi, W. Magnus, F. Oberhettinger, and F.G. Tricomi, Higher Transcendental Functions, Vol. 2 (McGraw-Hill, New York, 1953).

358

4. N.M. Atakishiyev and K.B. Wolf, Approximation on a finite set of points through Kravchuk functions. Rev. Mex. Fis. 40, 366-377 (1994). 5. E. Wigner, On the quantum correction for thermodynamic equilibrium, Phys. Rev. 40, 749-759 (1932); M. Hillery, R.F. O'Connell, M.O. Scully, and E.P. Wigner, Distribution functions in physics: fundamentals, Phys. Reports 106, 121-167 (1984); H.-W. Lee, Theory and application of the quantum phase-space distribution functions, Phys. Reports 259, 147-211 (1995). 6. K.B. Wolf, Wigner distribution function for paraxial polychromatic op tics. Opt. Commun. 132, 343-352 (1996). 7. N.M. Atakishiyev, S.M. Chumakov and K.B. Wolf, Wigner distribution function for finite systems, J. Math. Phys. 39, 6247-6261 (1998). 8. M. Ank, N.M. Atakishiyev and K.B. Wolf, Quantum algebraic structures compatible with the harmonic oscillator Newton equation, J. Phys. A 32, L371-L376 (1999). 9. P. Feinsilver and R. Schott, Operator calculus approach to orthogonal polynomial expansions, J. Comp. Appl. Math. 66, 185-199 (1996). 10. N.M. Atakishiyev, L.E. Vicent and K.B. Wolf, Continuous vs. discrete fractional Fourier transforms. J. Comp. Appl. Math. 107, 73-95 (1999). 11. N.Ja. Vilenkin and A.U. Klimyk, Representation of Lie Group and Spe cial Functions Vol.1, (Kluver Academic Publ., Dordrecht, 1991). G. Gasper and M. Rahman, Basic Hypergeometric Series (Cambridge Uni versity Press, Cambridge, UK, 1990); A.F. Nikiforov, S.K. Suslov and V.B. Uvarov, Classical Orthogonal Polynomials of a Discrete Variable (Springer-Verlag, Berlin, 1991). 12. S.M. Chumakov, A. Frank and K.B. Wolf, Finite Kerr medium: Schrodinger cats and Wigner functions on the sphere. Phys. Rev. A 60, 1817-1823 (1999). 13. K.B. Wolf, N.M. Atakishiyev and S.M. Chumakov, Wigner distribution function for finite signals. SPIE Proceedings Vol. 3076 (Photonic Quan tum Computing, Orlando, Fla., 23-24 April, 1997). The International Society for Optical Engineering, 1997, pp. 196-206; N.M. Atakishiyev, S.M. Chumakov, L.M. Nieto and K.B. Wolf, Wigner function on groups for various optical models. Proceedings of the Fifth International Confer ence on Squeezed States and Uncertainty Relations (Balatonfured, Hun gary, May 27-31, 1997). Ed. by D. Han, J. Janszky, Y.S. Kim and V.I. Man'ko, NASA/CP-1998-206855 (May 1998), pp. 301-308; K.B. Wolf, N.M. Atakishiyev and S.M. Chumakov, Wigner distribution function for some dynamical models. Proceedings of the V Wigner Symposium (Vi enna, Aug. 25-29, 1997). World Scientific Publ. Co. (1998), pp. 370-372.

359

14. S.T. Ali, N.M. Atakishiyev, S.M. Chumakov and K.B. Wolf, The Wigner function for general Lie groups and the wavelet transform. Ann. Inst. Henri Poincare / Physique Theorique , Submitted (2000). 15. L.M. Nieto, N.M. Atakishiyev, S.M. Chumakov and K.B. Wolf, Wigner distribution function for Euclidean systems. J. Phys. A 3 1 , 3875-3895 (1998). 16. N.M. Atakishiyev, E.I. Jafarov, Sh.M. Nagiyev and K.B. Wolf, Meixner oscillators. Rev. Mex. Fis. 44, 235-244 (1998). 17. K.B. Wolf and A.L. Rivera, Holographic information in the Wigner func tion. Opt. Commun. 144, 36-42 (1997).

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