Jet Single-Time Lagrange Geometry and Its Applications

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JET SINGLE-TIME LAGRANGE GEOMETRY AND ITS APPLICATIONS

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JET SINGLE-TIME LAGRANGE GEOMETRY AND ITS APPLICATIONS

Vladimir Balan University Politehnica of Bucharest

Mircea Neagu University Transilvania of Brasov

WILEY A JOHN WILEY & SONS, INC., PUBLICATION

Copyright © 2011 by John Wiley & Sons, Inc. All rights reserved Published by John Wiley & Sons, Inc., Hoboken, New Jersey Published simultaneously in Canada No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning, or otherwise, except as permitted under Section 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, (978) 750-8400, fax (978) 750-4470, or on the web at www.copyright.com. Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., I l l River Street, Hoboken, NJ 07030, (201) 748-6011, fax (201) 748-6008, or online at http://www.wiley.com/go/permission. Limit of Liability/Disclaimer of Warranty: While the publisher and author have used their best efforts in preparing this book, they make no representations or warranties with respect to the accuracy or completeness of the contents of this book and specifically disclaim any implied warranties of merchantability or fitness for a particular purpose. No warranty may be created or extended by sales representatives or written sales materials. The advice and strategies contained herein may not be suitable for your situation. You should consult with a professional where appropriate. Neither the publisher nor author shall be liable for any loss of profit or any other commercial damages, including but not limited to special, incidental, consequential, or other damages. For general information on our other products and services or for technical support, please contact our Customer Care Department within the United States at (800) 762-2974, outside the United States at (317) 572-3993 or fax (317) 572-4002. Wiley also publishes its books in a variety of electronic formats. Some content that appears in print may not be available in electronic formats. For more information about Wiley products, visit our web site at www.wiley.com. Library of Congress Cataloging-in-PubUcation Data is available. ISBN 978-1-118-12755-1 Printed in the United States of America. 10 9 8 7 6 5 4 3 2 1

CONTENTS

Preface

xi

PART I THE JET SINGLE-TIME LAGRANGE GEOMETRY 1

Jet geometrical objects depending on a relativistic time l. I 1.2 1.3 1.4

2

1

d-tensors on the I -jet space J (R, M) Relativistic time-dependent semisprays. Harmonic curves Jet nonlinear connections. Adapted bases Relativistic time-dependent semisprays and jet nonlinear connections

3 4 6 11 16

Deflection d-tensor identities in the relativistic time-dependent Lagrange geometry

19

2.1 2.2 2.3

19 24 30

The adapted components of jet Γ-linear connections Local torsion and curvature d-tensors Local Ricci identities and nonmetrical deflection d-tensors

v

VI

CONTENTS

Local Bianchi identities in the relativistic time-dependent Lagrange geometry 3.1 3.2

The adapted components of /i-normal Γ-linear connections Deflection d-tensor identities and local Bianchi identities for d-connections of Cartan type

The jet Riemann-Lagrange geometry of the relativistic time-dependent Lagrange spaces 4.1 4.2 4.3 4.4

4.5

Relativistic time-dependent Lagrange spaces The canonical nonlinear connection The Cartan canonical metrical linear connection Relativistic time-dependent Lagrangian electromagnetism 4.4.1 The jet single-time electromagnetic field 4.4.2 Geometrical Maxwell equations Jet relativistic time-dependent Lagrangian gravitational theory 4.5.1 The jet single-time gravitational field 4.5.2 Geometrical Einstein equations and conservation laws

The jet single-time electrodynamics 5.1 5.2 5.3

Riemann-Lagrange geometry on the jet single-time Lagrange space of electrodynamics ¿"DL™ Geometrical Maxwell equations on SVL™ Geometrical Einstein equations on £VL™

Jet local single-time Finsler-Lagrange geometry for the rheonomic Berwald-Moór metric of order three 6.1 6.2 6.3 6.4

Preliminary notations and formulas The rheonomic Berwald-Moór metric of order three Cartan canonical linear connection, d-torsions and d-curvatures Geometrical field theories produced by the rheonomic Berwald-Moór metric of order three 6.4.1 Geometrical gravitational theory 6.4.2 Geometrical electromagnetic theory

33 33 37

43 44 45 48 50 50 51 52 52 53 57 58 61 62

65 66 67 69 72 72 76

Jet local single-time Finsler-Lagrange approach for the rheonomic Berwald-Moór metric of order four

77

7.1

78

Preliminary notations and formulas

CONTENTS

7.2 7.3 7.4 7.5

7.6

The rheonomic Berwald-Moór metric of order four Cartan canonical linear connection, d-torsions and d-curvatures Geometrical gravitational theory produced by the rheonomic Berwald-Moór metric of order four Some physical remarks and comments 7.5.1 On gravitational theory 7.5.2 On electromagnetic theory Geometric dynamics of plasma in jet spaces with rheonomic Berwald-Moór metric of order four 7.6.1 Introduction 7.6.2 Generalized Lagrange geometrical approach of the non-isotropic plasma on 1-jet spaces 7.6.3 The non-isotropic plasma as a medium geometrized by the jet rheonomic Berwald-Moór metric of order four

The jet local single-time Finsler-Lagrange geometry induced by the rheonomic Chernov metric of order four 8.1 8.2 8.3 8.4

Preliminary notations and formulas The rheonomic Chernov metric of order four Cartan canonical linear connection, d-torsions and d-curvatures Applications of the rheonomic Chernov metric of order four 8.4.1 Geometrical gravitational theory 8.4.2 Geometrical electromagnetic theory

vii 79 81 84 87 87 89 89 89 90 96

99 100 101 103 105 105 108

Jet Finslerian geometry of the conformal Minkowski metric

109

9.1 9.2 9.3 9.4

109 111 113

Introduction The canonical nonlinear connection of the model Cartan canonical linear connection, d-torsions and d-curvatures Geometrical field model produced by the jet conformal Minkowski metric 9.4.1 Gravitational-like geometrical model 9.4.2 Related electromagnetic model considerations

115 115 118

CONTENTS

PART II APPLICATIONS OF THE JET SINGLE-TIME LAGRANGE GEOMETRY Geometrical objects produced by a nonlinear ODEs system of first-order and a pair of Riemannian metrics 10.1 10.2 10.3 10.4

Historical aspects Solutions of ODEs systems of order one as harmonic curves on 1 -jet spaces. Canonical nonlinear connections From first-order ODEs systems and Riemannian metrics to geometrical objects on 1-jet spaces Geometrical objects produced on 1-jet spaces by first-order ODEs systems and pairs of Euclidian metrics. Jet Yang-Mills energy

Jet single-time Lagrange geometry applied to the Lorenz atmospheric ODEs system 11.1 11.2

Jet Riemann-Lagrange geometry produced by the Lorenz simplified model of Rossby gravity wave interaction Yang-Mills energetic hypersurfaces of constant level produced by the Lorenz atmospheric ODEs system

Jet single-time Lagrange geometry applied to evolution ODEs systems from Economy 12.1 12.2

Jet Riemann-Lagrange geometry for Kaldor nonlinear cyclical model in business Jet Riemann-Lagrange geometry for Tobin-Benhabib-Miyao economic evolution model

121 121 123 127

129

135 135 138

141 141 144

Some evolution equations from Theoretical Biology and their single-time Lagrange geometrization on 1-jet spaces 147 13.1 13.2 13.3

Jet Riemann-Lagrange geometry for a cancer cell population model in biology The jet Riemann-Lagrange geometry of the infection by human immunodeficiency virus (HIV-1) evolution model From calcium oscillations ODEs systems to jet Yang-Mills energies 13.3.1 Intracellular calcium oscillations induced by selfmodulation of the inositol 1,4,5-triphosphate signal

148 151 154 155

CONTENTS

13.3.2 13.3.3

14

Jet geometrical objects produced by linear ODEs systems and higher-order ODEs 14.1 14.2 14.3

15

Calcium oscillations in a model involving endoplasmic reticulum, mitochondria, and cytosolic proteins Yang-Mills energetic surfaces of constant level. Theoretical biological interpretations

Jet Riemann-Lagrange geometry produced by a nonhomogenous linear ODEs system of order one Jet Riemann-Lagrange geometry produced by a higher-order ODE Riemann-Lagrange geometry produced by a non-homogenous linear ODE of higher-order

Jet single-time geometrical extension of the KCC-invariants

¡X

161 167

169 169 172 175

179

References

185

Index

191

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PREFACE

The 1 -jet fiber bundle is a basic object in the study of classical and quantum field theories ([77], [8], etc.)· For this a reason, a lot of authors (Asanov [8], Martinez [51], Saunders [89], Vondra [98], [99], Väcaru [96], [97] and many others) studied the differential geometry of the 1 -jet spaces. Considering the geometrical studies of Asanov [8] and using as a pattern the Lagrangian geometrical ideas developed by Miron, Anastasiei, or Bucätaru in the monographs [55] and [24], the second author of this book has recently developed the Riemann-Lagrange geometry of 1-jet spaces [65], which is very suitable for the geometrical study of the relativistic time-dependent (rheonomic or non-autonomous) Lagrangians. In this framework, we refer to the geometrization of Lagrangians depending on the usual relativistic time [67] or of Lagrangians depending on relativistic multi-time [65], [70]. It is important to note that a classical time-dependent (rheonomic) Lagrangian geometry (i.e. a geometrization of the Lagrangians depending on the usual absolute time) was sketched by Miron and Anastasiei at the end of the book [55] and accordingly developed by Anastasiei and Kawaguchi [1] or Frigioiu [34]. For such a reason, we shall further describe the main geometrical and physical aspects which differentiate the two geometrical theories: the. jet relativistic time-dependent Lagrangian geometry [67] and the classical time-dependent Lagrangian geometry [55]. In this direction, we point out that the relativistic time-dependent Lagrangian geometry [67] naturally relies on the 1-jet space J ^ K , M), where R is the manifold xi

XII

PREFACE

of real numbers having the temporal coordinate t. This represents the usual relativistic time. We recall that the 1 -jet space J 1 (R, M) is regarded as a vector bundle over the product manifold R x M, having the fiber type R n , where n is the dimension of the spatial manifold M. In mechanical terms, if the manifold M has the spatial local coordinates (χι)ί=γ^, then the 1-jet vector bundle J1(R,M)^RxM

(0.1)

can be regarded as a bundle of configurations whose total space has the local coordinates (t, xl, y\); these transform by the rules [67] t = t{t), xl =

χ%(χΐ),

_,¿ _ dxl dt

yi =

(0.2) j

dxlJt'Vl-

We remark that the form of the jet transformation group defined by the rules (0.2) stands out for the relativistic character of the time t. Comparatively, denoting by TM the tangent bundle of the spatial manifold M, we note that in the classical time-dependent Lagrangian geometry the bundle of configurations is the vector bundle [55] R x TM -» M, whose local coordinates (t,xl,yl)

(0.3)

transform by the rules ' t == <, x1 = x*(x-''),

. vl

dxl ~

Α,ν,ι

(0.4)

■yi.

Remark 1. The form of the transformation group (0.4) stands out for the absolute character of the time t. We emphasize that the jet transformation group (0.2) from the relativistic timedependent Lagrangian geometry is more general and more natural than the transformation group (0.4) used in the classical time-dependent Lagrangian geometry. This is because the last one ignores the temporal reparametrizations, emphasizing in this way the absolute character of the usual time coordinate t. Or, physically speaking, the relativity of time is an well-known fact. From a geometrical point of view, we point out that the entire classical timedependent (rheonomic or non-autonomous) Lagrangian geometry of Miron and Anastasiei [55] relies on the study of the energy action functional Ei(c)= Í

Ja

L(t,x\yl)dt,

PREFACE

XÜi

where L : I x TM —> R is a Lagrangian function and y% = dxl /dt, whose EulerLagrange equations

xi + 2G%xi,yi)

=0

produce the semispray G* and the nonlinear connection ΛΠ = dGl/dy:>. In the sequel, the authors construct the adapted bases of vector and covector fields, together with the adapted components of the ΑΓ-linear connections and their corresponding torsions and curvatures. But, because L(t, xl, yl) is a real function, we deduce that the previous geometrical theory has the impediment that the energy action functional depends on the reparametrizations t <—► t of the same curve c. For example, in order to avoid this inconvenience, the Finsler case imposes the 1 -positive homogeneity condition ¿ ( ί , χ ' , λ ^ ) = \L{t,x\y%

VA > 0.

Alternatively, the relativistic time-dependent Lagrangian geometry from this book (see also [67]) uses the relativistic energy action functional L(t,x%,y\)y/hn(t)dt,

E2(c)=/ Ja

where L : J J (R, M) —> E is a jet Lagrangian function and /»n(i) is a Riemannian metric on the time manifold R. This functional is now independent by the reparametrizations t <—► t of the same curve cand the corresponding Euler-Lagrange equations take the form xl + 2H^n

(i,**,»?) + 2G^n

(í,x f c ,yí) = 0,

where the coefficients W*2V respectively ffi'i, represent a temporal, respectively spatial, semispray. These semisprays produce a jet canonical nonlinear connection

I

(1)1

(l)l'

(1)*:

Qyk

I

We further describe the local adapted components of connections, torsion, and curvature. We emphasize that the local adapted components of the jet geometrical objects involved in the present study obey the formalism used in the works [55], [24], [1], [34], [65], [67], [12]. In this respect, the authors of this book believe that the relativistic geometrical approach proposed in this monograph has more geometrical and physical meanings than the theory proposed by Miron and Anastasiei in [55]. As afinalremark, we point out that for numerous mathematicians (such as Crampin [31], Krupková [45], [46], de León [48], Sarlet [88], Saunders [89], and others) the time-dependent Lagrangian geometry is constructed on the first jet bundle 3ΧΈ of a fibered manifold π : Mn+1 —> R. In their papers, if (t, xl) are the local coordinates on the n + 1-dimensional manifold M such that t is a global coordinate for the fibers

XIV

PREFACE

of the submersion π and xl are transverse coordinates of the induced foliation, then a change of coordinates on M is given by ~

~, .

t

J

* = *(*), x =x'(x ,t),

dt ,

ä*°>

(0.5)

rank I ——7 1 = n. χσχΐI

Although the 1-jet extension of the transformation rules (0.5) is more general than the transformation group (0.2), the authors of this book consider that the transformation group (0.2) is more appropriate for their final purpose: the development of the relativistic time-dependent Lagrangian geometrical field theories. For example, in our monograph, starting with a non-degenerate Lagrangian function L : J X (R, M) —> K and an a priori given Riemannian metric h\\{t) on the relativistic temporal manifold K (these geometrical objects produce together the Lagrangian C = L^hn(t)), one introduces the jet single-time gravitational potential dt + gijdx* dxj + hn(t)gij(t,

x, y)5y\ Sy{,

(0.6)

where 9ϋϋ,χ,ν)

fen(t) d2L =- ^ T — J , dy\dy{

Sy\ = dy\ + M\^dt +

N^dxK

Note that the above jet single-time gravitational potential G is a global geometrical object on J 1 (R, M), with respect to the group of transformations (0.2). Moreover, it is characterized (as in the Miron and Anastasiei case [55]) by some natural geometrical Einstein equations [67]. These geometrical Einstein equations will be described in the next chapters of this book. At the same time, the transformation group (0.2) is more appropriate for the development of a relativistic time-dependent Lagrangian electromagnetic theory, whose jet single-time electromagnetic field is defined by ¥ = where

Fr w

F{{gSy\Adx\

_ I D^.-D^

d)j ~( i 2) the metrical deflection d-tensors D,J. — h11gimy7fi. being produced only by the jet Lagrangian £ = L^/hu{t), via its Cartan canonical Γ-linear connection. From such a perspective, the electromagnetic components F5J. are governed by some natural geometrical Maxwell equations. These geometrical Maxwell equations will be also presented in this book and they naturally generalize the already classical Maxwell equations from Miron and Anastasiei's theory [55], which has many applications in Theoretical Physics, such as Electrodynamics, Relativistic Optics, or Relativity and Electromagnetism.

PREFACE

XV

In our book, we will show that our jet single-time Lagrangian geometry also gives a lot of applications to various domains of sciences: Theoretical Physics (the gravitational theory produced by the Berwald-Moór metric or by the Chernov metric), Atmospheric Physics, Economy, or Theoretical Biology. As well, at the end of the book, there are presented the basic elements of the Kosambi-Cartan-Chern theory on the 1-jet space J 1 ( E , M), which extend the approach developed in [44], [26], [27], [24], [4]. Finally, the authors of this book express their gratitude to Professors R. Miron, M. Anastasiei, Gh. Atanasiu, C. Udri§te, D. Opris, D. G. Pavlov, M. Rahula, P.C. Stavrinos, K. Teleman, K. Trencevski, V. Obädeanu, Gh. Piti§, Gh. Munteanu, I. Mihai, V. Prepelifä, G. Pripoae, M. Crä§märeanu, E. Pältänea, I. R. Nicola, M. Postolache, M. Lupu, E. Stoica, M. Päun and C. Radu for stimulative discussions on the geometrical methods used in the applicative research from this book. Special thanks are addressed to the late Professor R. G. Beil who kindly provided us G. S. Asanov's essential paper [8], which settles on 1-jet spaces the geometrical foundations for a similar related framework. Vladimir Balan Mircea Neagu April 21, 2011

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PARTI

THE JET SINGLE-TIME LAG RANGE GEOMETRY

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CHAPTER 1

JET GEOMETRICAL OBJECTS DEPENDING ON A RELATIVISTIC TIME

The differential geometry of the 1 -jet space J 1 (R, M) was intensively studied by numerous authors: Crampin [31], Krupková [46], de León [48], Martínez [51], Sarlet [88], Saunders [89], Vondra [98], [99], etc. Compared to their approaches, our framework focuses on the local decomposition of the involved geometrical objects relative to adapted bases associated to a given nonlinear connection. In the present Chapter, developing further the geometrical studies initiated by Asanov [8] and using as a pattern the geometrical ideas developed by Miron and Anastasiei in [55], we study a collection of jet geometrical objects (d-tensors, relativistic time-dependent semisprays, harmonic curves, and nonlinear connections), together with the underlying fundamental geometrical relations which relate them. These geometrical concepts are essential for the subsequent construction of the geometrization (in the sense of R. Miron) of the 1-jet space J 1 (R, M). This geometrization on the 1-jet spaces will therefore assume the decomposition of the geometrical objects involved into their local adapted components.

Jet Single-Time Lagrange Geometry and Its Applications 1st Edition. By Vladimir Balan and Mircea Neagu. © 2011 John Wiley & Sons, Inc. Published 2011 John Wiley & Sons, Inc.

3

4

JET GEOMETRICAL OBJECTS DEPENDING ON A RELATIVISTIC TIME

1.1

d-TENSORS ON THE 1-JET SPACE J X (R, M )

Let us consider the 1 -jet fiber bundle -tRxM,

J\R,M) whose local coordinates (t, xl,y\)

(1.1)

transform by the rules t = t(t), x%(xj), Vi

d&dt dJdt

(1.2) ■y{-

It is well known that in the study of the geometry of a fiber bundle an important role is played by tensors. For such a reason, let us consider

the canonical basis of vector fields on the 1 -jet space J 1 (R, M), together with its dual basis of 1 -forms {dt,dx\dy\} C Χ*(^(Λ,Μ)). In this context, let us remark that, doing a transformation of jet local coordinates (1.2), the following transformation rules hold true: dtd_

+

dy{ d

dt'~ Jt~& ~dt~dtfx' d dxi d dy\ d dxi ~ ~dx1~dxTi ~dxi~dy^x d and dt

(1.3)

_ dxi dt d

dt ~ dt

-rddt,

ox1

dxl = ——dxi, dx¡

dy\ yi = ^dt+^dxi

(1.4) +

^djdy{.

dt dx> dxi dt yi Taking into account that the transformation rules (1.3) and (1.4) lead to complicated transformation rules for the components of classical tensors on the 1 -jet space J 1 (M, M), we consider that in the geometrical study of the 1-jet fiber bundle J X (R, M) a central role is played by the distinguished tensors (d-tensors).

d-TENSORS ON THE 1 -JET SPACE J1 (K, M)

5

Definition 2. A geometrical object D = i^jfcmm'") on the 1-jet vector bundle J X (R, M), whose local components transform by the rules n li(j)(l)-

U

lk(X)(l)...

d d r

_ ñlp(m)(l)...dí ÖX^ (^_^\ - ^lr(l)(.)... ¿ ^ p

\^Jm ¿ ^

**

d f öa; fc

(^Ξΐ^Λ

\^ βχ1 β )

"'

(\ K\

V1·3'

is called a d-tensorfield. Remark 3. The utilization of parentheses for certain indices of the local components ^lktiW)" 0 ^ t n e distinguished tensor D on J ^ R , M) will be rigorously motivated after the introduction of the geometrical concept of nonlinear connection on the 1 -jet space J X (R, M). For the moment, we point out that the pair of indices " 9A " or " ,J " behaves like a single index. Remark 4. From a physical point of view, a d-tensor field D on the 1-jet vector bundle J X (R, M) —> R x M can be regarded as a physical object defined on the space of events R x M, which is dependent by the direction or the relativistic velocity V — (y\)- Such a perspective is intimately connected with the physical concept of anisotropy, EXAMPLE 1.1 Let L : J 1 (R, M) —> R be a relativistic time-dependent Lagrangian function, where ^ ( Κ , Μ ) 9 (t.zSj/i) - ¿ ( Μ ' , ι / ί ) G R. Then, the geometrical object G = (Gv¿¿ J 1, where G(i)(i) =

{i){j)

1

d2L

2 dy\dy{'

is a d-tensor field on J J (R, M), which is called the fundamental metrical dtensor produced by the jet Lagrangian function L. Note that the d-tensor field

GWfrx'M) is a natural generalization for the metrical d-tensor field gij(t, xl, yl) of a classical time-dependent (rheonomic) Lagrange space [55]

RLn =

(M,L(t,xi,yi)),

where L : R x TM —> R is an absolute time-dependent Lagrangian function.

6

JET GEOMETRICAL OBJECTS DEPENDING ON A RELATIVISTIC TIME

EXAMPLE 1.2 The geometrical object C = ( C Y l ) , where C\ZX = y\, represents a d-tensor field on the l-jet space J 1 (R, M). This is called the canonical Liouville dtensor field of the l-jet vector bundle J 1 (R, M). Remark that the d-tensor field C naturally generalizes the classical Liouville vectorfield[55]

used in the Lagrangian geometry of the tangent bundle TM. EXAMPLE 1.3 Let h = (/in (*)) be a Riemannian metric on the relativistic time axis R and let us consider the geometrical object 3u = ( J j ^ . 1, where J

(i)ij

=

^11*?''

6j being the Kronecker symbol. Then, the geometrical object J/, is a d-tensor field on J X (R, M), which is called the h-normalization d-tensor field of the 1 -jet space J 1 (R, M). We underline that our ^-normalization d-tensor field J/, of the 1 -jet space J 1 (R, M) naturally generalizes the tangent structure [55] J = δ) —— ®j dxi j - —— ®dx\ d

d

constructed in the Lagrangian geometry of the tangent bundle TM. EXAMPLE 1.4 Using preceding notations, we consider the set of local functions L^ = Í L,ll 1A, where L

(i)ii

-/ιπ2/ι·

The geometrical object L^ is a d-tensor field on J X (R, M), which is called the h-canonical Liouville d-tensor field of the 1 -jet space J 1 (R, M). 1.2

RELATIVISTIC TIME-DEPENDENT SEMISPRAYS. HARMONIC CURVES

We point out that the notions of d-tensor and classical tensor on the l-jet space J 1 (R, M) are distinct ones. However, we will show after the introduction of the

RELATIVISTIC TIME-DEPENDENT SEMISPRAYS. HARMONIC CURVES

7

geometrical concept of nonlinear connection, that any d-tensor is a classical tensor on J 1 (R, M). Conversely, this statement is not true. For instance, we construct in the sequel two classical global tensors which are not d-tensors on J 1 (R, M). We talk about two geometrical notions: the temporal semispray and the spatial semispray on J 1 (R, M). These concepts allow us afterward to introduce the geometrical concept of relativistic time-dependent semispray on the 1 -jet space J 1 (R, M). Definition 5. A global tensor H on the 1-jet space J1(R, M), which is locally expressed by H = dt®^--2H¡J1]1dt®^>1, v et '

(1.6) dy\

is called a temporal semispray on J 1 (R, M). Taking into account that the temporal semispray H is a global classical tensor on J X (R, M), by direct local computations, we find the following: Proposition 6. (i) The local components H^L of the temporal semispray H transform by the rules 2H

(in-2H(m{s)

o^J-JtlK-

(L7)

(ii) Conversely, to give a temporal semispray on J 1 (R, M) is equivalent to giving a set of local functions H = I H^L ) which transform by the rules (1.7). EXAMPLE 1.5 Let us consider that h = (hn(t)) manifold R and let

is a Riemannian metric on the temporal

! _ h11 dhn "2"dT'

Kn

where hn = l/hn > 0, be its Christoffel symbol. Taking into account that we have the transformation rule x

dt dtdH ^dt^'dtdlß'

"11 — "Ί1~?τ+ ~~7^>

(1-8)

we deduce that the local components Hij)

-

--K1

iij

define a temporal semispray H = [W^L 1 on J X (R, M). This is called the canonical temporal semispray associated to the temporal metric hn(t).

8

JET GEOMETRICAL OBJECTS DEPENDING ON A RELATIVISTIC TIME

Definition 7. A global tensor G on the 1-jet space J 1 (M, M), which is locally expressed by G

= y±dt®^-2G(indt®]ri>

0-9>

dy{

is called a spatial semispray on J 1 (R, M). As in the case of the temporal semispray, by direct local computations, we can prove without difficulties the following statements: Proposition 8. (i) The local components GAL of the spatial semispray G transform by the rules

20« = 2GJ?U f i V ^ - | ^ | % (1)1

(1)1

1 1

(1.10)

\dt) dxJ dxi dx * (ii) Conversely, to give a spatial semispray on J 1 (R, M) is equivalent to giving a set of local functions G = I GAL) which transform by the rules (1.10). EXAMPLE 1.6 Let us consider that φ = (φ^(χ)) is a semi-Riemannian metric on the spatial manifold M and let ljk

~

2 \dxk

dxi

dxm

be its Christoffel symbols. Taking into account that we have the transformation rules dxP_dxi_d^_ dxP d2x: : yp = V-, —■' l 1 (] i l l q l dxr ^ adx dxqdx ' 'Qr ljk dx » dx ~„ f,~r „ ¿ f)~qfí~r K ■ > Λ

Ν

we deduce that the local components 1 ^(1)1 -

define a spatial semispray G = (ÖAIA

2lkiyiVl

on J 1 (R, M). This is called the

canonical spatial semispray associated to the spatial metric ψ^(χ). Remark 9. It is important to note that our notions of temporal and spatial semispray naturally generalize the classical notion of semispray (or semigerbe in the French terminology) which was defined since 1960's (Dazord, Klein, Foulon, de León, Mirón, and Anastasiei, etc.) as a global vector field. Comparatively^ we point out that our temporal or spatial semisprays can be regarded in the form H = dt®Hi,

G = dt®Gi.

RELATIVISTIC TIME-DEPENDENT SEMISPRAYS. HARMONIC CURVES

9

Obviously, the geometrical objects (similarly with the classical concepts of semisprays or semigerbes) {1)l

dt

dy{

and yi

(l)1

dx*

dy{

cannot be regarded as global vector fields on J 1 (R, M) because they behave as the components of some d-covector fields on the 1-jet space J X (R, M). In other words, taking into account the transformation rules (1.2), the geometrical objects Hi and G\ transform by the laws dt

ñ _

fT dt

r

-

d t

r

dt

In conclusion, if we work only with particular transformations (1.2) in which the time t is absolute one (i.e., t = t), then the geometrical objects Hi and Gi become global vector fields and, consequently, we recover the classical definition of a semispray or a semigerbe. Definition 10. A pair <S = {H,G), which consists of a temporal semispray H and a spatial semispray G, is called a relativistic time-dependent semispray on the 1-jet space .Ιλ{Έ.,Μ). Remark 11. The geometrical concept of relativistic time-dependent semispray on the 1-jet space J X (R, M) naturally generalizes the already classical notion of timedependent semispray on R x TM, used by Miron and Anastasiei in [55]. EXAMPLE 1.7 The pair S = (H,G), where H [respectively G] is the canonical temporal (respectively spatial) semispray associated to the temporal (respectively spatial) metric hn(t) [respectively φ^(χ)], is a relativistic time-dependent semispray on the 1-jet space J X (R, M). This is called the canonical relativistic timedependent semispray associated to the pair of metrics (hn(t), φ^(χ)). In order to underline the importance of the canonical relativistic time-dependent semispray S associated to the pair of metrics (/in(i), φ^(χ)), we give the following geometrical result which characterizes the relativistic time-dependent semisprays on 1-jet spaces: Proposition 12. Let (R, hu(t)) be a Riemannian manifold and let (M, φ^(χ)) be a semi-Riemannian manifold. Let S = (H, G) be an arbitrary relativistic timedependent semispray on the 1-jet space J 1 (R, M). Then, there exists on JX(]R, M) a unique pair of d-tensors T — (T^> ^ \ 1

~ !/(i)i'°(i)i;'

10

JET GEOMETRICAL OBJECTS DEPENDING ON A RELATIVISTIC TIME

such that S =

S-T,

where S — (H, G) is the canonical relativistic time-dependent semispray associated to the pair of metrics (hu(t), ψ^(χ)). Proof: Taking into account that the difference between two temporal (respectively spatial) semisprays is a d-tensor [see the relations (1.7) and (1.10)], wefindthe required result. Now, let usfixon the 1-jet space J X (R, M) an arbitrary relativistic time-dependent semispray

S = (H,G) = (H$)1(t^,yk1),G($n(t,xk,yfy

.

Definition 13. A smooth curve c : i e ! c l - > c(t) = (xl(t)) <£ M, which verifies the second-order differential equations (SODEs) ^

+2 <

μ

(*,**(*), ^ ) + 2Gf?n (t,x*(t),

^ ) = 0,

(1.12)

where i runs from 1 to n, is called a harmonic curve of the relativistic time-dependent semispray S = (H, G). Remark 14. The SODEs (1.12) are invariant under a transformation of coordinates given by (1.2). It follows that the form of equations (1.12), which give the harmonic curves of a relativistic time-dependent semispray S = (H, G), have a global character on the 1-jet space J ^ R , M). Remark 15. The equations of the harmonic curves (1.12) naturally generalize the equations of the paths of a time-dependent semispray from classical rheonomic (nonautonomous) Lagrangian geometry [55]. ■ EXAMPLE 1.8 The equations of the harmonic curves of the canonical relativistic time-dependent semispray S — (H, G) associated to the pair of metrics (hu(t), φ^(χ)) are d?xl

, . . dxl

i , . άχί dxk =

Ί*-*Μ-Έ + ^Ί*Ί* »·

(U3)

These are the equations of the affine maps between the Riemannian manifold (R,/in(t)) and the semi-Riemannian manifold (Μ,φ^(χ)). We point out that the affine maps between the manifolds (R,hu(t)) and (Μ,φ^(χ)) are curves which carry the geodesies of the temporal manifold (R, hu(t)) into the geodesies on the spatial manifold (M, φ^(χ)). Remark 16. Multiplying the equations (1.13) with h11 = l/hn φ 0, we obtain the equivalent equations h 11

d2xl

i , . dxl

, . x άχί dxk

0.

JET NONLINEAR CONNECTIONS. ADAPTED BASES

11

These are exactly the classical equations of the harmonic maps between the manifolds (R, /in (t)) and (M, φ^{χ)) (seeEells and Lemaire [33]). For such a reason, we used the terminology of harmonic curves for the solutions of the SODEs (1.12). Remark 17. The jet geometrical concept of harmonic curve of a relativistic timedependent semispray S = (H, G) is intimately connected by the concept of EulerLagrange equations produced by a relativistic time-dependent Lagrangian C = L^h\i(t), where L : J 1 (R, M) —♦ R. The connection is given by the fact that the Euler-Lagrange equations of any non-degenerate Lagrangian C can be written in the form (1.12). For example, the Euler-Lagrange equations of the jet Lagrangian

where y\ = dxl /dt, are exactly the equations of the affine maps (1.13). Equations (1.13) are in fact the equations (1.12) for the particular relativistic time-dependent semispray S = (H, G) associated to the pair of metrics (/in(t), φ^{χ)). In this context, using the notations from Proposition 12, we immediately deduce the following interesting result: Corollary 18. The equations (1.12) of the harmonic curves of a relativistic timedependent semispray S = (H, G) on the 1-jet space J 1 (R, M) can be always rewritten in the following equivalent generalized Poisson form: h1 where

dh

dt2

dx3 dx ~dt~~dt

. dx
J

t,r-it),^-

( l ) l "l"°(l)l

1.3 JET NONLINEAR CONNECTIONS. ADAPTED BASES We have seen that the transformation rules of the canonical bases of vector fields (1.3) or covector fields (1.4) imply complicated transformation rules for the local components of the diverse geometrical objects (as the classical tensors, for example) on the 1-jet space J1(R, M). For such a reason, it is necessary to construct the so-called adapted bases attached to a nonlinear connection on J X (R, M). These adapted bases have the quality to simplify the transformation rules of the local components of the jet geometrical objects taken in the study. In order to do this geometrical construction, let us consider an arbitrary point u € E = J1(R, M) and let us take the differential map π,,„:Τ„£^ΤΜ(ΚχΜ) produced by the canonical projection π : E ^Rx

M, TT(U) =

(t,x).

12

JET GEOMETRICAL OBJECTS DEPENDING ON A RELATIVISTIC TIME

The differential map π*τ„ generates the vector subspace Vu = k e r ^ , u C TUE, whose dimension is diiriR Vu = n, V u € E, because 7r*>M is a surjection. Moreover, a basis in the vector subspace Vu is given by

\dy\ It follows that the map V : u e E -> Vu C TUE is a differential distribution on J 1 (R, M), which is called the vertical distribution of the 1-jet space E = , / ^ Κ , Μ ) . Definition 19. A nonlinear connection on the 1 -jet space E — J 1 (R, M) is a differential distribution n-.ueE->HuC TUE, which verifies the equalities TuE =

Hu®Vu,VueE.

The differential distribution Ή is also called the horizontal distribution of the 1-jet space , / ^ Κ , Μ ) . Remark 20. (i) It is obvious that the dimension of a horizontal distribution is dimji Hu = t i + l , V t t € £ . (ii) The set X{E) of the vector fields on E = J X (R, M) decomposes in the direct sum X(E) = T{H)®X(V), (1.14) where Γ(Ή) [respectively #(V)] represents the set of the horizontal [respectively vertical] sections. Taking into account that a given nonlinear connection (horizontal distribution) Ή. on the 1-jet space E — J X (R, M) produces the isomorphisms 7r,, u | Hu : Hu -* TW(U)(R

xM),\/u€E,

by direct local computations, we deduce the following geometrical results: Proposition 21. (i) There exist some unique linearly independent horizontal vector fields δ/St, δ/δχι G Γ(Ή) having the properties (S\

π

*{Τί)

=

d

( δ\

d

οϊ> H f a s T ä ? ·

(115)

JET NONLINEAR CONNECTIONS. ADAPTED BASES

(ii) With respect to the natural basis {d/dt, d/dxl, d/dy\] zontal vector fields 6/St and δ/δχ1 have the local expressions δ_ _ d

(j) d

Tt~di~

^ ^'

j5_ _ _9_

1

13

C X{E), the hori(j) _9_

Ix* ~ d^ ~ Wdú'

(

}

where the functions Μ,^λ [respectively N^i] are defined on the domains of the induced local charts on E = J 1 (K, M) and they are called the temporal (respectively spatial) components of the nonlinear connection H. (iii) The local components M^L and N^i transform on every intersection of preceding induced local charts on E by the rules (1)1

(1)1

\dtj

dxi

dt dt

and #<*> - A ^ > - — — - — M i\ Χ9Λ U (i)' - "Wdtdxl dxJ dxl Ox*' ' (iv) To give a nonlinear connection Ή on the 1-jet space JX(IR, M) is equivalent to giving on E = J 1 (E, M) a set of local functions iV

Y _ ( AfU)

MU) \

which transform by the rules (1.17) and (1.18). ■ EXAMPLE 1.9 Let (R, hn(t)) be a Riemannian manifold and let (M, ψ^(χ)) be a semi-Riemannian manifold. Let us consider the Christoffel symbols κ}χ(ί) and Jj¡.{x). Then, using the transformation rules (1.2), (1.8), and (1.11), we deduce that the set of local functions where M%i = -*ιι!/ί.

# $ < = Tfmí/Γ,

(1.19)

1

represents a nonlinear connection on the 1-jet space J (K, M). This jet nonlinear connection is called the canonical nonlinear connection attached to the pair of metrics (hn(t), φ^{χ)). In the sequel, let us fix Γ = (Ml(L, Ν/Χ),

a nonlinear connection on the 1 -jet space

E = Jx (R, M). Then, the nonlinear connection Γ produces the horizontal vector fields (1.16) and the covector fields Sy[ = dy[ + M^dt

+ N$dxi.

(1.20)

14

JET GEOMETRICAL OBJECTS DEPENDING ON A RELATIVISTIC TIME

It is easy to see now that the set of vector fields δ

δ

d

\Tt'^w}cX{E)

(l.2l)

represents a basis in the set of vector fields on J^M, M), and the set of covector fields {dt,dx\Sy\}dX*{E) (1.22) represents its dual basis in the set of 1-forms on J 1 (R, M). Definition 22. The dual bases (1.21) and (1.22) are called the adapted bases attached to the nonlinear connection Γ on the 1-jet space E — J 1 (R, M). The big advantage of the adapted bases produced by the nonlinear connection Γ is that the transformation laws of their elements are simple and natural. Proposition 23. The local transformation laws of the elements of the adapted bases (1.21) and (1.22) produced by the nonlinear connection Y = ÍM^L , W3!A are classical tensorial ones: S_

dt δ

St ' _δ_

άΐ'δΐ'

δχ}

θ dy\ and

dxJ δ dxi δχί' dx*dt d θχ' dt dy{

dt ~ dt — —zdt, dt dxl 3 dx , dxl dxi

(1.23)

(1.24)

dx* dt Proof: Using the properties (1.15), we immediately deduce that we have d_ dt

7Γ*

dt d _

_ ¡Μδ_

Λ ^ " " π * [dtöt

In other words, the temporal horizontal vector field δ St

dt δ dt St er(H)nx(V)

is also a vertical vector field. Taking into account the decomposition (1.14), it follows the required result.

JET NONLINEAR CONNECTIONS. ADAPTED BASES

15

By analogy, we treat the spatial horizontal vector fields δ/δχι Finally, let us remark that we have the equalities

* (£) *+*¡(¿) <*+*(=W dy[

h\

r i

=

Syl l

/ d i a

j r

dt+

. i (dxk

δ \ „..

dx3+Syl

(itTt) ^ {w**)

dx dt c^j

Corollary 24. Any d-tensor field D = ( D^^J."

. i (dxk

dt d \

,

6yi

{mniy»)

1 on the 1-jet space J1 (R,M) ¿5

a classical tensor field on J 1 (R, M). Proof: Using the adapted bases attached to a nonlinear connection Γ and taking into account the transformation rules (1.5) of a d-tensor, it follows that a d-tensor D — (^ifc(i)(z) j can be regarded as a global geometrical object (a classical tensor) on the 1 -jet space J 1 (R, M), by putting D

= Di$)lfc. ¡¡®¿®^j®dt®dxk®Sy[®

....

Remark 25. The utilization of parentheses for certain indices of the local components D j S w , ' " of the distinguished tensor D on J*(R, M) is suitable for contractions. To illustrate this fact, we give the following examples: (i) The fundamental metrical d-tensor produced by a relativistic time-dependent Lagrangian function (see Example 1.1) produces the geometrical object

(ii) The canonical Liouville d-tensor field of the 1 -jet space J 1 (R, M) (see Example 1.2) is represented by the geometrical object C-Cil)—

-«<

9

(iii) The h-normalization d-tensor field of tiie 1-jet space J X (R, M) (see Example 1.3) has the representative object

a)u¿[®^®^ ==hu—^¿

Λί\Ι.,·ΤΓ-Γ ®dt®dxj

®dt®dxi.

(iv) The h-canonical Liouville d-tensor field of the 1-jet space J X (R, M) (see Example 1.4) is equivalent to the geometrical object Lft = L^L.—r ^í'Lídy\

®dt®dt

= hny\—^ ® dt ® dt = C ® h. oy\

16

JET GEOMETRICAL OBJECTS DEPENDING ON A RELATIVISTIC TIME

1.4

RELATIVISTIC TIME-DEPENDENT SEMISPRAYS AND JET NONLINEAR CONNECTIONS

In this Section we study the geometrical relations between relativistic time-dependent semisprays and nonlinear connections on the 1 -jet space J1 (K, M). In this direction, we prove the following geometrical results: Proposition 26. (i) The temporal semisprays H =

ffi,

J and the sets of tem-

poral components of nonlinear connections M = ( M^L J are in one-to-one correspondence on the 1-jet space J 1 (R, M), via XfU) -2HU)

H{j)

--M{i)

(ii) The spatial semisprays G = (G/^L J and the sets q/ spatial components of nonlinear connections N = via the relations

JVl'i J are connected on the 1-jet space J 1 (R, M),

_ ^(1)1 (l)fe Qyk '

N(3) JV

- I i y r Ü ) „n, ( l ) l - 2 i V (l)r„yi '

G(i) U

Proof: The result is an immediate consequence of the local transformation laws (1.7) and (1.17), respectively (1.10), (1.18), and (1.2). Dennition 27. The nonlinear connection Y$ on the 1-jet space J 1 (R, M), whose components are TS = | M{g = 2 H & , N$k

dG\j) = - #

| ,

(1.25)

is called the canonical jet nonlinear connection produced by the relativistic timedependent semispray S = (H,G) =

(HtÍ]vG%).

Definition 28. The relativistic time-dependent semispray <Sr on the 1-jet space J 1 (R, M), whose components are *

= (
= \M%1

G% = \N$)my?) ,

(1.26)

is called the canonical relativistic time-dependent semispray produced by the jet nonlinear connection p _ (MU) NU) \

RELATIVISTIC TIME-DEPENDENT SEMISPRAYS AND JET NONLINEAR CONNECTIONS

17

Remark 29. The canonical jet nonlinear connection (1.25) produced by the relativistic time-dependent semispray S is a natural generalization of the canonical nonlinear connection N induced by a time-dependent semispray G in the classical rheonomic (non-autonomous) Lagrangian geometry [55]. Remark 30. Formulas (1.26) may offer us other interesting examples ofjet relativistic time-dependent semisprays. It is obvious that the equations (1.12) of the harmonic curves of the canonical relativistic time-dependent semispray (1.26) produced by the jet nonlinear connection p _ (MU)

NU)

\

have the form d?xj

—

„,m (

+

ki x dxk\

M(% (t,xHt), -a)

ij)

(

L.. . dxk\

+ N%m (t,XHt), -^)

dxm

-^

= 0.

(1.27)

Deñnitíon 31. The smooth curves c(t) = {xl(t)) which satisfy the equations (1.27) are called the autoparallel harmonic curves of the jet nonlinear connection Γ. Remark 32. The geometrical concept of autoparallel harmonic curve of a jet nonlinear connection Γ naturally generalizes the concept of path of a time-dependent nonlinear connection N from the classical non-autonomous (rheonomic) Lagrangian geometry [55] or that of autoparallel curve of a nonlinear connection 7Y from the classical autonomous (time-independent) Lagrangian geometry [24]. ■ EXAMPLE 1.10 The autoparallel harmonic curves of the particular jet nonlinear connection (see Example 1.9) attached to the pair of metrics (hn(t),ipij(x)) are exactly the affine maps between the manifolds (R, Λιι(ί)) and (M, φ^(χ)).

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CHAPTER 2

DEFLECTION d-TENSOR IDENTITIES IN THE RELATIVISTIC TIME-DEPENDENT LAGRANGE GEOMETRY

This Chapter provides an extension of the Miron-Anastasiei geometrical framework from [55] and describes on the 1-jet space J X (R, M) the adapted components of the Γ-linear connections, together with their d-torsions and d-curvatures. For an arbitrary Γ-linear connection, the local Ricci identities, together with their corresponding nonmetrical deflection d-tensor identities, are also determined. We point out that the nonmetrical deflection d-tensor identities are necessary for the description of the geometrical Maxwell equations which govern our jet relativistic time-dependent electromagnetism.

2.1 THE ADAPTED COMPONENTS OF JET l-LINEAR CONNECTIONS Let us suppose that on the 1-jet space E = J 1 (M, M) is fixed a nonlinear connection

r=KWíi,).

(2-1)

where M^L are its temporal components and N^l. are its spatial components. Jet Single-Time Lagrange Geometry and Its Applications ¡at Edition. By Vladimir Balan and Mircea Neagu. © 2011 John Wiley & Sons. Inc. Published 2011 John Wiley & Sons, Inc.

19

20

DEFLECTION D-TENSOR IDENTITIES IN RELATIVISTIC TIME-DEPENDENT LAGRANGE GEOMETRY

Let

WJ¿>~dy[)CX{E) and

{dt,dx\5y\}
be the dual bases adapted to the nonlinear connection (2.1), where

I _ 1 _ Μω A (1)1

%f

st'dt

A = A _ Νω A ¿x*

(l)i

ftr'

dy{'

5y\=dy\+M^)ldt

+ N^)jdxK

In order to develop a theory of the Γ-linear connections on the 1 -jet space E = 7Χ(Μ, Μ), we need the following simple result: Proposition 33. (i) The Lie algebra X(E) of the vector fields on E = J 1 (K, M) decomposes in the direct sum X(E) =

X(Hu)®X(HM)®X{V),

where X(H*) = Spanlj\,

X{HM)

= Span j A \

x(y)

= Span { A j .

(ii) 77¡e Lie algebra X*(E) of the covector fields on E = J 1 (R, M) decomposes in the direct sum X*{E) = X*{HR) Θ X*(HM) Θ **(V),

A"*(WM) = 5pon{
A,*(WAí) = 5pan{c/x ¿ },

**(V) =

Span{Sy\}.

Denoting by h^, hu, respectively v, the ^-horizontal, M-horizontal, respectively vertical canonical projections associated to the above decompositions, we get the following: Corollary 34. (i) Any vector field on E = J 1 (R, M) can be uniquely written in the form X = hRX + hMX + vX, VXeX(E). (ii) Any 1-form on E = J 1 (R, M) can be uniquely written in the form ω — /IRW + hjifu + νω,

\/ω£Λ"(£).

THE ADAPTED COMPONENTS OF JET Γ-LINEAR CONNECTIONS

21

Definition 35. A linear connection V : X(E) x X(E) —> X(E), which verifies the Ehresmann-Koszul axioms V/IR

= 0,

VhM = 0,

Vv = 0,

is called a T-linear connection on the 1-jet vector bundle E = J X (R, M), Using the adapted basis of vector fields on E = J 1 (R, M) and the definition of a Γ-linear connection, we prove without difficulties the following: Proposition 36. A T-linear connection V on E = J 1 (R, M) is determined by nine local adapted components Υ7Γ - Ir1

r> (i)(1)

rk

r1

rWW MW riW (i)0)fe' °i(fc)' °t(fc)'

rk

L/ rmi

(i)0)(fe),

;AÍC/Í are uniquely defined by the relations (/¡■R)

v

4=

V δr

- Ck

δ il

5xk'

^ δ -

St

ft

-L

(ΛΜ)

δχ*

1

δί

(fc)(i) d = G (W)\Qyk' d

¿ 0

υ

δζ*

Sxk

r (fc)(i) _ # _ ^(í)Wj^fc'

δχί

δχί

rfe(i)J_

»Wft' dyi

_ö_

v

(fe)(i)(D

c;(i)(¿)0)

a

dyffe·

%1

Taking into account the tensorial transformation laws of the adapted basis of vector fields on E = J 1 (R, M), by laborious local computations, we deduce the following: Theorem 37. (i) Under a change of coordinates (1.2) on the 1-jet space E = J 1 (R, M), the adapted coefficients of the T-linear connection V modify by the rules (

~ i _ ^ i dt -Gndt

G n

(h

rk

=

ΛΓ

" r

(l)(¿)l

+

dtdH 7tdt*'

dx^dx^dt >ιθχΓθχ*άί'

= G'(P)(l)

9a:fe <9íJ d£

fc

dt2'

L· -L1 — d2á -(fc)(i) "(i)(*b'

k p s = Z ( r ) ( 1 ) dx dx dx

(l)(p)sßxr

Qxi gxj

dxk

T Q~r +

d2xr QxiQxJi

22

DEFLECTION D-TENSOR IDENTITIES IN RELATIVISTIC TIME-DEPENDENT LAGRANGE GEOMETRY

Λΐ(ΐ) _ Λΐ(ΐ) dxJ dt

~s(\)dxk dxp dxr dt

cfe(i) =

(v)

iU) -

P(r) 0J.S Qxi Qxj ¿f

¿(r)(i)(i) dxk dxp dxfl dt

c(fc)(i)(i) =

(D(t)Ü) (i)(p)(g) öfr ο χ ι ö x j d ¿· (ii) Converse/y, ío give a T-linear connection V on the 1-jet vector bundle E = J*(R, M) is equivalent to giving a set o/nine adapted local components VI\ which transform by the rules described in (i). ■ EXAMPLE 2.1 Let hn(t) be a Riemannian metric on R and let φ^(χ) be a semi-Riemannian on M. We denote by κ\λ (t) [respectively ~fkj{x)] the Christoffel symbols of the metrics hn(t) and φ^(χ). Let us consider on the 1-jet space E = J 1 (R, M) the canonical nonlinear connection Γ associated to the pair of metrics (hn(t),
BT = (<5i1; 0, G\\]HJV 0, 4 , L^g,

0, 0, 0) ,

where r1

- K1

"-Ί1 —

K

1H

r(fc)(1) -

tT

fikKl

,v ( (l l) )((ii))ll ~~ ~°i" ¿ K l ll l>i

Tk --v fe

lj~

ij ij—!ij>

/¿j>

(k (fc)(i) _„,fc

T Tut ^^ (( 1l ) ( i ) j — '¿J

defines a Γ-linear connection on the 1-jet space E. This is called the Berwald connection attached to the metrics h\i(t) and φ^(χ). Remark 38. In the particular case (R, h) = (R, S) our Berwald linear connection naturally generalizes the canonical iV-linear connection induced by the canonical spray 2G% = Jjky-'yk from the classical theory of Finsler spaces (see Bao, Chern, and Shen [20] or Miron, Anastasiei, and Bucätaru [24], [55]). Now, let us consider that V is a fixed Γ-linear connection on the 1-jet space E = J J (R, M), which is defined by the adapted local coefficients VI -

^ 1 1 , O

u

, 0 ( 1 ) ü ) 1 , L-íj,

L·^,

^(i)(j)fe, 01{fc), Oi(fc), 0(1)(3.)(A;)j .

\l.L)

Then, the Γ-linear connection V naturally induces a linear connection on the set of the d-tensors of the 1 -jet vector bundle E, in the following way: Starting with a vector field X e X(E) and a d-tensor field D on E = J 1 (R, M), locally expressed by X=Xl*+XrS+X(r)d

St

D

= Dlk$)tLÍ

5xr

{1)

dyl'

®-^®-^-i®dt®dxk® Sy[,

THE ADAPTED COMPONENTS OF JET Γ-LINEAR CONNECTIONS

23

we introduce the covariant derivative

X^gD

VXD

+ X'V

Jt -

D + xfgV

δ

ΙχΡ

d

D

dy\

/yin^iKi), Yp r)i»0')(i), Y(P) nii(j)W-\M\ \ Λ - L 'ifc(i)(i).../i + A '1Jik(i)(i)...\p+JÍ(i)1Jik(i)(i)..Mp)J δ_ _s_ d >dtdxk ®Sy[, Sx% 'st dyi

where

lt(j)(l)...

SDlfc(l)(i)..

nlt(j)(l)...

^lfc(l)(i).../l

(hv

St

,r> l r (j)(i)-·.™

J£)li(j)(l).. 0U

ik(l)(l).

5χΡ

,

l*Ü)(l)-fl Χn

+

Λλ:(1)(ί)...^1ρ

^lMlja)-·

'ifc(i)(0- ^ r p

(l)(r)p

li(i)(l)·· D lr(l)(¿)... UfeP„ - i ?

ar)»(i)(l)··· ^lfcgXQ... ,

D lfe(l)(/)...l(p)

nlr(j)(l)...^i(l) + Iy 0 +

H(j)(l)... r ( r ) ( l ) Ι/Ο(Ι)(Γ)..ΛΙ)(/)Ρ

niiO)(i)...^i(i)

+

92/?

lfc(l)(i)... r(p)

1<(J)(1)·· ^ ( r ) ( l ) £>lfe(l)(r). ■ (1)(01

,liO')(l)··· - ^ 1 Γ ( 1 ) ( 0 - . fel

nH(j)(l)...fl "■^lfe(l)(i)- !P

(V)

ik(W)...uu

lfc(l)(i)..."(l)(r)l

"-L/lfc(l)(/)...°'H

(ftjw)

u

j - n i ^ X i l - r O X 1 ) 4-

'lk(l)(¿)..·

it(j)d)... £>lfc(l)(Z)...|p

+

^ifc(i)(0...°i(p)

ηΗ(Γ)(1)...^ϋ)(1)(1) ^1Λ(1)(ί)...°(1)(τ·)(ρ)

U(j)(l)...rr(l)

η1ίϋ)(1)...Λΐ(1)

D lr(l)(i)...°fc(p)

"^lfe(l)(/)...°l(p)

+

'

D.i»0)(D

W(i)(i) C. lfc(l)(r)...^(l)(i)(p)

Definition 39. The local derivative operators " / i , " "| p ," and <«|(1).. ¡, ■. are called the R-horizontal covariant derivative, the M-horizontal covariant derivative, and the vertical covariant derivative associated to the T-linear connection V I \ These apply to the local components of an arbitrary d-tensor field on the space E = J1(R,M). Remark 40. (i) In the particular case of a function f(t,xk,yk) J1 (R, M) the above covariant derivatives reduce to J/1 -

at St -

a+ dt

m

{i)ig

k>

/|p

df 5χΡ

Nw

on the 1-jet space

K.

f\w

df

(ii) Starting with ad-vector field!) = y on the 1-jet space!? = J ^ E , M ) , locally expressed by

st+

sx>+

w at/i'

the following expressions of the local covariant derivatives hold true:

24

DEFLECTION D-TENSOR IDENTITIES IN RELATIVISTIC TIME-DEPENDENT LAGRANGE GEOMETRY

SY1 St

1 Y r

/i

YL= 7i w

(ha

Y'GL·

SY*

+ YrGU, (i)

SY,

Y,ω

+γ;σ (1) " ( l ) ( r ) l '

St

(DA"

SY1 Y1\P = —— +V l f l § P X

SY*

{hM)

Y

W -

+

δχΡ

Υ

L

(1) | y M r ( ' ) P )

Y,((i) i)lp' vl|(l) J

r

l(p) dY*

y |(D KP)

(v)

^

y-(*)|(l) _

G

KP)'

+ YrC,r ( p ) '

dY, (i) (1) w

( D H P ) ~~ dy\

,C-Wi)(i)(i) i -y-vv/^k*;

(D°(i)(r)(p)·

Denoting generically by " : ^ " one of the local covariant derivatives "/χ, " "| p ," or "|; ?," we obtain the following properties of the covariant derivative operators: Proposition 41. IfT and S\ are two arbitrary d-tensors onE = J1 (R, M ) , then the following statements hold true: (i) The local coefficients T"\A represent the components of a new d-tensor field

onJHR.M). (ii)(Tv/ + S;;;):/l (iii) (T;;; ® S;;;):A 2.2

T::.A + S;:::A. = T;

>S-~+T~

LOCAL TORSION AND CURVATURE d-TENSORS

In the sequel, we will study the torsion tensor T : X{E) x X(E) associated to the Γ-linear connection V, which is given by the formula

T(X,Y)

= X7XY-X7YX-{X,Y},

X{E)

\/X,YeX(E).

In order to obtain an adapted local characterization of the torsion tensor T of the Γlinear connection V , we first deduce, by direct computations, the following important result:

LOCAL TORSION AND CURVATURE d-TENSORS

25

Proposition 42. The following identities of the Poisson brackets are true: \S S] 77 St' St

's

a'

,(r) R

TV S~x~i S ' 'S Sx*' Sxi

3<)i d

st: dy{

' s

5 '

'S

=0,

dy{

dy\

9NÜI d

a

SxV~dy~{

dy{

' d dy\

dy\

9 d

(r)

R (l)iJ'öyr'

d '

= 0,

where Ai( [?j and ΛΓ,^Λ are ífte /oca/ coefficients of the nonlinear connection Γ, while (r) ?(r) ?Λβ components R),L ■ and ?RrÁ^ are d-tensors given by the formulas (i)ij (Dü

<5JV,(r)

(5M,(r)

(1)J

(1)1

Sxi (r)

R (r)

(D'J

(2.3)

(r) SN 0JV (i)j

SN,(l)i

fe*

In these conditions, working with a basis of vector fields, adapted to the nonlinear connection (2.1), by local computations, we obtain Theorem 43. The torsion tensor T of the Y-linear connection (2.2) is determined by the following ten adapted torsion d-tensors: ,

S S

hKTl

Tt'Tt

= 0,

hMT

δ_ S_ St' St

^'Λ' VT

Ί«Τ

—7.-ΓΤ 5χ·?' ¿x* vT

ΛΜΤ

\ ¿ Λχ~ίϊ '' Λ+ 5t ) ~

=0,

hMT

δχϊ'δχ*) 5K1) i .

huT

υΤ

0,

η

5

m l

S

vTl

Tt>Tt

lj r Sx ν ^ ' Α±\=τ, ί

(1)1ί^Γ'

Sxi' áx*

'Sx*·'

W'dyi

a

J L A 1 - pM ί1) a

¿

77

=0,

o,

26

DEFLECTION D-TENSOR IDENTITIES IN RELATIVISTIC TIME-DEPENDENT LAGRANGE GEOMETRY

h»T

l

dy{' 5x

0,

T

* (¡sr£)"°· vT(—

δ

Pr(l)

hMT

áMT

tó'a!

-OMÜHDA

—λ

where ι

υ

τ1 lj'

Gr

T
Ll

pr(l)

=

^r(l)

dM{r) p(r)

ÖM

(1) _

(1)10)

(1)1

dy\ (r)

and í«e d-tensors Rnn

i-pr

rr

? M(1)(1)

(r)(l)(l) C,(1)(*)0)

'(l)(«)(j)

rW(i)

(r)

are

/"il(l

10)'

(r)(l)(l) ■ c,

(1)0)(¿) '

(r) aTv; (l)i

3(r)(l)

^(1)0)1'

· and -^au, (l)i¿

pl(l) 10)

rr

(l)i(j)

9j/i

-(0(1) J (i)0)¿'

given ¿y (2.3).

Corollary 44. 77¡e torsion tensor T o/cm arbitrary Γ-linear connection V on ine / -y'eí space E = J1 (R, M) /s determined by ten effective adapted local torsion d-tensors, which we arrange in the following table: hn /IR/IR

0

„

hu

o

o p(r) Λ (1)1ί

huh®.

%

htih-M

0

vhu

pi(i) ^l(j)

0

p(r)(l) ^(1)1(¿)

0

pr(l) ιθ')

p M (1) ^(IWj)

vhM

Ϊ«

^

Λ

(ΐ)ύ·

?W(i)(D 5

(l)(i)Ü)

LOCAL TORSION AND CURVATURE d-TENSORS

27

■ EXAMPLE 2.2 In the particular case of the Berwald Γ-linear connection ΒΓ, associated to the metrics hu(t) and ψ^(χ), all torsion d-tensors vanish, except DC 0 ) _ fr>k „,m (1)Ü ~ A n i j i / l >

where ÍH„¿j ( x ) aTe t n e classical local curvature tensors of the spatial semi-Riemannian metric φ^(χ). In order to study the curvature of the Γ-linear connection V, we recall that the curvature tensor R of V is given by the formula

R(X, Y)Z = VxVYZ

e X(E).

- VYVXZ - V[X>Y]Z, VX,Y,Z

Using again a basis of vector fields adapted to the nonlinear connection Γ, together with the properties of the Γ-linear connection V, by direct computations, we obtain the following: Theorem 45. The curvature tensor R associated to the T-linear connection (2.2) is determined by fifteen effective adapted local curvature d-tensors:

R

U'¿¡)^=0' Ά

(δ

δ\δ

\ TTfc' 72 I 72 ~ δχ^δί/δί

R

U'^)^=O'

m

rtl l f c

δ

iifc7T' Jí'

\Sxk'6t)dy[

Ά

f δ

R δ\

U'^)^[ =o ' δ _

k, x¿ Ii TU ~ \ \5x TTfc' StJSx

WWkdy['

t

δ δχι

Λ ίΙΗ α

28

DEFLECTION D-TENSOR IDENTITIES IN RELATIVISTIC TIME-DEPENDENT LAGRANGE GEOMETRY

d

R

d

? /(i)(i)_£_

?1(1)(1)_0_

'i(j)W¿t'

d

R

i(j)(k) §xl '

_d_\ _d_

d k

V\

dy{)

(i)(i)(i)(i)

d

= "s:U K W H * ) ^

dy\

!

whose local components we arrange in the following table:

hRh®

\

hu

1 h,M \

v

|

0

|

Ó

0

/ϊ Αί/iR

R\ik

^ilfc

/lAí/lM

R-ijk

Rijk

vhji

p i (1) Ml W p i (1) r ij(k) ci(i)(i) °iO)(fc)

pi (1) ^il(fc)

vflM

vv

p i (1) ij(k) o¡(l)(l) °i(j)(k) r

|

p(¡)(l) (l)(i)lfc

rt

p(i)(l) (i)(i)jk p(0(i)(D n

p(')K> (1) Ul)(i)j(k) c(0(i)(i)(D °(i)W(i)(fc)

Moreover, by laborious local computations, we deduce the following result: Theorem 46. The components of the preceding local curvature d-tensors are as follows: • h^-components 1

ffi

h

^ p i (i)

r

áL

lfc

, Λΐ(1)ηΜ +°l(r)-tt(l)lfc

¿í

+0

Sxj

öÖJi

1(Γ)Λ(1)^

Λ,.Α;

Λΐ(1) °l(fc)/l

ij(«

dyk

c¡( f c ) | j

l(l)d)

1(1) ÖC.10)

1(1) dC l(fc)

%1

dy{

M1W

4 P1 4

"

sL

-i

°

SÖ

(1)

5.5.iü)(fc)

1(1

, p l ( l )r p W (1) °l(r)- (l)l(fc)

+

I Λ^l(l) Ί ' ί η Μ (!) °l(r)^(l)j(fe)

+

liM-components 6.ÜÍ lfe

« ^ <5xfc

¿L
(r)"(l)lfc

LOCAL TORSION AND CURVATURE d-TENSORS

RÍ - SLV *J'fe ~ Jx*

7

β

i TT Tl ϋ rk

ík 5xi

Tl Tr Ll L ik 'rj

, r l { l ) R(r) + °¿(r)- f t (l)jfc

(1) __ dG\l _ nKl) _j_ r¿W Γ » i 1 ) · ^ 1 ( ί : ) ~ Q,.fc °t(fe)/l + ° i ( r ) - r ( l ) l ( f e )

o p i (i) _ *-Uj(k) ~

in 1U

•

SL

_ r 1 ^ ir¡(1)p(rl W °i(fc)|j + ° i ( r ) ^ ( l ) j ( f c )

dL

ij Qyk

C«1»!) _ · öi(3)(k) ~

¿(j) _ Qyk

°'W dyj

+

, ^(1)^(1) _ ^(1)^(1) ° ¿ 0 ) °r(fc) °¿(fc) ° r ( j )

v-components ΛΓ(0(Ι) W(i)k

xr(i)(i) 0<J

n

p(«)(i) _ li --K(l)(i)lfe -

01j

(i)Wi ¿a .fc

_ r(r)(l)

r,(/)(l)

, ^(0(1)(1) D W

+

^(1)(»)^(1)(Γ)1

1 9 L£

p(l)(l) (i)(i)jk

n

... -

0iy

(l)(»)j ί χ *

0¿¿

(l)(i)fe

faj

_ r(r)(l) r(i)(D L '(l)(i)kLl(l)(r)j ar(0(l) -,ο ρ(ί)(1) (1) _ " ^ ( I X » ) ! 1 í / « - (l)(«)l(fc) Q,.fe

9yf

-, c
alJ

°(l)(i)(r)-"(l)lfe

+

, r(r)(l) r(l)(l) ^(iXOj^UHr)*

,Γ(Ι)(1)(1)ρΗ + ° ( l ) ( i ) ( r ) rt(l)jfc

^(ί)(1)(1) , r . ( 0 ( l ) ( l ) p ( r ) (1) υ (1)(ί)(*)/1+υ(1)Μ(Γ)Γ(1)1(ΐ!)

(ί)(ΐ)

ΛΓ

p ( 0 ( i ) (i) _ *-*■ Um)j(k) 1A

. r(r)(i) Γ (ί)(ΐ) + °(1)(ί)1^(1)(Γ)Λ

¿ í

(i)(¿b· Qyk

Λ Γ .(0(1)(1) a

°(l)(»)(j) «„fe

öyf

^(0(ΐ)(ΐ)

, Γ ( ΐ ) ( ΐ ) ( ΐ ) ρ Μ (i) ^{1)(ι)(ν\ϊ^^(1)(ϊ)(τ)Γ(\)3{ν

α^(0(1)(1) _ °°(l)(»)(fc) , r(r)(l)(\)r{l)(l)(í) fsj "·" °(1)(«)0·) °(D(r)(fe)

dy{

r(r)(l)(l)r(l)(l)(l) '°(l)(t)(k)U(l)(r)Oy

EXAMPLE 2.3 In the case of the Berwald Γ-linear connection ΒΓ, associated to the pair of metrics (hu(t),

1 ijk

n

_ nil — -^ijk'

p(0(l) _«>/ (1)(ϊ)ί*; Hjfc>

Λ

where 9t¿ -fc(x) are the classical local curvature tensors of the spatial semi-Riemannian metric

φ^(χ).

29

30

2.3

DEFLECTION D-TENSOR IDENTITIES IN RELATIVISTIC TIME-DEPENDENT LAGRANGE GEOMETRY

LOCAL RICCI IDENTITIES AND NONMETRICAL DEFLECTION d-TENSORS

Using the properties of a Γ-linear connection V given by (2.2), together with the definitions of its torsion tensor T and its curvature tensor R, we can prove the following important result which is used in the subsequent Lagrangian geometrical theory of the jet relativistic time-dependent electromagnetism, in order to describe its geometrical Maxwell equations. Jl{R,M),lo-

Theorem 47. IfX is an arbitrary d-vectorfield on the ¡-jet space E 1

J

cally expressed by X = X - -fX^+X^^-, -^iJ_ j . V « ) _ 8 _ then the following Ricci identities of the Y-linear connection V are true: yl ^ /Vil* l|fe \j\k

y l |(1)

A

yl|(l)

_ Λy i rp

1

xt,(D

,(»·) (i) (r)Vl)l(fe)

/=Vl(l)

yl

A

/ri(fc) yi r,r'1) A |r°j(fc)

(1)

yl|(l)p(r) \(r)n(X)lV

Λ

l(r) J (l)jfe'

(i)

n(fc)

ylpl

(i) (k)\j

νΐττ \r1lk

Ji

XLT:jk

R\jk

l(fe)/i -

X1

X

|(1) /il(fe)

x

|j'(fc)

\

X'

- y r p»

X |fc/i

/life

X Ijlfe

XrR

X |fe|j X1

X¡

vllWnW Λ

(1)

\(rV(l)j{k)'

(i) _ χί

cr{1) -X

_ -

Λ

i(i) j(r) (1) l(r) J (l)j(fc) '

r)(l)(l) X< |(1) l(r)^(l)(j)(fe)'

yWnW(l)

Λ

v i | ( l ) p W (1) l(r) (l)l(fe) "

¡r^jik)

¿(1)(1)

(i)

(i) |(1) X,(i)/il(fc)

P /„l ^Cl ( 1(1) fc)

xrs; (j)(fc)

X,(l)|fe/l

(l)p(r) (r)n(l)lfc'

Y ¿ | ( 1 ) p ( nr ) * \(r) (l)jk'

rj(k)

|(1)|(1)

(i) X,W\k\j

i (1) rl(fc)

X

X\rT{k

\r1jk

XrP

(i) X,(i)|j|fe

τΊ

Ä

Xrp

l(fc)/i (fe)|j "

|(1)|(1) HjV(k)

y*

rjk

|(1)

|(D . X*\ (1)

(i) X (i)/i|fc

(v)

v\ Φ1 ^ /Vi lMfc

^1|(1)|(1) _ V l c l ( l ) ( l ) yllWc-MUKl) 1|(1)|(1) | , t J , , , - A 5 1 ( ) ( f e ) - A |(r)^(l)0')(fe) (r)Ö(1)(,-)(fc), Ü)'(fc) - A WO)

X

(hM) <

x

ii(i)

-X

/il(fc)

X

ylpl n llk

Λ

-

X, k\j

X

(AR)

yl ^\k/l

γ"(ν

y(i) φΐ ^(lí/l-'lfe

(1)Λ(1)(Γ)1Α:

rpr

Λ

(1)|Γ-ΊΛ

L

(l)l(r)"(l)jfe'

y(¿)|(i)p(r) (l)l(r)-"(l)lfe'

A

v(r)p(i)(l)

(1)

r(¿)|(l) L (i)l(fc)/i

Λ

(1)(Γ)^

(í) y r _ - X , (l)|r jfc

v ( r ) p ( t ) ( l ) (1) (1)-νΐ)(Γ)1(Α0

X^l^R^

1(1) X ((¿)l ) / l "c l(fc)

Λ

X ( l ) ll((r l) ^) (pl «) l ( f(1) c)' (i) |(1) X,(líb'Kfc)

YWp(i)(l) Λ

(1) ( 1 Γ (l)(r)j(fc)

(¿)|(1) X,(i)kfe)b'

- X , (K (i)|(l)|(l) X (i)l(j)l(fc)

Wi(i)i(i) X (DHfcJKj)

( x,(•l)|r^j(fe) íLc r(l)

| ( l ) p ( r ) (1)

y(Oc(0(l)(l)(l)

Λ

(Γ(ΐ)Μϋ)(*)

y ( « ) l ( l J) o W ( l ) ( l ) (l)l(r) (l)(j)(fc)·

A

LOCAL RICCI IDENTITIES AND NONMETRICAL DEFLECTION d-TENSORS

31

Proof: Let (YA) and (ωΑ), where A € < 1, iSJ >, be the dual bases adapted to the nonlinear connection Γ and let X = XFYF be a distinguished vector field on the 1 -jet space E = J1(R, M). In this context, using the equalities VYaYB=TFcYF,

1.

{YB,YC}=RF:CYF,

2.

T(Yc, YB) = TFBCYF = { r g c - TFB -

3.

R%B}YF,

R(YC,YB)YA=RFBCYF,

4. 5.

νΥι:ωΒ

=

-TBFCUJF,

6.

[R(YC,YB)X]

-ωΒ.ω°

= {VYcVYllX

- Vy B VYcX Β

-ν[Υ(;,ΥΒ]Χ}·ω -ω°, where " · " represents the tensorial product "(g)," we deduce by a direct calculation that X-.B-.C ~~ X-.C-.B = X R-FBC ~~ X-.F^BC' (2-4) where " :D " represents one from the local covariant derivatives "/χ," "y" or "|[ -j" produced by the Γ-linear connection V. Taking into account that the indices A, B, C,... belong to the set 11, iSJ >, by complicated local computations, the identities (2.4) imply the required Ricci identities. Now, let us consider the canonical Liouville d-tensor field dy\

yi

dy\

Definition 48. The distinguished tensors defined by the local components Γ>Μ (Di

u

— c^ ~ ^(i)/i'

η ^ {i)j

u

_ r*W - ^(l)b'

j(*)(i) _ ri(t)|(i) (i)(j) - ^(i)l(j)

a

-. n ^z·^

are called the nonmetrical deflection d-tensors attached to the T-linear connection V on the 1 -jet space E = J ^ R , M). Taking into account the expressions of the local covariant derivatives of the Γ-linear connection V given by (2.2), by a direct calculation, we find the following: Proposition 49. The nonmetrical deflection d-tensors of the T-linear connection V have the expressions ñ(i)

- ~M{i)

+r? ( i ) ( 1 ) ΉΤ

n(i)

- -Ν{ΐ)

4- τ{ί){ι)

ιΓ (2.6)

{m)

¿ (¿)(1)(1) r =á = ) +· Ο+(7 ?-/

d (!)(/) ' ΐ

'(1)('-)0·) ί·

32

DEFLECTION D-TENSOR IDENTITIES IN RELATIVISTIC TIME-DEPENDENT LAGRANGE GEOMETRY

In the sequel, applying the set (v) of the Ricci identities to the components of the canonical Liouville d-tensor field C, we get Theorem 50. The nonmetrical deflection d-tensors attached to the T-linear connection V on the ¡-jet space E — J 1 (R, M) verify the following identities: f)(>) _ r>( ¿ ) — ,,»·ρ(0(1) _ ñ(») Ί'1 ^(l)l|fc ^(l)fc/l — Ϊ Ί Λ ( 1 ) ( Γ ) 1 Λ ^(ljl^l*

nW

r><0

_„r R (»)(i)

nWi»

D ( i ) 7'Γ _ Λ<ί>(1) í ? ( r ) a - (l)r-£lk (l)(r)n(l)l/c'

x ,

.JWWBW

a ■^(Iblfc ^(l)fc|j - 2 Ί Λ ( 1 ) ( τ · ) ^ ^(Dr-'jfc (.D(.r)n(l)jk' ñ ( i ) |(1) _ j ( i ) ( l ) _ „ r p ( » ) ( l ) (1) _ ñ ( t ) Λ ΐ ( 1 ) _ . ( i ) ( l ) p ( r ) (1) a ^(l)ll(fc) (l)(i!)/l - Wr(l)(r)l(fc) "(l)ri(fc) "(l)(r)^(l)l(fc) '

r>(¿) |(1) ^UbKfc)

(Í)(1) _ „ r p ( i ) ( l ) (1) _ r > « r r ( 1 ) _ W P(r) U (l)(fc)|j - yiUl)(r)j(k) ^(l)rj(k) Ci)(r)^(l)j(k)

aJ(*)(l)

j(«)(D |(D _ j ( 0 ( l ) |(1) _ „ r o(»)(l)(l)(l) _ J W ( 1 ) o ( r ) ( l ) ( l ) a a a (l)0')l(fc) (D(fc)IW _ » l i 5 ( l ) ( r ) U ) ( f c ) (l)(r)°(l)Ü)W

(1)

'

, , .,, <- ¿ ·'»

CHAPTER 3

LOCAL BIANCHI IDENTITIES IN THE RELATIVISTIC TIME-DEPENDENT LAGRANGE GEOMETRY

Because of the huge number of nontrivial adapted components (nine sets) which characterize a general Γ-linear connection V, the aim of this Chapter is to introduce the geometrical concept of h-normal T-linear connection on the 1-jet space J X (K, M), whose local adapted components reduce only tofour sets. This new geometrical object naturally generalizes the notion of normal N-linear connection from the theory of Finsler spaces [20] or Lagrange spaces [55] on the tangent bundle. On the other hand, the importance of Bianchi identities in the geometrization of time-dependent Lagrangians is well known (see Miron [55], [56], Crampin [29], or Neagu [63]). The present Chapter describes the local Bianchi identities attached to an /i-normal Γ-linear connection of Cartan type. 3.1

THE ADAPTED COMPONENTS OF h-NORMAL Γ-LINEAR CONNECTIONS

Let us suppose that the 1 -jet space E = J 1 (E, M) is endowed with a nonlinear connection

^Klt'O" Jet Single-Time Lagrange Geometry and Its Applications 1st Edition. By Vladimir Balan and Mircea Neagu. © 2011 John Wiley & Sons, Inc. Published 2011 John Wiley & Sons, Inc.

(3-D 33

34

LOCAL BIANCHI IDENTITIES IN RELATIVISTIC TIME-DEPENDENT LAGRANGE GEOMETRY

Let us consider also that we have on the 1 -jet space E a fixed Γ-linear connection V, defined by the adapted local coefficients (f=,\

vr

nk

rl

η(ΐ)(\)

r(i)(l)

rk

^l(l)

ri(l)

n(i)(\)(\)\

,-,-,

Definition 51. A Γ-linear connection V on E, whose local components (3.2) verify the relations

where h = (hn(t)) is a Riemannian metric on R, κ\λ is its Christoffel symbol, and Jfc is the /i-normalization d-tensor field (see Example 1.3), is called an h-normal Γ-linear connection on E — J1 (R, M). Remark 52. The condition VJ^ = 0 is equivalent with the local equalities jd) _ Q (l)lj/1 - U'

J

_ A - U>

T(»)

J

(l)lj\k

J

T(») id) _ Q (l)ljl(fc) - U '

where "/i," "|fc," and "|L\" represent the R-horizontal, M-horizontal, and vertical local covariant derivatives produced by the Γ-linear connection VT. In this context, we can prove the following important local geometrical result: Theorem 53. The components of an h-normal T-linear connection V verify the identities u

n

—

K

L,

ii>

\j

1

r ^ X ) _ r
^(ΐ)(ΐ)ΐ — ^ ¿ i

itKi

ö K

i n>

-

u

'

i(k)

fc

h

r( )(!) _ rfc

(i)(i)j

- ^j^

~

u

'

^(fc)(i)(i) _ ^fe(i)

°(D(0(j) - ° i ü ) '

(3

·3)

Proof: The first three relations from (3.3) are a direct consequence of the definition of an /i-normal Γ-linear connection V. The condition VJ^ = 0 implies the local relations hnG(1)(i)l¿)(i) ft L

» (l)(j)fc

= /iiiGj·! + δ) « n i

- "H^jfe.

ft

HG(l)(i)(fc)

-

dhu di h C

^ j{kV

where Km = n^hn represent the Christoffel symbols of first kind attached to the Riemannian metric ftn(i). Contracting the above relations with the inverse h11 — l//iii, we obtain the last three identities from (3.3). Remark 54. Theorem 53 implies that an /i-normal Γ-linear connection V is determined by four effective local components (instead of two effective local components for a normal Af-linear connection in the Miron's case [56], p. 21), namely, VT = ( « > , , G^L^C™).

(3.4)

The other five components of V r cancel or depend on the above four components, via the formulas (3.3).

THE ADAPTED COMPONENTS OF ft-NORMAL Γ-LINEAR CONNECTIONS

35

■ EXAMPLE 3.1 The canonical Berwald Γ-linear connection associated to the pair of metrics (/iii(i), yjjj(x)) is an /i-normal Γ-linear connection, defined by the local components B f = ( « ! ! , 0,7$, 0 ) . The study of adapted components of the torsion tensor T and curvature tensor R of an arbitrary Γ-linear connection V on E — J 1 (R, M) was completely done in the preceding Chapter, where we proved that the torsion tensor T is determined by ten effective local d-tensors, while the curvature tensor R is determined by fifteen effective local d-tensors. In the sequel, we study the adapted components of the torsion and curvature tensors for an /i-normal Γ-linear connection V r given by (3.4) and (3.3). Theorem 55. The torsion tensor T of an h-normal T-linear connection V on E is determined only by the following eight adapted local d-tensors (instead often in the general case): | flR |

flM

|

V

hnhn

| 0 |

0

|

0

h.Mhu

0

flMtlM

0

rt-tr

vhm

0

0

vllM

0

vv

0

Ä(r)

where 1. T\ 2. R o

δΜ, (r)

(r)

τ~τ

(r)

4 R

Tr

-

5. P,d ) l ( j(1) ) r(l)

6. Pi(j)

(i)j

(1)1

δχί

l)lj

δΐ

'

Tr (r) δΝ,(l)i

δΝ,

dM, (r)

1)1

dy[ ■r(l)

1)J

5a; *

δχί

cHi) '

(r)

^

¿J/ill,

R(r) n

(l)ij

p M (1) p(r) (1)

0

<j(r)(l)(l)

(3.5)

36

LOCAL BIANCHI IDENTITIES IN RELATIVISTIC TIME-DEPENDENT LAGRANGE GEOMETRY

(r) (1) (l)i(i)

Ί.Ρ «

dN,

(r) ¿«'

oM(D(l) _

rr(l)

■r(l)

cJ(i) · °(W)U) ~ Hi) Proof: Particularizing the general local expressions from Theorem 43 (which give those ten components of the torsion tensor of a Γ-linear connection V, in the large) for an /i-normal Γ-linear connection V, we deduce that the adapted local components Tlj and P-uJ vanish, while the other eight ones from Table 3.5 are expressed by the preceding formulas. β

Remark 56. The torsion of a normal iV-linear connection in the Miron's case is characterized only by five effective adapted components (see [56], p. 24). Remark 57. For the Berwald Γ-linear connection BY, associated to the metrics hn(t) and φ^(χ), all adapted local torsion d-tensors vanish, except n( fc ) (l)ij

_ mk ~ ^mijVl

n

m

'

where ÍK^ÍJ {x) are the classical local curvature components of the semi-Riemannian metric φ^(χ). The expressions of the local curvature d-tensors of an arbitrary Γ-linear connection, together with the particular properties of an /i-normal Γ-linear connection, imply a considerable reduction (from fifteen to five) of the effective local curvature d-tensors that characterize an /i-normal Γ-linear connection. In other words, we have the following: Theorem 58. The curvature tensor R of an h-normal T-linear connection V on E is characterized by five effective local curvature d-tensors (instead of fifteen in the general case): h-n

flM

V

0

1 1

/IR/IR

o

0

hhihn

0

Ruh

flMflM

0

vhn

0

p i (1) ^il(k)

p ( 0 ( l ) (1) _ rp i (1) ^(i)(<)i(fc) - il(k)

vflM

0

r

p'(l) ij(k)

p(0(l)(l) Mi)Wj(fc)

VV

0

oi(i)(i)

B(l)(l)

_

D(0(I)

_

■Rjifc Rijk

_ p'(l) ~

c(0(i)(i)(i) _ o¡(i)(i) °(i)(¿)ü)(fc) ~ °i(¿)(fc)

where 1

R

nk

5Gla

- Sxk

SL\ik St

i f i r rl rr /~
(3.6)

DEFLECTION D-TENSOR IDENTITIES AND BIANCHI IDENTITIES FOR CARTAN D-CONNECTIONS

Δ

-

H

ijk

-

§xk

o pi (1) _ «*■ Ul(k) .

*·

pl

§xj

dG

a

dyk

+

L

'ijL'Tk

r c/(l)(D _ °- °i(j)(fc) ~

°»(fe)|j

'(j) __ g y fc

0

ί(Γ)Λ(1ΗΑ;'

,rl(Dp(r) (1) ^i(k)/l+L/i(r)Ul)l(k)>

_rKi)

(1) _ ^ i ¿ _ rl(l)

u(fc) ~~ gyfc

~ L'ihL'rj +

37

rl(l)

+u

'(fc) Qy3

p (r)

(1)

<Wr(i)iW +

, rr{l)rl(l) _ ^(1)^(1) tO) r(fc) ^¿(fe)0^)·

Proof: The general formulas that express those fifteen local curvature d-tensors of an arbitrary Γ-linear connection (see Theorem 45 and Theorem 46), applied to the particular case of an /i-normal Γ-linear connection V o n £ ' = J 1 (R, M), imply the preceding five formulas and the relations from Table 3.6. Remark 59. The curvature of a normal iV-linear connection in the Miron's case is characterized only by three effective adapted components (see [56], p. 25). Remark 60. For the Berwald Γ-linear connection BT, associated to the metrics h\\ (t) and φ^(χ), all local curvature d-tensors vanish, except n(')(i) — _ pi¿ fe —_ ml (i)(*)jfc J yfc' where 9Ί' -fc(x) are the local curvature tensors of the semi-Riemannian metric
DEFLECTION d-TENSOR IDENTITIES AND LOCAL BIANCHI IDENTITIES FOR d-CONNECTIONS OF CARTAN TYPE

Because of the reduced number and the simplified form of the local torsion and curvature d-tensors of an /i-normal Γ-linear connection V on the 1-jet space E, the number of attached local Ricci and Bianchi identities considerably simplifies. A substantial reduction of these identities obtains considering the more particular case of an h-normal V-linear connection ofCartan type. Definition 61. An ft-normal Γ-linear connection on E = J 1 (K, M), whose local components Vr=(/ti1,G?1,L*,C*¿1))) verify the supplementary conditions ZA = Z& and Cj.J h-normal Γ-linear connection ofCartan type.

= CjJ,

is called an

Remark 62. In the particular case of an h-normal Γ-linear connection ofCartan type, the conditions L^ = Lj?¿ and CjJ = CjJ imply the torsion equalities J

ü-

U

'

ö

(i)(i)Ü)

~υ·

38

LOCAL BIANCHI IDENTITIES IN RELATIVISTIC TIME-DEPENDENT LAGRANGE GEOMETRY

Rewriting the local Ricci identities of a Γ-linear connection V (described in detail in Theorem 47), for the particular case of an /i-normal Γ-linear connection of Cartan type, we find a simplified form of these identities. Consequently, we obtain the following: Theorem 63. The following local Ricci identities for an h-normal T-linear connection of Cartan type are true:

x

/m 1

(M

X

y l |(1) /l\(k)

-X1

|(1)

yUW

-X1

(1)

A

A

|(1)

X

X l

l

-X l

X,W

(i)l¿|fc

y(')

A

rj(fc)

l

i(fc)'(i)

(i) X (l)|fc/l

(¿) X,(l)/l|fe

-X

|(1)

(i)/ikfc)

(¿) i(l) X WW(k)

l|(l)p(0(l)

yi|(l)pW l(r)-"-(l)lfc'

Λ

— !V* f , r ^

= Xr P^ ^

¿ (1) (1)

k¿)'(fe)

Y

,(1) _ vrpi (1) _ v i | ( l ) p ( r ) (1) Λ l(fc)/l ~~ Λ rl(fc) 1(0 (l)l(fc) ' '(fc)|j

¿ (1) (1)

-K

x r #rife *|Α

X 1 id)

|j!(fc)

l | ( l ) p ( r ) (1) ! W r ( i ) (fc)'

-X

i = X r i ?rjki , l . - Xl(r)"(l)jfc' lMr)

|fcb'

-X'

/il(fc) X Id)

(l)lfe

'Wim ~ u '

X,|fe/l

|j|fc

)

(r) \(r)H (i)jfc'

=

=

(k)\i

i(j)l(fc)_A

x /l|fc

(«)

l(fc)/ 1

-χψ

1 rpr

UJ = - X

X

x

"~X 1

Y X

(ΛΜ)

—

1

\k/

|r

j"(fe)

— V* ( l ) rp ( O ( D

(r) (l)j(k)'

-¿(1)(1) xrsi{l)W r(j)(fc)'

= v(r)

(«) rpr X (i)Kife

pi

(¿)| : i r ) X (l)l(r)-"-(l)lfe' ^

(¿) y(¡)|(l)pW — V< r ) I?« Λ (1)Ι(Γ)Λ(1)^' (i)|fc|j - v i ( 1 ) i t r j f e y ( 0 | ( l ) p ( 0 (1) y(i)\W _ y(r)pi (1) A A ^(\)^rl(k) (l)l(r)^(t)l(fe)' (i)W/i -

_ y(0|(l) A

. Ay(r) pi (1) _ y(i) rr{\) A (l)|rS - (fc) (i)kfe)ij" (l)rj(k)

X,W|(1)|(D - X . W | ( l ) | ( l )

■X,(¿)|(l)p(r)(l)

(l)l(r)^(l)j(fc)'

(r)c¿(l)(l)

x (l)^r(j)(fc)' s,

X^'A+X'A + X<;»A is an arbitrary distinguished

vector field on the 1-jet space E = J 1 (R, M ) .

In what follows, let us construct the nonmetrical deflection ñ(')

— fW

^(i)i -

^(i)/i

l/i/i.

n(¿)

n

Wi

_ r(i)

-

Si)|j

_

_2/l

i

b''

J(¿)(1)

d-tensors _

n

(i)|(l) (i)i(j)

yi

|(D

DEFLECTION D-TENSOR IDENTITIES AND BIANCHI IDENTITIES FOR CARTAN D-CONNECTIONS

39

attached to the h-normal Y-linear connection (3.4). Then, a direct calculation yields the following: Proposition 64. The nonmetrical deflection d-tensors attached to the h-normal T-linear connection V given by (3.4) have the expressions (1)1 + °>i?/i

^(i)i ~

K

(i)j ~~

uVn

(i)j

^jVi'

j(i)(l) _ ri , /^*(l),.r

Applying now the general nonmetrical deflection d-tensor identities (2.7) to the /i-normal Γ-linear connection of Cartan type V given by (3.4), we find the following: Theorem 65. The following five identities of the nonmetrical deflection d-tensors associated to an h-normal T-linear connection of Cartan type (instead of three in the Miron 's case [56], p. 80) are true: ñ(»)

— „rm

_ r>W

Tr

_ n^

^(l)l|fe (l)fc/l "" yi-^rlfc ^(lJr-Mfc /-»(*) _ Γ)(*) - , , Γ βΛί _ αj(*)(l) r>(r) Λ

^(lb'lfe

^(l)fc|j - ί / ΐ Γ ^

|(D _ J ( ¿ ) ( 1 )

f)(i)

U

^(l)ll(fc)

_

„rp.

α

AVW n(r)

(1)(Γ) Λ (1)1Α:'

(1)(Γ) (1)^'

(1) _

(l)(fc)/1 ~ » l r l ( i )

|(1) _ J ( < ) ( 1 ) _ „ r p ¡ (1) _ n W ^(l)jl(fc) °(l)(fc)|j -y^rrj(k)

j(t)(l) p(r)

U

(1)

(l)(r)r(l)l(t)'

n(i) U

^r(l) _ (\)r^j(k)

,,

s

,

^0''

,(t)(l) p ( r ) (1) a (l)(ry(l)j(ky

aj(t)(D

|(1) _ j(i)(D |(D _ „r Dc?i(l)(l) (l)(j)Hk) a(l)(k)\(j) -yi r{J)(k)·

Remark 66. The identities (3.8) are used in the local description of geometrical Maxwell equations that govern the jet single-time electromagnetic 2-form produced by a relativistic time-dependent Lagrangian on the 1-jet space E = J 1 (R, M). The use of /i-normal Γ-linear connections of Cartan type in the study of differential geometry of the 1 -jet vector bundle E = J 1 (R, M) is also convenient because the number and the form of the local Bianchi identities associated to such connections are considerably simplified. In fact, we have the following: Theorem 67. The following nineteen effective local Bianchi identities for the hnormal Y-linear connection of Cartan type V given by (3.4) are true:

l-^i,*}{^i f c +^|t+ÄSl!yCj3}=0, o* V" ¿

/

R' U

■ 2^{i,j,fc} \ ijk

-i A //?(') °- "*-{j,k} \n(l)lj\k -

n

(l)ij^k(r)j

U

-

'

r

_LT>-Í?(') _i_ f?< ) p W ( i ) \ "^ - t l j / l ( l ) f c r "'" Λ (1)1.Γ (l)fe(r) J n

""(lb'fc/l

,* V fp(i) " Ls{i,j,k} \n(l)ij\k

4

p( r ) r W \ — n

(\)jk^(\)l(r)>

, p(r) p(l) (1)1 _ n ( l ) i j r ( l ) f c ( r ) J - U'

+il

40

LOCAL BIANCHI IDENTITIES IN RELATIVISTIC TIME-DEPENDENT LAGRANGE GEOMETRY r T l |(1) _ r . « ( l ) T r , pi (1) , ^ ( 1 ) , rr(l)Tl _ rKl) p ( r ) (1) _ n u °- J lfcl(p) ° r ( p ) J l f c ~l~-rfcl(p) " " " ^ ( p ) / ! " l "°fc(p) J lr k ( r ) r ( l ) l ( p ) ~~ U ' 6

■ -^O'-fc} {^(pilfc + ^ ( Γ ) Ρ ( 1 ) 7 ( Ρ )

p ( 0 (1) '■Ul)l(p)\k

_ p W (1) Ul)k(p)/1

7

+

P

Q* 9

°'

, p ( 0 W p W (1) _ p ( 0 ( l ) p ( r ) (1) + r(l)k(r)r(l)l(p) ^(l)l(r)^(l)fe(p)

_n(l) |(1) _ Di , D ( D r.'-(l) _ ~~ Λ(1)1&Ι(ρ) """pife ~*~ -"-(lHr^fcip) s* A β · ^{j,k}

=

jfc(p) /

/»<<> ^ W o - p O d l p M C ) \n(l)jr^k(p) + r(l);(r)r(l)fc(p)

+

T

r

J

p(¡) (1) 1Λ^(1)Γ(Ρ) '

, p ( 0 ( i ) \ _ pi _ p(') |(i) Λ ^(l)fc(p)|jj ~ ^ Ρ ^ ( 1 )jfcl(p)'

A /^(llid) , rr(l)rKDl •■A{J,fe}\Ciü)l(fc)+Gi(fc)Cr(j)/ - j - Λ{ίΜ

J S

(1) cí,»ü)

, ^»-(1)^(1) l _ "+" °¿(j) °r(fe) r -

Qyk

<J(i)(i) ¿(j)(fc)'

J

l (1 m A / P ( Í ) (^Ι^) -i- P >1 - n i o . ^ , f c } \y{1)Hj) \{k) + -P 1(fe) ) - o,

1 1 * Á, ; 1 / p ¡ ( i ) , iJ - · ·*{]*} Yji(k) 12*

9Í(1)(1)

V

IO /( f p¡ l á . ^ { j i f c } \tlpljlk 1 4

+

p(0 ( D r r ( i ) _ ρ(ί)(ΐ),(ΐ)·| _ „ Ml)r(j)°í(fc) ^(l)i(fc)l(j) / ~~ ' - 0

i rrr p / i pM ρ' (1)\ _ pi + J jjitpfcr + « ( i j y ^ r ) J - ~Upjk/l

/?(*") p ' (1) n (l)jk%l(r)i

· E{ 2 ,j,fc} \ ñ p i j | f c + ^ ( l í i j ^ p f c í r ) / = ° '

ir ÍO -

p i (1) Ul(p)\k -

_ pi (1) f !Kp)/l

+

, p ( r ) (1) pi (1) _ p ( r ) (1) p í (1) r (l)l(p)rft(r) Ml)fc(p) ^ i l ( r )

P¡ l' 1 ' ± R(r> ci(l)(l) , ^ ( l ) p / _ ί1*;Ι(ρ) "+" - n -(l)lfc 0 i(p)(r) -1" '-"feipj-^tlr

Λ

J

i (1) lfc^iríp) '

Ifi* Ar ,/pi í ^ 1 ' 4- P1 ( 1 ) P< r ) ( 1 ) 4- P ' ( 1 ) \ J.O . ^{j,k} ^• í l ijr°fc(p) "+" ^ij(r) r (l)fc(p) "+" ife(p)|j J _ 17 1'· TR* 1 0

iq*

_ci(l)(l)p(0 _ „I ,(1) °ί(ρ)(Γ) Λ (1)ί*: -"-¿jfcl(p)'

4 / p i (l)l(l) , p W ( l ) o i ( l ) ( l ) \ _ _ ς ί ( 1 ) ( 1 ) a ^Ü,fe} \rpl{j) \{k) i" ^ ( l ) l ( j ) a p ( * ) ( r ) J p(j)(fc)/l'

·

A í Pl Wr^V { ^ } \ ν ϋ ) υ # )

-

V

_

Λ

c'(D(l)|(l)

qlWWpir) p(j)(rrWHk)

0

■Í(D(1) (1) _ p ¡ ( l ) i ( l ) \ _ _ a> ^pi(k)ki)} íp(j)(fe)|i'

n

where ΣΗ j k} represents a cyclic sum and A{i¿\ represents an alternate sum. Proof: Let (XA) = (5/St, δ/δχι, d/dy\) be the adapted basis of vector fields produced by the nonlinear connection (3.1). Let V be the /i-normal Γ-linear connection

DEFLECTION D-TENSOR IDENTITIES AND BIANCHI IDENTITIES FOR CARTAN D-CONNECTIONS

41

of Cartan type given by the components (3.4). For the linear connection V, the following general local Bianchi identities are true (see [55] or [63]): / j {A,B,C}

{R-ABC

~ ^AB-.C

~~ ^AB^CGf

/ , {R-DAB-.C + ^AB^DCGf {A,B,C}

~ 0'

= 0>

where R(XA,XB)Xc = ^CBA^D, T(XA,XB) = T^AXD and " :A " represents one from the local covariant derivatives "/χ," "|f," or "ÚJ" Obviously, the components Τ ^ β and RABC are the adapted components of the torsion and curvature tensors associated to the /i-normal Γ-linear connection of Cartan type V r given by (3.4). These components are expressed in Tables 3.5 and 3.6. Then, replacing A, B, C,... with indices of type

{>.C}· by laborious local computations, we obtain the required Bianchi identities. Remark 68. The above eleven "star"-Bianchi identities are exactly those eleven Bianchi identities that characterize the canonical metrical Cartan connection of a Finsler space (see [56], p. 48). Remark 69. The importance of preceding Bianchi identities for an /i-normal Γlinear connection of Cartan type V on the l-jet space J 1 (K, M) comes from their use in the local description of the geometrical Maxwell equations that characterize the jet single-time electromagnetic field F from the background of our relativistic time-dependent Lagrange geometry.

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CHAPTER 4

THE JET RIEMANN-LAGRANGE GEOMETRY OF THE RELATIVISTIC TIME-DEPENDENT LAGRANGE SPACES

This Chapter develops the Riemann-Lagrange geometry (in the sense of canonical nonlinear connection Γ, Carian canonical T-linear connection, together with its dtorsions and d-curvatures) produced only by a given Kronecker h-regular Lagrangian function on the 1-jet space J 1 (K, M). This naturally extends the Miron-Anastasiei geometrical framework from [55]. At the same time, we construct a geometrical jet relativistic time-dependent Lagrangian theory of physical fields, which blends the electromagnetic and the gravitational theories. The developed generalized-theoretical field theory is governed by natural geometrical Maxwell and Einstein equations, which generalize the classical ones. In order to have a clear exposition of our ideas, we point out that, in our geometrical development, we use the following two distinct notions:'

1. time-dependent Lagrangian function = asmooth function L on J 1 ( K , M ) ; 1 We note that if L is a Lagrangian function on J1 (R, M), then C = L\Jh\\, metric on K, represents a Lagrangian on J 1 (M, M).

Jet Single-Time Lagrange Geometry and Its Applications 1st Edition. By Vladimir Balan and Mircea Neagu. © 2011 John Wiley & Sons, Inc. Published 2011 John Wiley & Sons, Inc.

where hu is a Riemannian 43

44

THE JET GEOMETRY OF THE RELATIVISTIC TIME-DEPENDENT LAGRANGE SPACES

2. time-dependent Lagrangian (in PJ. Olver's terminology [77]) = a smooth local function £ on J X (R, M), which transforms by the rule C = £ \dt/dt\. We also underline that we generally use the terminology "time-dependent" instead of the equivalent "non-autonomous" or "rheonomic" concepts used by other authors. 4.1

RELATIVISTIC TIME-DEPENDENT LAGRANGE SPACES

To develop our jet time-dependent Lagrange geometry, we start the study considering L : £ - t R a smooth Lagrangian function on E = J1(R:M), which is locally expressed by E 3 (t, xl, y\) —» L(t, xz,y\) € R. The vertical fundamental metrical d-tensor of L is defined by r (D(D

1

^~2dy\dy[

d2¿

(

,4n '

Let h = (ftii(i)) be a Riemannian metric on the time manifold R. Definition 70. A jet Lagrangian function L : E —» R, whose vertical fundamental metrical d-tensor (4.1) is of the form (1)(1 "'WOT >(t,x*,rf) := i - ^ =Ä»(*) fly (i,**,rf),

■dy\dy[

where/i 11 = l / / i n > Oandg¿j(í,x fc ,yi) is a distinguished tensor on E, symmetric, of rank n = dim M, and having a constant signature, is called a Kronecker h-regular Lagrangian function with respect to the temporal Riemannian metric h = {h\\). In this context, we introduce the following concept: Definition 71. A pair RRUl = ( ^ ( Ε , Μ ) , ^ , where n = d i m M and L is a Kronecker Λ,-regular Lagrangian function, is called a relativistic time-dependent (nonautonomous or rheonomic) Lagrange space or a jet single-time Lagrange space. EXAMPLE 4.1 Suppose that the spatial manifold M is endowed with a semi-Riemannian metric ψ = (ipij(x)). Then, the time-dependent Lagrangian function BS: defined by

J1(1,M)^R,

BS = Λ11 (*)?>« (a:)i/ii/i,

is a Kronecker /i-regular Lagrangian function. Consequently, the pair

BSL^ =

(J\R,M),BS)

THE CANONICAL NONLINEAR CONNECTION

45

is a relativistic time-dependent Lagrange space. It is important to note that the Lagrangian BS = BSy/KTi = hn{^ij{x)y\y{^/hri is exactly the energy Lagrangian whose extremals are the harmonic maps between the manifolds (R, h) and (M, φ) (see Eells and Lemaire [33]). At the same time, the Lagrangian BS is a basic object in the physical theory of bosonic strings. EXAMPLE 4.2 Let us consider the Kronecker ft-regular Lagrangian function NED:

J1(R,M)^R,

given by NED =

hn{t)giJ{t,xk)y\y{+U§{t,xk)y\+<S>{t,xk),

where gij(t, xk) is a d-tensor field on E = J 1 (R, M), symmetric, of rank n, and having a constant signature, U,J{t, xk) are the local components of a dtensor on the 1 -jet space E = J 1 (M, M), and Φ(ί, xk) is a smooth function on the product manifold R x M. Then, the pair MSVV\ = ( J 1 (R,Af ),#££>) is a relativistic time-dependent Lagrange space, which is called the non-autonomous jet single-time Lagrange space of electrodynamics. Physically speaking, the dynamical character of the spatial gravitational potentials gij(t,xk) (i.e., the dependence of the temporal coordinate t) forced us to use the term nonautonomous in the preceding definition. 4.2 THE CANONICAL NONLINEAR CONNECTION Let us consider a relativistic time-dependent Lagrange space RRL" = (J r (R, M), L), where L is a Kronecker /i-regular Lagrangian function. Let [a, b] C R be a compact interval in the time manifold R. In this context, we can define the energy action functional of the space RRL™, setting E:C°°([o,6],M)-»R,

E(c)=/

Ja

L(t,x\yi)v7m(t)di,

where the local expression of the smooth curve c on M is t —> (x l (i)), and we have j/j = dx^dt.

46

THE JET GEOMETRY OF THE RELATIVISTIC TIME-DEPENDENT LAGRANGE SPACES

The extremals of the energy action functional E verifies the Euler-Lagrange equations 2G

(i)(1)dV

d)U)

dt2

d2L + dxJdyi

dxi

d2L dL x _ + Qtdyl + Qyi « n " °'

dL

dt

dx}

where 1

V i - 1, n, (4.2)

_ h11 dhn

Kn

~T~dT

is the Christoffel symbol of the Riemannian metric hu. Taking into account the Kronecker /¿-regularity of the Lagrangian function L, we can rearrange the Euler-Lagrange equations (4.2) of the Lagrangian £ = Ι/\Αϊ7 in the Poisson form (see also [73]) Ahxk

+ 2gk(t,xm,yT)=0,

Vfc=M,

(4.3)

where ■fi^k

J^k

~\

^χη

f =

^-■»"{ΤΡ—'Ι'ΤΓ}· «„«,

9

í

2

dL

i

dt '

2

dL

dL

dL

,

nl11

.

,-Ί

(4.4)

Theorem 72. The geometrical object G

= (Ga)i) .

where G^L = hnGr, is a spatial semispray on the l-jet space E. Proof: By a direct calculation, we deduce that the jet local geometrical entities 9ki Í d2L 2 \dxidy[yi

2,k

g * [ d'L

2Hk

2Jk =

,

dL } dx¿/' dL

\

hnKlnyk

verify the following transformation rules:

J

-

J

dx"

dV dt dt ■

(4.5)

THE CANONICAL NONLINEAR CONNECTION

47

Consequently, the local entities 2QP = 2SP + 2W + 2JP modify by the rules

Multiplying now the relation (4.6) by hu, we find the transformation rules (1.10), which characterize a spatial semispray on J X (R, M). Taking into account the harmonic curve equations (1.12) of a time-dependent semispray on J 1 (R, M), we can give the following natural geometrical interpretation to the Euler-Lagrange equations (4.2) and to their Poisson form (4.3), produced by the Lagrangian C = L^/hu'Theorem 73. The extremals of the energy action functional E (produced by the Kronecker h-regular Lagrangian function L of the space RRL™) are harmonic curves of the relativistic time-dependent semispray <S¿ = (if, G), which is defined by the temporal components

(?n = -|«iiyi

(4·7)

dL d¿L dL k + τΐΈττ + -3-r«u + 2Λ"κίι W i dx dtdy^ dyl

(4.8)

H

and the spatial components n(i)

_

hug1

d¿L , ßxidy$ylM

In other words, these extremals verify the equations of the harmonic curves hn j ~ - + 2#{f)1 + 2G^n | = 0 ,

V i = M : , n = dim M.

Definition 74. The relativistic time-dependent semispray i>L - ^ ( 1 ) 1 , G ( 1 ) 1 j , defined by the components (4.7) and (4.8), is called the canonical time-dependent semispray of the jet single-time Lagrange space RRL™. Using now Proposition 26, we can introduce the following concept: Definition 75. The nonlinear connection Γ, — Γ - ÍM{i)

ΛΓ(ί) ^

given by

dGY} M

(i)i ~

z

-"(i)i -

K

uVi'

iV

(i)j -

Q.j

dy{

'

<- 4 - y >

where H^l χ and G¡*X χ are defined by formulas (4.7) and (4.8), is called the canonical nonlinear connection of the relativistic time-dependent Lagrange space RRL™.

48

4.3

THE JET GEOMETRY OF THE RELATIVISTIC TIME-DEPENDENT LAGRANGE SPACES

THE CARTAN CANONICAL METRICAL LINEAR CONNECTION

The main result of this Chapter is the Theorem of Existence and Uniqueness of the Cartan canonical h-normal T-linear connection CT, which allows the subsequent development of ourjet single-time Lagrangian theory ofphysicalfields on a relativistic time-dependent Lagrange space RRU[. Theorem 76 (Cartan canonical linear connection). On the relativistic timedependent Lagrange space RÜL™ = ( J ^ R , M),L) endowed with its canonical nonlinear connection Γ [see formulas (4.9)] there exists an unique h-normal T-linear connection CT = [nn, Gn, Lljk, C.{k)j having the metrical properties 1· 9ij\k = 0 ,

gijl^] = 0 ,

-) r
°9mj

_

Ti

^¿(1) _ ^,¿(1)

Ti

Proof: Let us consider that CT = (Gn, Gjl, L)k, Cj (fc) j is an ft-normal Γ-linear connection whose coefficients are defined by the relations G1 ^ ~i

nk _ t_m S9rn¿ 2 St Sgkm

V

jk

Sgjk N

Then, by direct computations, one verifies that CT satisfies conditions 1 and 2. Conversely, let us consider that the /i-normal Γ-linear connection CT = I Gn, GjX, L)k,

Cj(k)

satisfies conditions 1 and 2. It follows directly that we have 7? The condition g^

_ ,1

= 0 is equivalent to

rk

-

gkm

δ9πι

>

THE CARTAN CANONICAL METRICAL LINEAR CONNECTION

49

Applying a Christoffel process to the indices {i, j , k} and using the symmetry relations £}* = ££,■. we find ji

9^_ fSgjm 2 V <^fe

=

jk

Sgkm _ 5gjk\ ?>xi 5xm) '

By analogy, using the relations Cl<λ = C%Sl and gij\\k\ process applied to the indices {i, j , k} leads to £«(i)

'<*>

dgkm _

9^_ (dgjm

=

2 \dyk

— 0, a Christoffel

dgjk\

dy{

dy?J-

In conclusion, the existence and the uniqueness of the Cartan canonical linear connection CY is clear. Remark 77. Replacing the canonical nonlinear connection Γ by an arbitrary one, the preceding Theorem still holds true. Remark 78. As a rule, the Cartan canonical linear connection of the relativistic time-dependent Lagrange space RRL™ verifies also the metrical properties ^ n / i = Ήΐ|Α: = '"'iil(fc) — 0)

9ij/\ = 0.

Remark 79. In the particular case gij(t,xk,yll) = gij(xk) := φ^{χΗ), the coefficients of the Cartan linear connection CT are the same with those of the Berwald linear connection BY, namely, CY = BY=

(κΐ,

0, 7 j f c , 0) .

However, it is important to note that the Cartan connection is a Γ-linear connection, where Γ is the canonical nonlinear connection of the space RRL™, while the Berwald connection is a Γ-linear connection, where Γ is the canonical nonlinear connection produced by the pair of metrics (hi i (t), φ^ (χ)). Consequently, the Cartan connection and the Berwald connection are distinct ones in this case, even if they have the same local adapted coefficients. Remark 80. The torsion T of the Cartan canonical connection of the relativistic timedependent Lagrange space RRL" is determined only by six effective local components, because the properties of the Cartan canonical connection imply the equalities (see Theorem 55) Tr

_ n

ij - υ '

1

o(r)(l)(l) _ ö

(D(i)ü)

-

n

υ

·

At the same time, it is important to note that the number of the curvature local d-tensors of the Cartan canonical connection does not reduce. Five d-curvatures remain, their formulas being given by Theorem 58. Definition 81. The torsion and curvature d-tensors of the Cartan canonical connection of the space RRL" (given by Theorem 55 and Theorem 58) are called the torsions and curvatures of the relativistic time-dependent Lagrange space RRL".

50

THE JET GEOMETRY OF THE RELATIVISTIC TIME-DEPENDENT LAGRANGE SPACES

4.4

4.4.1

RELATIVISTIC TIME-DEPENDENT LAGRANGIAN ELECTROMAGNETISM The jet single-time electromagnetic field

Let us consider that RRL™ = (J 1 (R, M),L) is a relativistic time-dependent Lagrange space and Γ

=

(M(i)

Nd)

\

is its canonical nonlinear connection. At the same time, let

CT - \κη, Gjl, L'jk, Clj{k)J be the Cartan canonical connection of the space RRL™. In this context, using the canonical Liouville d-tensor field C:

= c u«t JL-„< o = ^ ¡Vl A \

dyV

we can introduce the following nonmetrical deflection d-tensors: ^(1)1 - *-(l)/l - J/l/1.

^ ( i W - <-(l)|¿ - Vl\j,

«(1)0) - ^ ( l ) l ( j ) - 1/1 l ( j ) '

where "/i," "| 7 ·," and "|; .?" are the local covariant derivatives induced by the Cartan canonical connection CT. It follows that, by direct calculations, wefindthe following: Proposition 82. The nonmetrical deflection d-tensors ofthe relativistic time-dependent Lagrange space RRL" have the expressions jai)

_ 9l bgim

D(i)

m

_

, ri

N(i)

m

J « ( 1 ) _ ri , ^«(1) ,.m

In the sequel, using the vertical fundamental metrical d-tensor

we construct the metrical deflection d-tensors of the relativistic time-dependent Lagrange space RRL": -hll59im,.m

T¡W . _ r ( i ) ( i ) - ñ M D

(i)i

d

:== G

(i)(r)D(l)i

d)(j) '— Gd)(r)d(im

=

h

-

9i Λ

-iV(1)j +

Ljmy1

" [9ij + 9irCrm{j)yT\ ■

Definition 83. The distinguished 2-form o n £ = J 1 (R, M), given by

RELATIVISTIC TIME-DEPENDENT LAGRANGIAN ELECTROMAGNETISM

51

where 1 rD„(D m _D„W] eil 1

(1) F;(i)j

mm 1 f„(1)(1)

j(l)(l) _

(¿)(i) ~ 2 α

2

J(1)(1)'

ωω "-um

is called the ye? single-time electromagnetic field of the relativistic time-dependent Lagrange space RRL™. Using the above Definition, by a direct calculation, we obtain the following: Proposition 84. The expressions of the electromagnetic components are (D _ * " ' 9jrN[[)i (t)j 2 f(i)(i) _ J

(Í)Ü)

υ

^

ftr^fä

+ (9irLrjm

- 9jrLrim)

j/f

(4.10)

n

-

Remark 85. The above Proposition shows us that the jet single-time electromagnetic field of the relativistic time-dependent Lagrange space RRL™ has, in fact, the following simpler form: W=

F¡V.oy[AdxJ,

where the electromagnetic components F, J. are given by (4.10). 4.4.2

Geometrical Maxwell equations

The main result of the relativistic time-dependent Lagrangian electromagnetism is the following: Theorem 86. The electromagnetic components F^J, ofthe relativistic time-dependent Lagrange space RRL" are governed by the following geometrical Maxwell equations. p(l) _ ! A, (i)k/T. ~ 2Λ^Μ

t

V ^{i,j,k}

(~ñW , \^(í)l|fe +

rpm , j(l)(l) p(m) (i)m1-ík + d(i)(m)H(l)lk

¡Tp , rp(l) fí(m) ] 1 \ ~ [ i li|fc + Gfc(m)K(l)l¿J Vp ) ■

n m L>

F^1» - _ V --.(IXIXI) pM m (í)i\k ~ ¿^{i,j,k}^(i)(,m)(T)n-l,l)jkyi '

where A^^ means an alternate sum, ]T\ ¿ · k, means a cyclic sum, and „i _

A

n . „e

yp - «

rd)(i)(i)

_Aii.

°(¿)(m)(r)-

5MJ/I,

n

gmq

r

g(D

^i(r)

-

!

g 3

¿

4dy\dyfdy\'

Proof: First, we point out that the Ricci identities applied to the spatial metrical d-tensor g¿j lead us to the following curvature d-tensor identities: -ftmilfe + tiim\k

=

0,

t^raijk ~r t^ivajk

=

«,

*mij(k)

imj(k)

=

>

52

THE JET GEOMETRY OF THE RELATIVISTIC TIME-DEPENDENT LAGRANGE SPACES

where Rmilk = gipRpmlk, Rmijk = gipRPmjk, and P m i J ¡ ^ = gtpP^y Now, let us consider the following nonmetrical deflection d-tensor identities attached to the Cartan canonical connection: U

\a\) -^(χ)!^ ÍA \ Γ)(

p

n< )

-

a

^(l)jl(fc)

LJ

(l)m1\k

a

(l)(m)n(\.)lk>

J(P)(1)

—„mnP

n(m)

a

{l)(m)K(l)jk>

^(i)fc|j - 2/1 "-mjk

[a2) V(mk ^

- Vl Kmlk

(l)k/\

ρ)

(l)(fc)b' ~ y l V W

a

^(l)m°j(fc)

(l)(m)- r (l)j(fc) ·

Contracting these nonmetrical deflection d-tensor identities by G L , J = we obtain the following metrical deflection d-tensor identities: (™ i ñ W m

l U -^(¿Jllfc Irr,

n

(i)k/\X

Λ η ^

im \ η^

r>(1)

u

n< '

ι<υ

ym3) U{i)j\(k)

— hn„mn n

Vl milk

,

A^W

-

r

w(1)(1) p("i)

1

a

(i)(m)n(l)lk'

(i)m lk

Λ^Μ )

ΛΙΙ„"»Ρ

(i)(k)\j - "

n ( 1 ) T" 1 1

_ U l „ m D

a

u

p(

m

)

W _ o*1) rm(V U

Vl mij(k)

(i)m^j(k)

- J^(1) a

p<m> W

(i)(m)Ul)j{k)

At the same time, we recall that the following Bianchi identities are true: (61) A{tM

{tfilk + T*i]k + Clk\VR$ni}

hngip,

■

= 0,

(fc) E{<j,fc} \Rlm - c\{r)R(\)jk} = °' (03) -4{»,j} j ^

( f c )

+ C. ( f c ) | j + O j ( r ) P ( 1 ) i ( f c ) j - U.

In order to obtain the first Maxwell equation, we do an alternate sum A{i
Σ {*»*<&} = 0. Doing an alternate sum A{i¿} in (m^) and combining with the Bianchi identity (63), we obtain a new identity. A cyclic sum by the indices {i, j , k} applied to this last identity implies the third Maxwell equation. Note that we also use in these computations the symmetry relations ^(l)(!) _ J(1)(1)

%)(*) ~ a(k){jV 4.5

4.5.1

^»"(1) _ ^rn(l)

S'(fe) ~ ° f c ü ) '

p (m)

(1) _

p (m)

(1)

MiW(fc) ~Mi)fc(j)·

JET RELATIVISTIC TIME-DEPENDENT LAGRANGIAN GRAVITATIONAL THEORY The jet single-time gravitational field

Letusconsiderarelativistictime-dependentLagrange space RRL™ = (J X (R, M), L), whose Kronecker /i-regular Lagrangian function L produces the vertical fundamental

JET RELATIVISTIC TIME-DEPENDENT LAGRANGIAN GRAVITATIONAL THEORY

53

metrical d-tensor WO)

3

2dy\dy{

Let us consider that Γ

_

(M(i)

N(i)

\

is the canonical nonlinear connection of the space RRL™ and let

CT = [κη, GjV L)k, Clj(k)J be the Cartan canonical connection of the space RRL^. In order to develop on J1 (R, M) the relativistic time-dependent Lagrangian theory of gravitational field, we introduce the following geometrical concept with physical meaning: Definition 87. The adapted metrical d-tensor G on the 1-jet space J 1 (R, M), locally expressed by G = hudt ®dt + 9ijdxl ® dx? + hngijSy\

<8> Sy{,

is called the jet single-time gravitational field of the relativistic time-dependent Lagrange space RRL™.

4.5.2 Geometrical Einstein equations and conservation laws We postulate that the geometrical Einstein equations, which govern the jet singletime gravitational field G of the space RRL™, are the geometrical Einstein equations attached to the Cartan canonical connection CT, namely, R i c ( C r ) - S c ( C r ) G = A:T,

(4.11)

where Ric(CT) represents the Ricci d-tensor of the Cartan connection, Sc(CT) is its scalar curvature, K, is the Einstein constant, and T is a given intrinsic d-tensor of matter, which is called the stress-energy d-tensor. In this context, working with the adapted basis of vector fields

™-{έ·έ·4}·

where A (or any other capital Latin letter, like B, C, D,...) is an index of type 'Ί,' ) or (i\," the curvature d-tensor of the Cartan connection has the folowing adapted "¿,"or"^ ) ,"thecu local components: H(XC,XB)XA

=

R-ABC-XD-

Therefore, the Ricci d-tensor Ric(CT) has the adapted components RAB

= Ric (XA, XB) = Trace {Z ~ R(Z, XB)XA]

=

R^BM-

54

THE JET GEOMETRY OF THE RELATIVISTIC TIME-DEPENDENT LAGRANGE SPACES

the scalar curvature takes the form Sc(CT) =

for A= 1, B = 1,

' hn, GAB

for A = i,B = j ,

g»,

= <

where

GABHAB>

(4.13)

tj

hng , forA={l¡,B={¿],

, 0,

otherwise.

Remark 88. Because of the form of the Bianchi identities that characterize the Cartan canonical connection CT of the space RRL7} (see Theorem 67), the Ricci d-tensor Ric(CT) is not necessarily symmetric. Taking into account the expressions of the local curvature d-tensors R ^ ß C of the Cartan connection CT (see Theorem 58), together with the form (4.13) of the d-tensor GAB, we find the following: Proposition 89. The Ricci d-tensor Ric(CT) is determined by the following six effective adapted local components: R\\

Rfl

(i)l

/ill — 0, (1) := P,(¿)1 <■

Ril = Rilm, J

,m(l) ¡l(m)·

Am ÍÁ

Ri;

R (1)

'"'(¿b ·

(1) R.Hi)

._ p(l) pm(l) ' ' Hi) p(l)(l) . __ c(l)(l) _ om(l)(l) n (')U) · °(')U) ~ °i(j)(m) '

_ p(i) _ p"»U) r (i)j y(m)'

wftere K n = 0 « the classical Ricci tensor of the Riemannian metric hn. = 0, i? = g^Rij, and 5 = ftn5tJ'5L/ ·Λ we obtain the

Denoting κ = hllKn following:

Proposition 90. 77ie scalar curvature of the Cartan canonical connection has the expression Se (CT) = K + R + S = R + S. Using the above results, we can establish the main result of the relativistic timedependent Lagrangian theory of gravitational field: Theorem 91. The local geometrical Einstein equations, which govern the jet singletime gravitational field G of the space RRL™, have the form R + S, -hn = fCTn, ■"'ij j(l)(l)

"(i)ü)

0 = Tu, u

-

1

i(i)

o

R+S 2

hn9ij

Rn = KTn, U(j) - ^

J

(4.14)

*J'

"tí ~

i(iV

= Κ,Τ (i)(i)

(i)U)'

p(i) _

r

(i)l

D(I)

- ^

1

ICT^

(i)l'

/CT ( i )

(4.15)

JET RELATIVISTIC TIME-DEPENDENT LAGRANGIAN GRAVITATIONAL THEORY

where TAB-, Α,ΒΕ d-tensor T.

55

<. 1, i, ,J >, are the adapted components of the stress-energy

Proof: We locally describe the global Einstein equations (4.11) in the adapted basis (4.12). Remark 92. In order to have the compatibility of the Einstein equations (4.14) and (4.15), it is necessary as certain adapted local components of the stress-energy d-tensor T to be a priori equal to zero. It is well known that, from a physical point of view, the stress-energy d-tensor of matter must verify the local geometrical conservation laws — u>

I

A\M

where T f =

V.4G

{ i . ·,!,',»},

GBDTDA.

Theorem 93. in the relativistic time-dependent Lagrangian geometry of the gravitational field the following geometrical conservation laws of the Einstein equations must be true: R+S — TV71 _1_ p(m)\W - IXl\m "+" ^(1)1 l(m) 2 /I RT-

R + S,

? (m)(l)

where R? = gmrRri,P[n S,(m)(l)

mr

(m)

=

R + S,

m)|(l)

= -P,( l ) i l ( m ) l(m)

hngmrP?(i) R?--

hng Sll]¡¡J.andPym(l) (i)

pm(l) " ^ (t)|m'

|(1)

p(l) r(t)·

„mr D

g

pi7")

Hri,r'^i

i,

„r,

— nng

o(i) (r)¿>

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CHAPTER 5

THE JET SINGLE-TIME ELECTRODYNAMICS

The aim of this Chapter is to develop the Riemann-Lagrange differential geometry (in the sense of Cartan linear connection, d-torsions, d-curvatures and geometrical Maxwell and Einstein equations) for the jet single-time Lagrange space of electrodynamics, which is governed by the quadratic Lagrangian function Lje, : ^ ( Κ , Μ ) ->R, given by Ljet(t,xk,ykx) = hn(t) φφ«)

y\yÍ + U§(t,xk) y\ + Φ(ί,ζ*),

(5.1)

where hu (t) [resp. φ^ (xk)] is a Riemannian (respectively, semi-Riemannian) metric on the temporal manifold R. (resp. spatial manifold M), wJ(t,xk) are the local components of a d-tensor on the 1-jet space JX(M, M) and <&{t,xk) is a smooth function on the product manifold R x M. In order to justify our "electrodynamics" terminology, we recall that, in the study of classical electrodynamics, the Lagrangian function on the tangent bundle -^autonomous '· Í M.

► K,

Jet Single-Time Lagrange Geometry and Its Applications 1st Edition. By Vladimir Balan and Mircea Neagu. © 2011 John Wiley & Sons, Inc. Published 2011 John Wiley & Sons, Inc.

57

58

THE JET SINGLE-TIME ELECTRODYNAMICS

that governs the movement law of a particle of mass m φ 0 and electric charge e, which is displaced concomitantly into an environment endowed both with a gravitational field and an electromagnetic one, is given by £autonomous(z, y) = mc9ij{x)yty:i

•

2e + —Ai{x)y< + F(x).

(5.2)

Here, the semi-Riemannian metric %(.x) represents the gravitational potentials of the space of events M, Ai(x) are the components of an 1-form on M representing the electromagnetic potential, J-(x) is a smooth potential function on M, and c is the velocity of light in vacuum. Remark 94. The classical Lagrange space Ln = (M,L(x,y)), where L is the Lagrangian function (5.2), is known in the literature of specialty as the autonomous Lagrange space ofelectrodynamics. A geometrical study of this space was completely done by Miron and Anastasiei in the book [55]. More generally, we point out that, in the study of classical time-dependent (nonautonomous or rheonomic) electrodynamics, a central role is played by the nonautonomous Lagrangian function of electrodynamics, Lnon-autonomous : RxTM->R, expressed by ■ ·

¿non-autonomous (t, x, y) = rnceij(x)y,y'1

H

2e

Ai(t, x)yl + F{t, x).

(5.3)

Remark 95. The differential geometry produced by the non-autonomous Lagrangian function of electrodynamics (5.3) is also given in the book [55]. As a consequence, in our jet relativistic single-time approach of the electrodynamics, we consider that a central role is played by the following jet relativistic time-dependent Lagrangian of electrodynamics: C = (πΐΓ^^θφΜυί

+

2

-^A{^{t,x)y\ + r(t,x)\

y/KTi,

(5.4)

where /in (i) is a Riemannian metric on the time manifold R, A,J (t, x) is a d-tensor on J1(R, M) and F{t, x) is a smooth function on R x M. 5.1

RIEMANN-LAGRANGE GEOMETRY ON THE JET SINGLE-TIME LAGRANGE SPACE OF ELECTRODYNAMICS £T>L?

In such a context, by a natural extension of Lagrangian functions (5.2), (5.3), and (5.4), we introduce Definition 96. The pair SVL™ = (J 1 (R, M), Ljet), where L¡et is a quadratic singletime Lagrangian function of the form (5.1), is called the autonomous jet single-time Lagrange space of electrodynamics.

RIEMANN-LAGRANGE GEOMETRY ON THE SPACE OF ELECTRODYNAMICS £ X > L f

59

Remark 97. The non-dynamical character of the spatial metrical d-tensor φ^(χΗ) (i.e., the independence on the temporal coordinate t) determined us to use the term "autonomous" in the preceding definition. In the sequel, we apply the general Riemann-Lagrange geometry of the relativistic time-dependent Lagrange spaces from the preceding chapters to the particular autonomous jet single-time Lagrange space of electrodynamics £X>_L". To initiate the development of the Riemann-Lagrange geometry of the jet singletime Lagrange space SVL", endowed with the Lagrangian of electrodynamics SV

— i/jetV/ill

=

hll{t)Vij{xk)y\y{ + U$(t,a*)y[ + Φ(ί,χ*)

/ill,

let us consider the energy action functional, defined by pb

pb

E£V{c) = / SVdt =

E£v:C°°([a,b},M)^R,

Ja

L^y/h^dt,

(5.1)

Ja

where the [o, b] c R is a compact interval, the local expression of the smooth curve c on M is t —> (xl(t)), and y\ = dx% jdt. In this context, a preceding general result implies (see also [65] and [66]) Theorem 98. The extremals of the energy action functional Έεν (produced by the autonomous single-time Lagrangian of electrodynamics ET>) are harmonic curves of the relativistic time-dependent semispray Sgv = (H, G), which is defined by the temporal components H(i)

-

-"(l)i ~

--K

2

1

JI* nVl

and the spatial components G

(i)i = 2%yiyl + — j -

. J >+■ dU(-m)

UTW

(m)pyi

dt

■U rr(D

i

ΚΛ i (mril

9 Φ

dx

where ^lpq{x) are the Christoffel symbols of the semi-Riemannian metric φ^(χ), the Christoffel symbol of the Riemannian metric /in (i) is !

ft11

Kll(í) = 11

where h

(¿/in "2"*-·

= l / / i n > 0, and we have the following: 1 TJ( ) ._ (m)p ■

din\ βχΡ(

m

du9\

) __

(P) ■ Qxm

In other words, these extremals verify the equations of the harmonic curves

hu | ^

+ 2ff((f}1 + 2G¡; ) )1 1=0,

V i = M , n = dim M.

60

THE JET SINGLE-TIME ELECTRODYNAMICS

Definition 99. The relativistic time-dependent semispray <Sf t> = (H, G) is called the canonical semispray of the autonomous jet single-time Lagrange space of electrodynamics SVL". In the sequel, following our Riemann-Lagrange geometrical ideas from the preceding chapters, the canonical time-dependent semispray Ssv = (H, G) naturally induces on the 1 -jet space J 1 (R, M) a nonlinear connection r^:=r=(M((1í¡1,iV((1i¡.),

(5.2)

which is called the canonical nonlinear connection of the autonomous jet single-time Lagrange space of electrodynamics EVU[. In this way, we have the following: Theorem 100. The canonical nonlinear connection (5.2) of the autonomous jet single-time Lagrange space of electrodynamics ET>U[ is determined by the temporal components M¡¡>n = 2H$n = -κ\ιυ\ (5.3) and the spatial components M) MW

{l)i

_ -

9" ^ 0 (i)i

D i -_ V ..ii Li¿± j

mVl

~~ dy

^^

, ..m™ m i,

v

Λ\) *η<Ρ™ í í r η\\ψ·-Τ jpL>

4

>M'

(

Cid)

'

Now, let us consider the adapted bases associated to the canonical nonlinear connection (5.2), which are given by

{Ι·έ·4Η μ ' (Κ · Μ)) and

{dt,dx\Sy\}

C X*{Jl{R,M)),

where

St

dt

_i_

i

Sx

=

Wdyf

J L _jv(m)— dx

i

^

dyf

(5-5)

Sy\=dy\ + M§1dt + Nl?)mdxm. Following again the geometrical ideas from the preceding chapters, by direct computations, we can determine the adapted components of the generalized Carian canonical connection of the jet autonomous single-time Lagrange space of electrodynamics £PL™, together with its local d-torsions and d-curvatures. Theorem 101. (i) The generalized Cartan canonical connection CT =

[Hn,Gjl,L)k,C%.{k)]

GEOMETRICAL MAXWELL EQUATIONS ON SVL^

61

of the ¡et autonomous single-time Lagrange space of electrodynamics SVL™ has the following adapted components: i?111 = Ai}1,

G,i=0,

7(fe)

(5.6)

(ii) The torsion T σ/ί/ie generalized Cartan canonical connection of the space SVL'i is determined by two effective adapted components, namely, ftnV fcm

p(fc) Λ (ΐ)υ

a ■"■flHi (1)»J — r m i i / l

+

(1)

«hc/gL (m)j + /»IIP'fern

au:("t)j dt

i/ (1)

(5.7)

+ t/ (1)

U

+

(m)i\j

U

(m)j|¿

w/iere ÍH^¿J (x) are f/*e /oca/ curvature tensors of the semi-Riemannian metric φ^ (χ) and "|j " represents the local M-horizontal covariant derivative produced by the generalized Cartan canonical connection CT. (iii) The curvature R of the generalized Cartan canonical connection of the space £VU[ is determined by a single effective adapted component, namely, p(')(l)

_ pi

__ foi

that is exactly the curvature tensor of the semi-Riemannian metric φ^(χ). 5.2

GEOMETRICAL MAXWELL EQUATIONS ON £T>L*N

In order to describe our Riemann-Lagrange electromagnetic theory on the autonomous jet single-time Lagrange space of electrodynamics 5DL™, let us remark that, by a simple direct calculation, we obtain the following: Proposition 102. The metrical deflection d-tensors of the space £T>L™ are expressed by the formulas:

D (i) 1

(i)U)

= [h^imyTh [h'WimVT

1.

= -

3

(1)

^ .

(5.1)

'ω

where "/i," "\j," and "|; l" are the local covariant derivatives induced by the generalized Cartan canonical connection CT of the space SOL™. Moreover, taking into account the general formulas which give the electromagnetic components of a general relativistic time-dependent Lagrange space RRL™, we find the following:

62

THE JET SINGLE-TIME ELECTRODYNAMICS

Theorem 103. The local electromagnetic d-tensors of the autonomous single-time Lagrange space of electrodynamics SVU{ have the expressions rW

1

U

U)i

f(l)(l) _ I Γ^(1)(1) '(*)(J)

\u(1) ~u(1)}

■•(i)(i)

j(l)(l) *0)W

---u{1)

(5.2)

0.

Now, particularizing to EVU[ the geometrical Maxwell equations of the electromagnetic field of a general relativistic time-dependent Lagrange space RRL™, we obtain the main result of the single-time electromagnetism on the space £T>U{: Theorem 104. The electromagnetic components F^J. of the autonomous single-time Lagrange space ofelectrodynamics EVL™ are governed by thefollowing geometrical Maxwell equations ' p(l)

_ I Λ, , Γι,ΙΙ,?W

p(m) 1 (5.3)

=0

P(1)|( '{i.J.fc} r (i)j'C

where A{i¿} represents an alternate sum and J \ . . fc, represents a cyclic sum.

5.3

GEOMETRICAL EINSTEIN EQUATIONS ON SOL**

To expose our generalized Riemann-Lagrange gravitational theory on the autonomous single-time Lagrange space of electrodynamics £T>L\, we recall that the fundamental vertical metrical d-tensor of the single-time Lagrange space 5DL", given by

and the canonical nonlinear connection (5.2), given by (5.3) and (5.4), produce a single-time gravitational /i-potential G on the 1-jet space J 1 (R, M), which is locally expressed by G = hudt ®dt-\- ipijdx1 dx3 + hL^ijSy\

® 8y[.

(5.1)

In order to describe the local geometrical Einstein equations of the single-time gravitational /i-potential G (together with their geometrical conservation laws) in the adapted basis {XA}

~\St'Sx^dyJj'

let us consider

CT= Κ , Ο ^ , Ο ) ,

GEOMETRICAL EINSTEIN EQUATIONS ON £ £ > L f

63

the generalized Cartan canonical connection of the space ET>L\. Taking into account the expressions of the adapted curvature d-tensors of the space EVL", we immediately find the following: Theorem 105. The Ricci tensor Ric(CT) of the autonomous single-time Lagrange space of electrodynamics EVL" is characterized only by a single effective local Ricci d-tensor, Ri % :5R? which is exactly the Ricci tensor of the semi-Riemannian metric U[. Theorem 107. (i) The local geometrical Einstein equations, which govern the jet single-time gravitational h-potential G of the space EVL™, have the form -2-Λιι

0"ιι,

«R ;Ψυ = 0-,;

9t

/ i ' V i j = κ>τ\

(5.2)

i)(i)

0 - r (1)

o = x(!),

(5.3)

where TAB, A,B E < 1, «,L? \, are the adapted local components of the stressenergy d-tensor of matter T. (ii) The geometrical conservation laws of the geometrical Einstein equations of the space ET>U[ are expressed by the classical formulas k

m where
ipkmu\my

^¿k

|fc

0,

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CHAPTER 6

JET LOCAL SINGLE-TIME FINSLER-LAGRANGE GEOMETRY FOR THE RHEONOMIC BERWALD-MOÓR METRIC OF ORDER THREE

It is a well known fact that, in order to create the Relativity Theory, Einstein was forced to use the Riemannian geometry instead of the classical Euclidean geometry, the first one representing the natural mathematical model for the local isotropic space-time. But, there are recent studies of physicists which suggest a non-isotropic perspective of the space-time. For example, according to D. G. Pavlov [80], the concept of inertial body mass infers the necessity of study of local non-isotropic spaces. Obviously, for the study of non-isotropic physical phenomena, the Finsler geometry is very useful as mathematical framework. The studies of Russian scholars (Asanov [7], Mikhailov [54], and Garas'ko and Pavlov [36], [37]) emphasize the importance of the Finsler geometry which is characterized by the total equivalence in rights of all non-isotropic directions. For such a reason, the frameworks developed by Asanov and Pavlov underline the important role played by the Berwald-Moór metric models (whose classical Finsler geometry background was studied by Matsumoto and Shimada [53]), whose fundamental function

Jet Single-Time Lagrange Geometry and Its Applications 1st Edition. By Vladimir Balan and Mircea Neagu. © 2011 John Wiley & Sons, Inc. Published 2011 John Wiley & Sons, Inc.

65

66

THE RHEONOMIC BERWALD-MOÓR METRIC OF ORDER THREE

has the form2 F(y) = ( j / V ...y») i

F : TM -+ R,

(n > 2),

in the theory of space-time structure and gravitation, as well as in unified gauge field theories. Because any tangent direction can be related to the proper time of an inertial reference frame, Pavlov considers that it is appropriate to generically call such spaces as "multi-dimensional times" [79], [80]. For such geometrical and physical reasons, this Chapter is devoted to the development on the 1 -jet space J 1 (R, M 3 ) of the Finsler-Lagrange geometry (which serves as an extended model within a theoretical-geometric gravitational and electromagnetic field theory) for the rheonomic Berwald-Moór metric of order three,

F(t,y) = VhW) ■ <Jyhiyl

F:J\R,M3)-+R,

where hu(t) is a Riemannian metric on R and ( t , χ ι , χ 2 , χ Ά , yj,y2,y3) are the coordinates of the 1-jet space J J (R, M3). Consequently, in the sequel, we apply the general geometrical results from Chapter 4 to the particular rheonomic Berwald-Moór metric of order three F, in order to obtain the so-called jet local Finsler-Lagrange geometry of the three-dimensional time. 6.1

PRELIMINARY NOTATIONS AND FORMULAS

Let (R, /iii(i)) be a Riemannian manifold, where R is the set of real numbers. The Christoffel symbol of the Riemannian metric ftn(i) is , _h^_dhu_ - 2 dt '

Kn

hll_J_ h ~ hn

0

°-

Let also M3 be a real manifold of dimension three, whose local coordinates are (x1, x2, x3). Let us consider the 1-jet space J 1 (R, M 3 ), whose local coordinates are

(t,x1,x2,x3,y\,y¡,yf). These transform by the rules (the Einstein convention of summation is assumed) ~ ~ t = t(t),

p

p

x =x (x"),

dxp dt y?=—J.jy?,

p , 9 = l,3,

(6.1)

where dt/dt Φ 0 and rank (dxp/dxq) = 3. We consider that the manifold M 3 is endowed with a tensor of kind (0,3), given by the local components Gpqr(x), which is totally symmetric in the indices p, q and r. Suppose that the d-tensor Giji = 2

6GijPy1,

For n even, we either assume all the ^-components as being positive, or, alternatively, consider the product below the root taken in absolute value.

THE RHEONOMIC BERWALD-MOÓR METRIC OF ORDER THREE

67

is non-degenerate, that is, there exists the d-tensor Gjkl on J 1 ( R , M 3 ) such that GijiGikl = 6k. In this geometrical context, if we use the notation G m = G^^g^yl, we can consider the third-root Finsler-like function [90], [12] (it is 1-positive homogenous in the variable y) F(t,x,y)

3

=

sjGpqr{x)ylylyl

· x / f t ü ( t ) = ^/Gin(x,y)

■ y/h^ij,

(6.2)

where the Finsler function F has as domain of definition all values (t, x, y) which verify the condition Gm(x,y) Φ 0. If we denote Gm = SGipgy^yf, m e n t n e 3" positive homogeneity of the "¿/-function" G m (this is in fact a d-tensor on the 1-jet space J 1 (R, M 3 ) ) leads to the equalities

ac?i 11

Gn

G,ji

dGni

d Gin

dy[

dy\dy{

The fundamental

= 3Gm,

GiiiVi

dy\

^

Gijiy{

% i

=

2Gm,

an

^G^i dyf

metrical d-tensor produced by F is given by the formula u

9ij(t,x,y)=

hn(t)

d2F2

¿

dy\dy{

y

By direct computations, the fundamental metrical d-tensor takes the form η~ίι 1/3

9ij(x,y)

°111

Giji

-GuiGju ZGi 11

(6.3)

Moreover, taking into account that the d-tensor G y i is non-degenerate, we deduce that the matrix g = (
G{G\

on1/3 Qjkl 3(G

where G{ = &^Gpll

6.2

and 3 0 m =

m

—Qui)

(6.4)

G^GpuGgn.

THE RHEONOMIC BERWALD-MOOR METRIC OF ORDER THREE

Beginning with this Section we will focus only on rheonomic Berwald-Moór of order three, which is the Finsler-like metric (6.2) for the particular case —,

{p, q, r } - distinct indices,

0,

otherwise.

metric

68

THE RHEONOMIC BERWALD-MOÓR METRIC OF ORDER THREE

Consequently, the rheonomic Berwald-Moór metric of order three is given by

F(t,y) = V&Ht) ■ i/ylyhl

(6.5)

Moreover, using preceding notations and formulas, we obtain the following relations: 1 2 3

r-i

r~i

G\\\

C m = VxVxVx-, Gin = — r , y\ SI

Giji = (1 - Sij) —j (no sum by i or j), y\y{ where ¿¿¿ is the Kronecker symbol. Because we have det(G«i) i i > = T 3 = 2 G u l ^ 0 , we find Gjkl = ^-¿p;

-ylVi (no sum by j or k).

It follows that we have (? 1U = ( l / 2 ) G m and G{ = (l/2)y{. Replacing now the preceding computed entities into formulas (6.3) and (6.4), we get (2 - My) G\{\ 9ij = ^—— -f^j (no sum by i or j) (6.6) 9 y\y[ and g* = (2 - 3δ^)Οηψνίνϊ

(no sum by j or k).

(6.7)

Using the general formulas (4.7), (4.8), and (4.9), wefindthe following geometrical result: Theorem 108. For the rheonomic Berwald-Moór metric of order three (6.5), the energy action functional

É(t,x(t))= Í F^y/hT1dt= [ Ja

Ja

1

\Z{ylylyi}2-huy/hüdt

3

produces on the 1-jet space J (R Í M ) the canonical nonlinear connection r=(M$1

= -K\1y\,

N$.=0).

(6.8)

Because the canonical nonlinear connection (6.8) has the spatial components equal to zero, it follows that our subsequent geometrical theory becomes trivial, in a way. For such a reason, in order to avoid the triviality of our theory and in order to have a certain kind of symmetry, we will use on the 1-jet space J 1 (R, M 3 ), by an a priori definition, the following nonlinear connection (which does not curve the space):

CARTAN CANONICAL LINEAR CONNECTION. d-TORSIONS AND d-CURVATURES

r = ^ ; i = -«i1yl,^ =- ^ j .

69

(6.9)

Remark 109. The spatial components of the nonlinear connection (6.9), which are given on the local chart U by the functions N JV

- i i ¡y« V ~

'

--fnlßij

-

(i)j

2

have not a global character on the 1-jet space J^M, M 3 ), but has only a local character. Consequently, taking into account the transformation rules (1.8) and (1.18), it follows that the spatial nonlinear connection JV has in the local chart U the following components: £(*) _ N

«hdl j. Afc +

w - ~T

6.3

l £dH

&

d k l J, *

2didP

d2xm

π

dx^dxmrVx-

CARTAN CANONICAL LINEAR CONNECTION. d-TORSIONS AND d-CURVATURES

The importance of the nonlinear connection (6.9) is coming from the possibility of construction of the dual adapted bases of d-vector fields

Tt = Έ + Κ » * Θ %

;

**= ^+-fm

;

mc *(E)

(6 10)

·

and d-covector fields idt

; dz ¿ ; Sy{ = dj/j - κ\^[ά± - ^γάχΛ

c **(£),

(6.11)

where i? = J X (R,M 3 ). Notethat, under a change of coordinates (6.1), the elements of the adapted bases (6.10) and (6.11) transform as classical tensors. Consequently, all subsequent geometrical objects on the 1-jet space J 1 (R, M 3 ) (such as Cartan canonical connection, torsion, curvature, etc.) will be described in local adapted components. Using the general result of the Theorem 76, by direct computations, we can give the following important geometrical result: Theorem 110. The Cartan canonical T-linear connection, produced by the rheonomic Berwald-Moór metric of order three (6.5), has the following adapted local components: CT-ÍK1

nk

-n

Ti - 1ΐλΓί{1)

ri(1)\

70

THE RHEONOMIC BERWALD-MOÓR METRIC OF ORDER THREE

where, if we use the notation

W5jk

3<Sj + 3<% + 36jk

A jk

(no sum by i, j or k),

9

then _

r*Hi)

\i

(no sum by i, j or k).

viví

Proof: Via the Berwald-Moór derivative operators (6.10) and (6.11), we use the general formulas which give the adapted components of the Cartan canonical connection (see Theorem 76): 9km Sg,m,} St

G*

im

9

i(l)

c:3(h)

2

Ögjm

J

jk

/ dgjm \ dy*

2

Sxk

dgkr, dy{

+

_ Sgjk Sxm

Sßk. SxJ

d

9jk dy? \

2

dgjk dyf

■»(i) Remark 111. The below properties of the d-tensor C.\. \ are true (sum by m):

J'(fe)

^»(1) _

ri(l)

cfKyT = o,

(6.12)

For similar properties, see also the papers [12], [53], or [90]. Remark 112. The coefficients A\^ have the following values: 2 9

4, = <

i = j Φ I or i = l φ j or j = l φ i, -,

(6.13)

= 1.

i

Theorem 113. The Cartan canonical connection CT of the rheonomic BerwaldMoór metric of order three (6.5) has three effective adapted local torsion d-tensors: K

p(fc)(l) _ (i)i(i) ~

r

R

dK

n dt

(k) (l)lj

D fc(i)

Urk(l) Hi)

9

fc(l)

Hi)

a

_ κι „i

Sk.

Η1Ή1

Hi)

Proof: A general /i-normal Γ-linear connection on the 1-jet space J 1 ( R , M 3 ) is characterized by eight effective d-tensors of torsion (see Theorem 55). For our Cartan canonical connection CT these reduce to the following three (the other five cancel): D(k)

(1)

βΑΓ^) OJV

(i)¿ -

dy{

k h T ii>

R

(k) (1)1.7

(fc) (1)1

6M,

SxJ

SN:((fc) i)j St

,fc(l)

P Hi)

fc(l)

c:Hi) ·

CARTAN CANONICAL LINEAR CONNECTION. d-TORSIONS AND d-CURVATURES

71

Theorem 114. The Cartan canonical connection CT ofthe rheonomic Berwald-Moór metric of order three (6.5) has three effective adapted local curvature d-tensors: pi _ « í l « í l yt(l)(l) i3k 4 °t(¿)(k)'

pÍ(l)_«llJ(l)(l) ^*j(fc) - 2 «(J)(fe)'

n

r)r*í(1)

<Λ(1)(1) _ °¿(j)(fc) _

«r/(1)

»(j) _ ßyk

ßj

»(fc) , ^ ( 1 ) ^ ( 1 ) _ ,νη(Ι) ~í(l) + ° i Ü ) °m(*) °¿(fe) ° m O ) ·

Proof: A general /i-normal Γ-linear connection on the 1-jet space J1(R,M3) is characterized by^ve effective d-tensors of curvature (see Theorem 58). For our Cartan canonical connection CT these reduce to the following three (the other two cancel): p/ ijk

_ °^ij $Lik | TmTi - -fak ~ -J~T + L'iiL'mk

U

-

L

Tmrl L

ife mj.

(1) _ <^Al __ rl(l) ,rl(l) Urn) (1) U(fe) ~ ßyk ^i(k)\j+^i(m)Ul)j(k) '

pl

nrl{1)

U

<j/(l)(l) _

^i(j)

r>rl{V>

_ ^»(fc)

, ^171(1)^(1)

_

^(1)^(1)

where ^i(i)

_"^i(fc)

(fe)b "" ¿ ^ '

¿7"·»(ΐ)£* . - c ' ( 1 ) L1" - C z(1) X

"^^iffe) ^ m j

^mik^ij

^iim^kj-

Remark 115. The curvature d-tensor ^ ! .w J has the properties „¿(Dd)

9 i(i)(i)

_

0


n

,

,

..

Theorem 116. The expressions of the curvature d-tensor S"¡^!)tl are as follows J i(j)(k) 1. S ,. wfe? =

2. S-, .{fJ =

9

\— (i φ k φ l φ i and no sum by i); (l/Í) »i \—r (i φ j φΐ φϊ and no sum by i);

3. Sf, . ¿ J = 0(^φjφkφi

and no sum by i);

4. S ),}}.! = —r-7- (i Φ k ΦΙ Φ i and no sum by I); «(«)(&)

5. S^wiV =

gy*íyk i -r

-r

r

- T (ίφ j φΐφιαηάηο SylM

J>

sum by I);

72

THE RHEONOMIC BERWALD-MOÓR METRIC OF ORDER THREE

6. S-, ¿ n = 7. SSjyj

9(t/i)

=

Öß

Φ l αηα" no sum by i or I);

—2 (i Φ l and no sum by i or I); 9(i/i)'

} 8. S,(,¿.) i(D(k) =~ 0n (k Φ I and no sum by I);

9· SilKm) \m = 0 ( j φ I and no sum by I). Proof: For j φ k, the expression of the curvature tensor «SÍ. w J takes the form (no sum by i, j , k, or I, but with sum by m) o«(l)(l)

+

°»(j)(fc)

A\kkjy[

AAkú

M) vi

04) vi

y\y{y\' where the coefficients i4' ■ are given by the relations (6.13). 6.4

6.4.1

GEOMETRICAL FIELD THEORIES PRODUCED BY THE RHEONOMIC BERWALD-MOÓR METRIC OF ORDER THREE Geometrical gravitational theory

From a physical point of view, on the 1 -jet space J 1 (R, M 3 ), the rheonomic BerwaldMoór metric of order three (6.5) produces the adapted metrical d-tensor G — hndt ®dt + gijdxi ® dxj + Ηης^δι/\

® Sy{,

(6.14)

where g^ is given by (6.6). This may be regarded as a "non-isotropic gravitational potential" [55]. In such a physical context, the nonlinear connection Γ [given by (6.9) and used in the construction of the distinguished 1-forms 6y\] prescribes, probably, a kind of "interaction" between (f)-, (x)- and (t/)-fields. We postulate that the non-isotropic gravitational potential G is governed by the geometrical Einstein equations (

Λ

Sc

(Ct)

where Ric f CT 1 is the Ricci d-tensor associated to the Cartan canonical connection C r , Sc ( C r J is the scalar curvature, K. is the Einstein constant, and T is the intrinsic stress-energy d-tensor of matter.

GEOMETRICAL FIELD THEORIES PRODUCED BY BERWALD-MOÓR METRIC OF ORDER THREE

73

In this way, working with the adapted basis of vector fields (6.10), we can find the local geometrical Einstein equations for the rheonomic Berwald-Moór metric of order three (6.5). First, by direct computations, we find the following: Proposition 117. The Ricci d-tensor of the Cartan canonical connection CT of the rheonomic Berwald-Moór metric of order three (6.5) has the following effective adapted local Ricci d-tensors: n

_ n m

n

-

ij

_ « l l « j l a(l)(D

n

zjm

-

¿(i)(j)

4

V (1 Sf »-V = ¿ C O T = ^

p (1) _ p ( l ) _ >

ri(J)

.

, ■—

1

- r(i)j

p

m (1) _ « Π <j(l)(l)

- fij(m)

-

¿{i)ij)

,

(6.16) (no sum by i orj).

Remark 118. The local Ricci d-tensor S,^>-} has the following expression: 1

o(l)(l) _

ι

1

-

^

■

9

y\y{ 2 i

l=J.

Remark 119. Using the last equality of (6.16) and the relation (6.7), we deduce that the following equality is true (sum by r): SrnU

* /

rrs(m)

=

^ψ

. 1 ^

. VT

( m

s u m

b y

.Qr

m )

( 6 J ? )

Moreover, by a direct calculation, we obtain the equalities

í>rxS=o, ¿^p

m,r=l

m=l

yi

=

|.J-. G -»/». yi

(6.18)

Proposition 120. The scalar curvature of the Cartan canonical connection CT of the rheonomic Berwald-Moór metric of order three (6.5) is given by

(cf)=- 4ftll Y2llKíl .G- 2 / Proof: The general formula for the scalar curvature of a Cartan connection is (see Proposition 90) Sc (CT) = g™Rpq + hll9™S\(i)(i)

(P)(9)·

Describing the global geometrical Einstein equations (6.15) in the adapted basis of vector fields (6.10), we find the following important geometrical and physical result:

74

THE RHEONOMIC BERWALD-MOÓR METRIC OF ORDER THREE

Theorem 121. The adapted local geometrical Einstein equations that govern the non-isotropic gravitational potential (6.14), produced by the rheonomic BerwaldMoór metric of order three (6.5) and the nonlinear connection (6.9), are given by £n " ^ i i i K

' ^ιι — ^ίΐ'

K

Íl Íl o(l)(l) i r 4/C

1

(W)>i • qo(i)(i:

' °(i)(i) ¡C W<J)+

0 = Tn, η

,--2/3

ζ 1 1

^-2/3 ' °Ί11

,n

r

(6.19)

„ _r(i)(i) " UV ~ (i)(j) '

(1) 0 = T,(i)l'

0 = Tn,

_Τ-(1)

_

W(J>

ft

^Πο(1)(1) _ T - ( 1 )

ll

o(l)(l) _

r

(D

(6.20)

2JC (¿XJ' ~ ^ Λ ' 2Jt (')Ü) ~ (*W

iW

where

4/tn + κΐ!«;ίι 4£

6i =

(6.21)

Remark 122. The adapted local geometrical Einstein equations (6.19) and (6.20) impose as the stress-energy d-tensor of matter T to be symmetric. In other words, the stress-energy d-tensor of matter T must verify the local symmetry conditions TAB = TBA,

VA,Be{l,

i, {]]}.

By direct computations, the adapted local geometrical Einstein equations (6.19) and (6.20) imply the following identities of the stress-energy d-tensor (sum by r): , 11-7t f~<—2/3 T1-l def — n lu — ςιι · <^m ,

^(ΐΐ

= ' ^ 1 5 mr T ( ( r 1 ) ) 1 = 0,

"jy 7"m i —

T(m)

7

def

Ύτη(1)

-* (¿)

7

(i)(¿)

-

/"i—*/·*

rm

»


J

(i)

-

h

Λ

n

T

(1)

- Π

'l(i) - U

_ « n cmll

r(¿) - 2tc '■ '

Τ-Μίΐ) def i

V ) i - ~2JC~bi

mr
- y

.-.

-M — <J,

4/C

Uff

def

1t

mrn-Ct) _ ^ l l ^ l l omll

ft

(l)i -

mrsr

-Ί — 9

7-1 ^ / Λ»Τ« = 0,

omll

„mrf 1 y ri

rrm def

nmrq-(l)(l) h

n9

J{r){i)

_ hH em 11 , t - -^-í>¿

^"2/3 ™

+ξιι·<^ιιι

-d¿ ,

where the distinguished tensor S™11 is given by (6.17) and ξη is given by (6.21).

GEOMETRICAL FIELD THEORIES PRODUCED BY BERWALD-MOÓR METRIC OF ORDER THREE

75

Theorem 123. The stress-energy d-tensor of matter T must verify the following geometrical conservation laws (summation by m): ■Ί/1 +

i

I

7

i

l|m +

i / l "+" -S|m

+

i

m = ( l ) l >'-^ - ~~~

r— IZ 16/c dT \Δ dt2 ~ 7Γ77■\~dl

<>

r(m),(l) ( l ) i l(m)

r r-1(1) Hl) ,^-"1(1) rm(l) 7 (¿)/l + 1 (¿)|m

+

"Wll

'

o,

(mHlWl) _ n , ^/τ-(" υ ''(I)l)(i) l(m) ~ >

(6.22)

vv/iere (summation by m and r) r J

i def δΤ^ l / l — Af 1 Sxm

r(m),(l)

de/

I

i/\

n

*

"m Jy Um ~

"'(1)1 l(m)

,

r l

~

, _ ¿7Í ~~ ~~ÄT'

r l

n

!

ST?

i rr-r T r, + Jl Lr

δχη

¡γγ-(τη) fyr(rn) (l)l ^(l)! r ( r ) ~m(l) _ a m "f '(l)l"-V(m) ~ fl„m

Ui

92/Í

¿X 1 ~~ 5t

— St

. , A7" m a Tm üJ i i TT T m i\m δχτ im)

d e / dZ ^(ΐμ (1)> Km) Qym

r(m),(l)

1(1) d e /

0 j

(0/1 ^-m(l) d e / 1 (i)\m -

X (m)(l),(l) 1)(¿) !(m)

+

(i)

01

5xm

( Γ ) ^ ( 1 ) __ r ( m W ( l ) _ - í (l)¿ L y r(m) '(l)r°«(m)

Τ

+

r r 2

(l)rm (i) ljrm

*rN(l) d e / °^(1)(¿) -

2 %f ' ^(l] 9j/n

9Τ1(1)„1

Jí <5T•m(l) (i)

Tm T?

öy

m

+

_/7-m(l)/-r J ( r ) ^im

T(r)(l)^m(l)

7

(l)(i) ° r ( m )

~

T 7

K1 «2 U

9Τ7Λ] 9 y m(t) >

(m)(l)^(l) _

(1)(Γ)

°¿(m)

~

θ7-(»"Ηΐ; ^ 0 ] W _ ^ m

'

Proof: The geometrical conservation laws (6.22) are provided by direct computations, using the relations (6.12) and (6.18).

76

6.4.2

THE RHEONOMIC BERWALD-MOOR METRIC OF ORDER THREE

Geometrical electromagnetic theory

Using the general relations (4.10), we find for the rheonomic Berwald-Moór metric of order three (6.5) and the nonlinear connection (6.9) the electromagnetic 2-form F : = F = 0. In conclusion, our Berwald-Moór geometrical electromagnetic theory on the 1jet space ^(Μ.,Μ3) is trivial (this means that it contains tautological geometrical Maxwell equations). In our opinion, this fact suggests that the Berwald-Moór geometrical structure on the 1 -jet space J 1 (R, M3) contains rather gravitational connotations than electromagnetic ones.

CHAPTER 7

JET LOCAL SINGLE-TIME FINSLER-LAGRANGE APPROACH FOR THE RHEONOMIC BERWALD-MOÓR METRIC OF ORDER FOUR

It is obvious that our natural physical intuition distinguishes four dimensions in a natural correspondence with the material reality. Consequently, the four-dimensionality plays a special role in almost all modern physical theories [54]. Thus, taking into account the non-isotropic perspective of the space-time endowed with the Berwald-Moór metric (together with its total equivalence of the non-isotropic directions involved), on the four-dimensional time (i.e., on the four-dimensional linear space endowed with the Berwald-Moór metric), Pavlov's framework [37], [80] offers specific physical-geometrical interpretations such as: • physical events = points in the four-dimensional space; • straight lines = shortest curves; • intervals = distances between the points along of a straight line; • light pyramids = light cones in a pseudo-Euclidian space. In such a geometrical and physical context, the aim of this Chapter is to apply the general geometrical results from Chapter 4 to the rheonomic Berwald-Moór metric Jet Single-Time Lagrange Geometry and Its Applications 1st Edition. By Vladimir Balan and Mircea Neagu. © 2011 John Wiley & Sons, Inc. Published 2011 John Wiley & Sons, Inc.

77

78

JET APPROACH OF THE RHEONOMIC BERWALD-MOÓR METRIC OF ORDER FOUR

of order four

3

where hu(t) is a Riemannian metric on R and (t,x1,x2,x3,x4,y\,yf,yf,y4) are the coordinates of the 1-jet space J1(R, M 4 ), in order to obtain the so-called jet local Finsler- Lag range geometry of the four-dimensional time. As a consequence, a theoretic-geometric gravitational and electromagnetic field theory is produced from the given rheonomic Berwald-Moór metric of order four F. At the same time, at the end of this Chapter, a geometrical approach of the dynamics of plasma, regarded as an 1 -jet medium endowed with the Berwald-Moór metric of order four F, is constructed. 7.1

PRELIMINARY NOTATIONS AND FORMULAS

Let (R, /in(t)) be a Riemannian manifold, where R is the set of real numbers. The Christoffel symbol of the Riemannian metric hn(t) is 11

f^_dhn 2 dt

h^ =

^->0. hn

Let also M 4 be a real manifold of dimension four, whose local coordinates are (xl,x2,xz,x4). Let us consider the 1-jet space J ^ R , M 4 ), whose local coordinates are

(t,x\x2,x3,x4,y¡,yl,yi,yt)-

These transform by the rules (the Einstein convention of summation is assumed) _ _ t = t(t),

ßq?P df yP = —-g.yl

χΡ = χΡ(χ«),

p,q = l,A,

(7.1)

where dt/dt φ 0 and rank (dxp/dxq) — 4. We consider that the manifold M4 is endowed with a tensor of kind (0,4), given by the local components Gpqrs(x), which is totally symmetric in the indices p, q, r and s. Suppose that the d-tensor, Gijn -

\2Gijpgy\y\,

is non-degenerate, that is, there exists the d-tensor G^kl1 on J X (R,M 4 ) such that GijiiG' - * 11 = 5?. In this geometrical context, if we use the notation Gun = Gpqrsy\y\y\yl, we can consider the fourth-root Finsler-like function [90], [12] (it is 1-positive homogenous in the variable y): F(t,x,y) 3

= ¡jGpqrs(x)ypiy¡y[yt

■ ^/¥ψ)

= i/Gnnfav)

■ V^W,

(7-2)

We assume all the y-components as being positive, or, alternatively, consider the product below the root taken in absolute value.

THE RHEONOMIC BERWALD-MOÓR METRIC OF ORDER FOUR

79

where the Finsler function F has as domain of definition all values (i, x, y) which verify the condition G i m ( x , y) > 0. If we denote Gnu = AGipqr(x)y\y{y\, then the 4-positive homogeneity of the "y-function" G n u [this is in fact a d-tensor on J 1 (R, M 4 )] leads to the equalities Gjiii = —ö~i—'

Gjinj/J = 4 G n n ,

öGjiii

Gyi »¿11

ö Gun

,

Gynj/j = 3G¿m,

GijUy\y{

= 12G1111.

The fundamental metrical d-tensor produced by F is given by the formula gij{t,x,y)

= —— ¿

——dy\dy{

By direct computations, the fundamental metrical d-tensor takes the form 9ij(x,y)

4v/G 1111

-GiuiGjin 2G 1111

Giju

(7.3)

Moreover, taking into account that the d-tensor G y n is non-degenerate, we deduce that the matrix g = (#y) admits the inverse g~l = (<¡rjfe). The entries of the inverse matrix g are nJk

Ay/G 1111

where G\ = G™nGpUl

GJfcll

and2Öim =

+

G\G\ 2 ( G n u — Q\\\\)

(7.4)

G™nGpluGqlll.

7.2 THE RHEONOMIC BERWALD-MOOR METRIC OF ORDER FOUR Beginning with this Section we will focus only on the rheonomic Berwald-Moór metric of order four, which is the Finsler-like metric (7.2) for the particular case

{

—,

{p, q, r, s} - distinct indices,

0,

otherwise.

Consequently, the rheonomic Berwald-Moór metric of order four is given by F(t, y) = y/h^W) · ¡Ivlvlvlvl

(7-5)

Moreover, using preceding notations and formulas, we obtain the following relations: G n u = 2/iJ/i«/M-

Gnu

Gim y[

80

JET APPROACH OF THE RHEONOMIC BERWALD-MOÓR METRIC OF ORDER FOUR

Giju = (1 - Sij) —^j- (no sum by i or j), where δ^ is the Kronecker symbol. Because we have = -3 (Gnu)2

det(Gijn)i<J=iz

Φ 0,

we find Qjku

=

(___

l^yk

(n0 sum by

j

o r fc)

¿»1*1111

It follows that we have 0 i i n = ( 2 / 3 ) G m i and G{ = (l/3)j/f Replacing now the preceding computed entities into the formulas (7.3) and (7.4), we get (1 -25ij) % / G l m 1 . 9ij = ^ —■ (no sum by ι or j) (7.6) 8 ?/Í2/i and ,fc 2(1 - 2^'fc) ,· k , . . ,x „jfc _ _j v G i m ¿j^fe ( n o s u m by j or k). (7.7) Using the general formulas (4.7), (4.8), and (4.9), wefindthe following geometrical result: Proposition 124. For the rheonomic Berwald-Moór metric of order four (7.5), the energy action functional

É(t,x(t))= f F2sfhn~dt= Í Ja

Ja

y/ylyfyfyf-^y/hTidt

produces on the 1-jet space Jl(R, M4) the canonical nonlinear connection

r = ( M * ^ = -κ\ιν[, 7v((;¡. = o ) .

(7.8)

Because the canonical nonlinear connection (7.8) has the spatial components equal to zero, it follows that our subsequent geometrical theory becomes trivial, in a way. For such a reason, in order to avoid the triviality of our theory and in order to have a certain kind of symmetry, we will use on the 1-jet space J 1 (R, M 4 ), by an a priori definition, the following nonlinear connection (which does not curve the space): i

Γ = \M^t

= -n\iyl

i f Jt j . TV-. = - l ^lÄ

(7.9)

Remark 125. The spatial components of the nonlinear connection (7.9), which are given on the local chart U by the functions

CARTAN CANONICAL LINEAR CONNECTION. d-TORSIONS AND d-CURVATURES

81

do not have a global character on the 1-jet space J 1 (M, M 4 ), but have only a local character. Consequently, taking into account the transformation rules (L8) and (1.18), it follows that the spatial nonlinear connection N has in the local chart U the following components:

fj{k) _ _Άΐ,Η +, leí fit* (D' ~

7.3

3 '

3 dt dt* '

. d*k d2*m ~r Vl ÖX™ dxldxr

■

CARTAN CANONICAL LINEAR CONNECTION. d-TORSIONS AND d-CURVATURES

The importance of the nonlinear connection (7.9) is coming from the possibility of construction of the dual adapted bases of d-vector fields

and d-covector fields (dt

; dx' ; 6y[ = dy\ - Klny\dt - *ψάΑ

C X*(E),

(7.11)

where E = J 1 (M,M 4 ). Note that, under a change of coordinates (7.1), the elements of the adapted bases (7.10) and (7.11) transform as classical tensors. Consequently, all subsequent geometrical objects on the 1-jet space J 1 (K, M 4 ) (as Cartan canonical connection, torsion, curvature, etc.) will be described in local adapted components. Using the general result of the Theorem 76, by direct computations, we can give the following important geometrical result: Theorem 126. The Cartan canonical T-linear connection, produced by the rheonomic Berwald-Moór metric of order four (7.5), has the following adapted local components: CT = ( « h , G)x = 0, L)k = ^ i c j g ) , < $ > ) ,

(7.12)

where, if we use the notation 2^ + 2 ^ + 2 ^ - 8 ^ - 1 , . . Ai Aik = —(no sum by i, j or k), 8 then C

í(k)

= A

)k ■ —TT <no sum hh VxVi

j

or

k).

Proof: Via the Berwald-Moór derivative operators (7.10) and (7.11), we use the general formulas which give the adapted components of the Cartan canonical connection (see Theorem 76): nk ljl

_9kmHmj 2 St '

Ti jk

9im (5gjm 2 fe'

, Sgkm 6xi

Sgjk δχ

82

JET APPROACH OF THE RHEONOMIC BERWALD-MOÓR METRIC OF ORDER FOUR

j(fe)

dg.'jm Vdyf

2

, dgtm

9im dgjm 2 dy\k ■

dgjk

dy{

dy?

l) Remark 127. The below properties of the d-tensor Ci * (,Λ are true (see also the papers [10] and [53]):

3(2=^8)'

3 ( ^ = 0 ,

C^^Oisumbym).

(7.13)

Remark 128. The coefficients A\, have the following values: 1 :,

i ¿3 ¿I

1

A1

¿i,

: j ^lori

= / φ j o r j = l φ i,

(7.14)

Theorem 129. The Cartan canonical connection CT of the rheonomic BerwaldMoór metric of order four (7.5) has three effective local torsion d-tensors: 1

p(fe)(l) *(l)i(¿)

3

K

iio¿0) >

^¿(j)

-

d/íij

p(fc)

#.

n'ni

di

fe(l) ^¿(j)

Proof: A general /i-normal Γ-linear connection on the 1-jet space J 1 (K, M 4 ) is characterized by eight effective d-tensors of torsion (see Theorem 55). For our Cartan canonical connection CT these reduce to the following three (the other five cancel): „(fc)(l) (1)¿Ü)

dN,

(fc)

(fe)

rfe

¿M,(1)1

p(fe)

δχί

dy{

<W,(fc) (ib St

pfc(l) _ ^ f c ( l ) «(j) ~ ° i ( j ) '

Theorem 130. The Cartan canonical connection CT of the rheonomic BerwaldMoór metric of order four (7.5) has three effective local curvature d-tensors: fíi

,«(!)(!)

£i(j)(fc)

_

l i

K

i

o'(i)(i)

p'(i) _

1

i

c'(D(i) »(j)(fc)'

,¿(1)

(1) dC i(j')

9c:¿(fc)

3yf

9yj

+

„«(1)^(1) ° j ( j ) °m(fc)

_ ^(1)^(1) ^j(fc) ^ m ( j ) ·

Proof: A general /i-normal Γ-linear connection on the 1-jet space J X (R, M 4 ) is characterized byfiveeffective d-tensors of curvature (see Theorem 58). For our Cartan

CARTAN CANONICAL LINEAR CONNECTION. d-TORSIONS AND d-CURVATURES

83

canonical connection CT these reduce to the following three (the other two cancel): SLl. δχί

SU, Sxk

k

v~

, iJ

pi (1) _ d}hl _ rl(l) U'(fe) ~

U

oi(l)(l) _ i(j){k)-

ik p(m)

rl(l)

mp

(1)

°i(fe)|j+°t(m)^(l)j(fe)

Qyk

^iU) dyk

ö

,

mk

U

_

^i(k) dyj

+

'

rm(l)r,l(l)

°i(j)

_

°m(fe)

U

i(t)

rm(l)rl(l)

°m(j)'

where ^(1) _%(k) i(k)\j ~ §xj

+

_W(!) rm_r^(l)rm rra(l)fl °¿(fc) ^ m j ^mifcj^ij ^¿(m)^·

Remark 131. The curvature d-tensor S.; A s has the properties i0')(*)

9 i(D(i)

<j(i)(i) _

n

ς,ί(Ι). «JUXU _

n f

.

..

Theorem 132. The following expressions ofthe curvature d-tensor SSJSA hold true: 1. S'jLwfcj = 0/or {i, j , fc, /} distinct indices; 2. ^[¿WM = - —

l b (yt)

3

=

· ^ωω

Ϊ6

i\2

2

yk

-' (l^

(y\) yi

— r\ / 4. S 1 !.;',. =0(i^=j^=k^=i i(j)(k)

5. S'-¿(i)(i) jMWfe) == 6

- 5!((!!m Jl

=

7. SSJ\J

=

8. SS^yj

=

and no sum by i);

3 ^l^iand

and no sum by i);

^ Τ Τ Τ Τ (ιϊ

3 φΐ^ί

and no sum by I);

— fi ^ Z and no sum by i or I);

\y\Y 1

—2 (i φ I and no sum by i or I);

9. 5.; JA) = 0 (k Φ I and no sum by I); 10. S^J.J

no sum by i);

L—¡f (i Φ k φΐ φι and no sum by I); I6y\yk 16Í/I2/I

'

(ϊφΐϊφΐφί

— 0 (j φ I and no sum by I).

84

JET APPROACH OF THE RHEONOMIC BERWALD-MOÓR METRIC OF ORDER FOUR

Proof: For j -£ k, the expression of the curvature tensor S Í Λ J takes the form (no sum by i, j , k or /, but with sum by m)

4*3

viví

'¿(j)(fc)

+

y[yi

+

AlikSljy[

.(j/i) yí y[ ^■ik-^-mj] y my i

¿j rnk [A?A

4,Wi (yí)2^.

where the coefficients A\¿ are given by the relations (7.14). 7.4

GEOMETRICAL GRAVITATIONAL THEORY PRODUCED BY THE RHEONOMIC BERWALD-MOÓR METRIC OF ORDER FOUR

From a physical point of view, on the 1 -jet space J 1 (R, M 4 ), the rheonomic BerwaldMoór metric of order four (7.5) produces the adapted metrical d-tensor
Sy{,

(7.15)

where <jr¿j is given by (7.6). This is regarded as a "non-isotropic gravitational potential," while the nonlinear connection Γ [given by (7.9) and used in the construction of the distinguished 1-forms Sy\] is regarded as a kind of "interaction" between (£)-, (a:)-, and (y)-fields. We postulate that the non-isotropic gravitational potential G is governed by the geometrical Einstein equations ic (
Sc

(<*),

=¡CT,

(7.16)

where Ric (CT J is the Ricci d-tensor associated to the Cartan canonical connection CT, Se Í CT J is the scalar curvature, K. is the Einstein constant, and T is the intrinsic stress-energy d-tensor of matter. In this way, working with the adapted basis of vector fields (7.10), we can find the local geometrical Einstein equations for the rheonomic Berwald-Moór metric of order four (7.5). Firstly, by direct computations, we find the following: Proposition 133. The Ricci d-tensor of the Cartan canonical connection CT of the rheonomic Berwald-Moór metric of order four (7.5) has the following effective adapted local Ricci d-tensors: Ri

RYi i3m

±1K„ 1K a„1 o ( l ) ( l ) (i)(j) Q l\ ll

9'

1 (1)(1) /ti i J , 3 l l O (0Ü) ' 7δα 1 1 (no sum by i or j). y\y{

,m(l)

Pij(m) c(l)(l) _

qm(l)(l)

(7.17)

GRAVITATIONAL THEORY INDUCED BY 4-RHEONOMIC BERWALD-MOÓR METRIC

85

Remark 134. The local Ricci d-tensor S(J¡J has the following expression: (i)U) 1 1 8 ?(i)(i)

3

'(»)(J)

i^j,

y|yi' 1

i=3-

Remark 135. Using the third equality of (7.17) and the equality (7.7), we deduce that the following equality is true (sum by r): c m l l <*£/ „ m r n ( l ) ( l )

146?

_

4

Vi (no sum by i or m). Vi

VGnii

(7.18)

Moreover, by a direct calculation, we obtain the equalities dSlm i l

Σ sr 1 1 ^ = o. Σ ri dyf

m,r—l

VG^yl

(7.19)

Proposition 136. The scalar curvature of the Cartan canonical connection CT of the rheonomic Berwald-Moór metric of order four (7.5) is given by Sc

(cf).

9hu

+ KÍIKÍI

VU1111

Proof: The general formula for the scalar curvature of a Cartan connection is (see Proposition 90) Sc (ct)=g"Rpq + hilir'S$W. Describing the global geometrical Einstein equations (7.16) in the adapted basis of vector fields (7.10), we find the following important geometrical and physical result: Theorem 137. The local adapted geometrical Einstein equations that govern the non-isotropic gravitational potential (7.15), produced by the rheonomic BerwaldMoór metric of order four (7.5) and the nonlinear connection (7.9), are given by

VG1111

= r„

«ll«il q(l)(l) ,

ξii

' 9ij — hj i

(7.20)

1111

J_o(l)(l) j Í11 K WO) + V G 1111 -

7-(l)(l) . 1,11 . „ . . __< T-V n ytJ — J ( j

0 = TU,

0 = Tn,

(i) 0 = %' (i)l'

0 = %1 (1) «

ί|ι?(ΐ)(ΐ)_τ(ΐ)

K

n o(i)(i) _ -τ-(- ( i )

(7.21)

86

JET APPROACH OF THE RHEONOMIC BERWALD-MOÓR METRIC OF ORDER FOUR

where 9/ln + « n / i n 2ÍC

;n

(7.22)

Remark 138. The local geometrical Einstein equations (7.20) and (7.21) impose as the stress-energy d-tensor of matter T to be symmetric. In other words, the stressenergy d-tensor of matter T must verify the local symmetry conditions TAB = TBA,

WA,Be{l,i,^}.

By direct computations, the adapted local geometrical Einstein equations (7.20) and (7.21) imply the following identities of the stress-energy d-tensor (sum by r):

n d= hnTu T$l

)

d

=f hn9mrT™

-Ί

- 9

T (m)

TTi-

de/ ,

11 ξ 11 ^

j-m(l) de/

mrj-(l)

r(i)

(i)

T-(m)(l) def

T (1)(i)

+ -^==Λ

- ^ - b , fe

llKll

„mrr(l) _

=

cm 11

r

χπι 4

,

l ( D «W ü l

r

W

_

n

Kn 3IC

mrT(l)(l)

hu c m n _ "11

- ftuff T (p)(i) - - ^

where the d-tensor 5|™

rl — «i

7 ? d=f h"Tu = 0,

= 0,

/t-t i Pv def „rnrn- _ '"11"Ί1 Ττη. "±J cm 11 ι

Τ%

1

—9

N±I ;n

^m

+^ Π Ϊ Γ ,

is given by (7.18) and ξχχ is given by (7.22).

Corollary 139. The stress-energy d-tensor of matter T must verify the following geometrical conservation laws (summation by m): (m),(l) •Ί/1 "+" \\m "+" ((1)1 l ) l l(m) 1

i

(Ä n ) 2 dA 11 8/C

T I _i_rm , r ( m ) i ( 1 ) _ Miíil h/l

+ J i|m + J (l)i l(m)

1 8

d h\i dt2

¿t

3 (dh\\ h n \ dt

*

J_

v

1 J_ ^ ^ - ' y« '

1 \/Giin

· ^ / G ^ - ' yi >

^-1(1) , ^-mil) ,T-(m)(l)|(l) _ 6l_ J (¿)/l "+" 7 (¿)|m + J(l)(i) \(m) - 6 ·

(7.23) w/jere (summation by m and r)

SOME PHYSICAL REMARKS AND COMMENTS

!

def ¿Tj

J

l\m

6xm

r

i

i

L

+ Jl

r

, _ ¿7"l

i

™

87

Sxm

'

o^-(m) o/y-(m) ^ ( m W l ) de/ ^ ( 1 ) 1 ^(1)1 r(r) ^ ( 1 ) _ 7 (1)1 l(m) "+" -'(l)lW(m) ~ £ y m ' öym

l def δΤ^ ■'¿/i ~ <5i

T

T

m

'l

m

de/ ^

m

7

l(l) «)/i

*

¿1

+

~ ^ ~

rm

Γ

im

9T1(1)

Z i

(«)

_ «11 d^"*

3 dj/5» ' ^(1)» fl^m '

1

K

" '

c/y-m(l) m ( l ) de/ o i (¿) rr{l)Tm "* (¿)|m fem ~r ^ (¿) L ™

r

^ ( m K l M l ) def 7 (1)(¿) l(m) -

_ ^L ,rl 1 ~~ ¿í · n'

(1)» ^rO^mfl) _ ^ m W l ) _ 0 y m + y(l)¿°r(m) "'(l)r ° i ( m ) ~

d£/ ° ^ (i) -

j i _TiCr

, -T-rrm _ 7-m r r



T-(m),(l) def " i (l)i l(m) -

r

'

j _ «11 (r) ^¿m g

7-m(l)rr J

aT-(m)(l) (l)(i) r(r){l)rm(l) ßym "+" J(l)(i) ° r ( m )

U2

ftrr-m(l) (i) ^m >

ϋ 7

aq-(m)(l) ^(lH*) °t(m) ~ · öj/m

r(m)(l)^r(l)

>)(r)

_

Proof: The geometrical conservation laws (7.23) are provided by direct computations, using the relations (7.13) and (7.19).

7.5 SOME PHYSICAL REMARKS AND COMMENTS 7.5.1 On gravitational theory It is known that in the classical Relativity Theory of Einstein (which characterizes the gravity in an isotropic space-time) the tensor of matter must verify the conservation laws where ";" means the covariant derivative produced by the Levi-Civita connection associated to semi-Riemannian metric gij(x) (the gravitational potentials). Comparatively, in our non-isotropic gravitational theory [with respect to the rheonomic Berwald-Moór metric of order four (7.5)] the conservation laws are replaced with (?; = Τ7Ϊ)

88

JET APPROACH OF THE RHEONOMIC BERWALD-MOÓR METRIC OF ORDER FOUR

T,=

%

{hn)2dhn

^d2hn dt2

dt

8/C

3 ~ hn

c//i.i]

~dr~

T-(D

«1161

18

λ/Giin y j '

21

— *l/l~l~ Il\m~t

y/G 1111

_ in 6

A/GIIII

J/Í'

where q- de_f

T

i

T

"*' ~~ J i / 1 (1) de/ 2 (i) -

m

•I"'

f

I

(l)ll{m)>

/7-(m)|(l)

(1)· Ή '

1(1) rm(l) (¿)/l "+" ^ (t)|m

J

T(m)(l),(l) 7

+

(l)(i)

l(m)·

By analogy with Einstein's theory, if we impose the conditions (V i = 1,4) 71=0, Ti = 0, T (i) W

_

0,

then we reach to the system of differential equations dhu

J2hn

~dT

J

dt

2

~

3

d/li]

/HI

~dT

0,

(7.24)

9/lii -f « i i K i i = 0·

Obviously, because we have h\\ > 0, we deduce that the ODE system (7.24) does not have any solution. Consequently, we always have

pi]2 + \n2 + T,(i)a:

i2

^0,

Vi = l,4.

In our opinion, this fact suggests that our geometrical gravitational theory [produced by the rheonomic Berwald-Moór gravitational potential (7.15)] is not suitable for media whose stress-energy d-components are TAB = 0,

VA,Be{l,

i, g } .

However, it is important to note that at "infinity" (this means that y\ —* 00,

V i = 1,4),

our Berwald-Moór geometrical gravitational theory seems to be appropriate even for media characterized by a null stress-energy d-tensor of matter. This is because at "infinity" the stress-energy local d-tensors tend to become zero.

GEOMETRIC DYNAMICS OF PLASMA INDUCED BY 4-RHEONOMIC BERWALD-MOÓR METRIC

7.5.2

89

On electromagnetic theory

Using the general formula (4.10) in our particular case of the rheonomic BerwaldMoór metric of order four (7.5) and the nonlinear connection (7.9), we have the electromagnetic 2-form F := F = 0. Consequently, our Berwald-Moór geometrical electromagnetic theory of order four is again trivial (this means that we have tautological geometrical Maxwell equations). In our opinion, this fact also suggests that the rheonomic Berwald-Moór metric of order four (7.5) has rather strong gravitational connotations than electromagnetic ones. 7.6

7.6.1

GEOMETRIC DYNAMICS OF PLASMA IN JET SPACES WITH RHEONOMIC BERWALD-MOÓR METRIC OF ORDER FOUR Introduction

During the so-called "radiation epoch" (in which photons are strongly coupled with the matter), the interactions between the various constituents of the Universal matter include radiation-plasma coupling, which is described by the plasma dynamics. Although it is not traditional to characterize the radiation epoch by the dominance of plasma interactions, however, it may be also called the plasma epoch (see [43]). This is because, in the plasma epoch, the electromagnetic interaction dominates all four fundamental physical forces (electrical, magnetic, gravitational, and nuclear). Nowadays, the Plasma Physics is a well established field of Theoretical Physics, although the formulation of magnetohydrodynamics (MHD) in a curved space-time is a relatively new development (see Punsly [86]). The MHD processes in an isotropic space-time are intensively studied by many physicists. For example, the MHD equations in an expanding Universe are investigated by Kleidis, Kuiroukidis, D. Papadopoulos, and Vlahos in [43]. Considering the interaction of the gravitational waves with the plasma in the presence of a weak magnetic field, D. B. Papadopoulos also investigates the relativistic hydromagnetic equations [78]. The electromagneticgravitational dynamics into plasmas with pressure and viscosity is studied by Das, DeBenedictis, Kloster, and Tariq in the paper [32]. It is important to note that all preceding physical studies are done on an isotropic four-dimensional space-time, represented by a semi- (pseudo-) Riemannian space with the signature (+, +, +, —). Consequently, the Riemannian geometrical methods are used as a pattern over there. From a geometrical point of view, using the Finslerian geometrical methods, the plasma dynamics was extended on non-isotropic space-times by V. Girfu and Ciubotariu in the paper [40]. More generally, after the development of Lagrangian geometry of the tangent bundle by Miron and Anastasiei [55], the generalized Lagrange geometrical objects describing the relativistic magnetized plasma were studied by M. Girju, V. Girtu, and Postolache in the paper [41]. In such a physical perspective and because of all preceding geometrical and physical reasons, this paper is devoted to the development on the 1-jet space J 1 (R, M 4 ) of the geometric dynamics of plasma endowed with the relativistic rheonomic Berwald-

90

JET APPROACH OF THE RHEONOMIC BERWALD-MOÓR METRIC OF ORDER FOUR

Moór metric of order four F: JHR,M4)-^R,

y/hü(t)¡/y11y21y*y*,

F(t,y) =

where ftn(t) is a Riemannian metric on R and (ί,χ1 ,χ2,χ3:x4,y\,yf,yf,yf) the coordinates on the 1-jet space J 1 (R, M 4 ). 7.6.2

are

Generalized Lagrange geometrical approach of the non-isotropic plasma on 1-jet spaces

Let (R, hn(t)) be the set of real numbers endowed with a Riemannian structure. Let us suppose that the Christoffel symbol of the Riemannian metric hn(t) is 11

ft_»^n 2 dt

1 /in

Let us consider that Mn is a real manifold of dimension n, whose local coordinates are (x%) -¡—. Note that, in this Subsection, the Latin letters run from 1 to n, and the Einstein convention of summation is assumed. Let J 1 (R, Mn) be the 1-jet space of dimension 2n + 1, whose local coordinates are (t,xl,y\). These transform by the rules _ __ t = t(t), x% =

χι(χί),

_¿ 8xl dt j Vl = 1

^df '

where dt/dt Φ 0 and rank (dxp/dxQ) = n. Let RRGL^ = (j1(R,Mn), G{$$ = hngi:¡) be a generalized relativistic time-dependent Lagrange space (for more details, see Neagu [65]), where gij (t,xk,yi) is a metrical d-tensor on J 1 (R, Mn) (not necessarily provided by a jet Lagrangian function), which is symmetrical, non-degenerate, and has a constant signature. Let us consider that RRGL™ is endowed with a nonlinear connection having the form r=(M¡l = -n\1y\,N^j). The nonlinear connection Γ produces on J 1 (R, Mn) the following dual adapted bases of d-vectors and d-covectors:

{jt'í'w) C *(jl(R'M"))·

{*,^<,Äj/i}c*V1(R,M'1)),

where

i.-i. i„ m dyf — ¿f dt + K uyi SJ. — a . " T ^ l l i / l

«..mi

d J6x_i - Adx_i/ v ( m )W*dy?' l

r„,· —

η

-* v mj

„ί

Ay! = dy\ - K ny\dt + N¡»ac» (l)m C ■ ( * )

r

GEOMETRIC DYNAMICS OF PLASMA INDUCED BY 4-RHEONOMIC BERWALD-MOÓR METRIC

91

Moreover, it is obvious that the generalized relativistic time-dependent Lagrange space RRGL™ produces on the 1-jet space J 1 ( E , Mn) the global metrical d-tensor G = hudt ®dt + gijdx1 dxj + hngij5y\

Sy{,

which is endowed with the physical meaning of non-isotropic gravitational potential. Obviously, the d-tensor G has the adapted components f Λη,

for

A = l,

B = l,

9ij,

for

A = i,

B =j,

n

for

A

B

GAB

h 9ij,

I o,

"(O'

~U)

otherwise.

Following the geometrical ideas from the previous chapters, the above geometrical ingredients lead us to the Cartan canonical Γ-linear connection (see Theorem 76) CT - [κη, GjV L)k, Clj{k}) , where u _ 9km Sgmj St "

G£ = 2

ri(i)

_ 9

j

%

L

Sg]m Sxk

—

jk~

dgjm 2 \dyk

dgkm dy{

¿fffcm _ Sgjk \ δχΐ Sxm J ' dgjk dyf

(7.25)

In the sequel, the Cartan linear connection CT, given by (7.25), induces the horizontal (/IR—) covariant derivative ΐιωα)SD, ifc(i)(0St

H(j)(l)...

D lfe(l)(i).../l

l.(m)(l).

+£>lfc(l)(()..

li(j)(l).. 1 nlm(j)(l)...ri lfc(l)(/).../iU "^ -t^lfc(l)(i)... " m l ü

+

ml

lt(j)(l).. -£>lfc(l)(i).. ..i "11

v ^lfc(l)(i)...'"-11 ll "Γ

r<m

r^iii)^)--

■^1ηι(1)(ί)..· fcl n li(j)(l)... fim - M lfe(l)(m)..."il·",

n li(j)(l)..

"•^lfc(l)(i).. "11

the M-horizontal (/IAÍ—) covariant derivative H(j)(l)...

D ifc(i)(0-|p

0U

lk(l)(l). δχΡ

li(m)(l).

lmO')(l)V + £>lfe(l)(0... ■Ύηρ +1?.lfc(l)(i)... ¿L·

-D'lm(l)(í)...-"fcp "

n li(j)(l)... U

jm

\k(\)(m)...lv

and the vertical (t>—) covariant derivative n H0')(l)-.|(l)

^ifc(i)(0...l(p)

α η 1»0')(1)· öjL/ ifc(i)(i).

3y?

lm(j)(l)... r ,i(l)

+ Dlfc(l)(i)...

^mfr)

n lt(m)(l)... r ,j(l) -^lfefl)^)... ^m(p)

^mfl) n li(j)(l)..· ^"1(1) _ n li(j)(l)..· "Wlm(l)(l)...U/c(p) -L/lfe(l)(m)...°i(p)

92

JET APPROACH OF THE RHEONOMIC BERWALD-MOÓR METRIC OF ORDER FOUR

where D

,u(j)(i)...,.

,

,.. δ _

δ

_

d

= D\k(i)(/)'.'.'.(*- χΓ> yri)jt ®j-®—1®dt®dxk®

Sy[ ...

is an arbitrary d-tensor on J 1 (R, M). Remark 140. We recall that the Cartan covariant derivatives produced by CT have the metrical properties hn/i

= hl)x = 0,

9ij/i -
hn\k = h% = 0, gn\k = 9\

ftn|g

= Ä » | g = 0,

= 0, ffyljfc) = 9ij\\k) = °-

For the study of the magnetized non-viscous plasma dynamics, in a generalized relativistic time-dependent Lagrangian geometrical approach on 1 -jet spaces, we use the following geometrical objects (see also [41], [64]): 1. the unit relativistic time-dependent velocity-d-vector of a test particle, which is given by

where, if we take ε 2 = hngpqy^y1 > 0, then we put u\ = y\/e. Obviously, we have hnunu\ = 1, where un — gimu™; 2. the distinguished relativistic time-dependent 2-form of the (electric field)(magnetic induction), which is given by H = Hij(t,xk,yl[)dxi

Λ dxj;

3. the distinguished relativistic time-dependent 2-form of the (electric induction)(magnetic field), which is given by G = Gij(t,xk,yk)dxi

Adxj;

4. the relativistic time-dependent Minkowski energy d-tensor of the electromagnetic field inside the non-isotropic plasma, which is given by E = Eij(t,xk,yk)dxi

®dxj + h11Eij(t,xk,yk)5y\

® Sy{.

The adapted components of the relativistic time-dependent Minkowski energy are defined by h/ij = —gijiij-sLr

~\- g

iiir(jrjs,

where Grs = grpgsqGpg. Moreover, we suppose that the adapted components of the relativistic time-dependent Minkowski energy verify the non-isotropic Lorentz conditions [64]

E%mu\ = 0,

£ r C « Í = 0,

(7.26)

GEOMETRIC DYNAMICS OF PLASMA INDUCED BY 4-RHEONOMIC BERWALD-MOÓR METRIC

93

where E™ = gmpEpi. Obviously, if we use the notations if™ = gmpHpr and C¡ = grsGsi, we have Ern

=

^S™HrsGrs

_

JJ^QT^

where 5™ is the Kronecker symbol. In our jet generalized Lagrangian geometrical approach, the relativistic timedependent non-isotropic plasma is characterized by an energy-stress-momentum d-tensor T, which is defined by [41], [64] T = Tij(t,xk,yk)dxt

» ώ ' + hnTij(t,xk,yk)Sy\

® Vi,

where [32] %j = \P + ~2) hnuaUji

(7.27)

+ pgij + Eij.

The entities c = constant, p = p(t, xk, yk), and p = p(t, xk, yk) have the physical meanings of speed of light, non-isotropic hydrostatic pressure, and non-isotropic proper mass density of plasma. Note that the adapted components of the energystress-momentum d-tensor T, which characterizes the non-isotropic plasma, are for

C = i,

h Tijt ih

for

C=)

o,

otherwise.

T nn

TCF={

F =j, , F=i,

(7.28)

In the jet generalized Lagrange framework of plasma, we postulate that the following non-isotropic conservation laws of the components (7.27) and (7.28) are true: 1

A:M

—

u

i

V ¿ e { l , ¿, <J>},

(7.29)

where the capital Latin letters A, M,... are indices of kind 1, i or j Λ " : M " represents one of the local covariant derivatives /IR—, /IM—> or v—, and f 7" m , 1Λ

=

(LT

-'DA ¿DA =

<

L

for

Tm fnr h , ior 0, otherwise.

A = i, A —^ ~(¿)'

A

M M —^ _ (m)'

M

Note that the d-tensor T™ is given by the formula

77" = gmpTpi = (p + -g) ft11^«,! + p¿™ + ET. It is easy to see that the jet non-isotropic conservation laws (7.29) reduce to the following local non-isotropic conservation equations: 7 ^ = 0,

77ΊΪ)=0.

(7.30)

94

JET APPROACH OF THE RHEONOMIC BERWALD-MOÓR METRIC OF ORDER FOUR

Moreover, by direct computations, we deduce that the non-isotropic conservation equations (7.30) become h

w

" \(P + 4 ) <} L \

C /

-I |m

" + (P + 4 ) hnufuinm \

+

C /

- girTr = 0,

Piii

i) (m)

+ P'#(*) J m - ftr^7"1 = 0, where p ,¿ = δρ/δχ\ • Tr = ~9rs^^\m

(1)

plú\

(7.31)

= dp/dy\, and

' s t n e non-isotropic horizontal Lorentz force;

• .P"1 = — grsE™\,\

is the non-isotropic vertical Lorentz d-tensorforce.

Contracting now the non-isotropic conservation equations (7.31) with u\ and taking into account the non-isotropic Lorentz conditions (7.26), wefindthe non-isotropic continuity equations of plasma, namely, (p + 4 ) « H V

C /

J |m

+ P„m«r = 0, (7.32)

where we also used the equalities 1 = ^ (/i 11 M i i«l) )

0 = huunu\lm 0=

11hnunu\il(l)

(m)

=

h (1)

i (h"Uil„·) i

2 ^

'#(m)

un\mu\,

= -/^llíVí, li

l'

<(tn)

the symbols ",, m " and " # /J,\" being the derivative operators δ/δχ™ and d/dy™. Replacing the continuity laws (7.32) into the conservation equations (7.31), we find the non-isotropic relativistic Euler equations for plasma, namely, (p + § ) ^ 1 V : i | m < - P„m {hllufua

(p+J)

11

Λ ««^)«?

ftll

- p# £) ( «r«a - ¿r) - Ö™^" = o.

If we take now yf1 = dxm/dt, m

(

-
_ J _ d x ^ _ cfccTO £o dt ds

1

then we have 4 =

, dxl άχί hu(t)gij(t,x,dx/dt) dt dt '

(7.33)

GEOMETRIC DYNAMICS OF PLASMA INDUCED BY 4-RHEONOMIC BERWALD-MOÓR METRIC

95

where s is a natural parameter of the curve c = (xk(t)), having the geometrical property ds/dt = εο· Introducing this u™ into the non-isotropic Euler equations (7.33), we obtain the equations of the non-isotropic stream lines for jet plasma, which are given by the following ODE systems: • horizontal non-isotropic stream line ODEs: 2 d'2„fc x ~ds^

Lk

p + pc

N

(i)m dxm εο ds

+

hUN

:#P„n

hllN

h hue2 k 2 T P + pc

dxr dxm ds ds

gkmP,

dxr dxm dxk eis ds ds

(i)m9pr εο

dgpq dxp dx" dxm dxk dy\ ds ds ds ds

{i)m 2

vertical non-isotropic stream line ODEs: Hi) Cr(m)

dxr dx" ds ds

e ¿ r f2"rr#( p, (1) )

_|_

pc

m

hue2 J* 1 p + pc2

g

#(m)

h dgpq dxp dxq dxr dxk 2 dy\ ds ds ds ds Remark 141. If the metrical d-tensor <¡r¿j(í, x, y) is a Finslerian-like one, that is, we have . . hn d2F2 = — gij(t,x,y) ¿ dy\dy{ where F : J 1 (R, Mn) —> K+ is a jet Finslerian metric, then we use the canonical spatial nonlinear connection N = (NJJ. 1 of the jet Finsler space, whose general formula is given by (4.8) and (4.9). Consequently, the ODEs of the stream lines of plasma in non-isotropic jet Finsler spaces reduce to the following: • horizontal non-isotropic stream line ODEs: d¿2^.fc x ds2

p + pc(fc)

N,( i)m dxm

+ - £Q

;^P,,

h, Ι Ι Λ NΓ ( Ρ ) (i)m9pr

ds

£o

dxr dxm ds ds

tue

p + pc2

jrk

_

gkmp

dxr dxm dxk ds ds ds

• vertical non-isotropic stream line ODEs: P

(1)

hug

mk

dxm dxkΊ ds ds

huP fcl

(7.34)

96

JET APPROACH OF THE RHEONOMIC BERWALD-MOÓR METRIC OF ORDER FOUR

where ε0 = F and, if the generalized Christoffel symbols of g^ (t,x, y) are 9im fdgjm dxk

r}fc(i,aM/) = V

+

dgkm dxi

dgjk dxm

then we have N,

(k)

— rk

9

IAI«

&9mp

p

at v\-

More particular, if we have a jet Minkowski-like metrical d-tensor g^ = gij (y), then the horizontal non-isotropic stream line ODEs of plasma simplify as d2xk ds2

+

Tk rm

hn
■pk __

p + pc2 7.6.3

p + pc

r^rP„n

dxr dxm ds ds

(7.35)

akm

The non-isotropic plasma as a medium geometrized by the jet rheonomic Berwald-Moor metric of order four

As a particular case of the preceding geometrical-physics results, in this Subsection we work with afixedspatial manifold of order four M 4 , whose l-jet space J 1 (R, M 4 ) is endowed with the jet relativistic rheonomic Berwald-Moór metric of order four F(t,y)

\/νψ)·\Ιν\ύήύ·

(7.36)

Recall that the canonical geometrical objects produced by the Berwald-Moór metric of order four (7.36) and the nonlinear connection (7.9) were computed in the preceding Sections. Also, note that, throughout this Subsection, the Latin letters run only from 1 to 4. Consequently, the geometric dynamics of the non-isotropic plasma regarded as a medium geometrized by the jet rheonomic Berwald-Moór metric of order four (7.36) and the nonlinear connection (7.9) is obtained using the ODEs of stream lines (7.35) and (7.34) for the particular Berwald-Moór geometrical objects (7.6), (7.7), (7.10) and the relations (7.12) and (7.13). Therefore, we find the following geometrical equations for non-isotropic plasma: the ODEs of the horizontal Berwald-Moór non-isotropic stream lines of order four are given by d2xk ds2

p + pc-

rp„ m (1 - 4«5fcm)

dxm dxk ds ds

hue2 p + pc

where k e {1,2,3,4} is a fixed index and we do sum by m;

jk

GEOMETRIC DYNAMICS OF PLASMA INDUCED BY 4-RHEONOMIC BERWALD-MOÓR METRIC

97

• the ODEs of the vertical Berwald-Moór non-isotropic stream lines of order four are given by _ (i) (-,

Umk\

dxm dxk _

v

where k e {1,2,3,4} is a fixed index and we do sum by m. Remark 142. In the particular case when the hydrostatic pressure is dependent only by t and x [i.e., we have an isotropic hydrostatic pressure p = p(i, xk)], the ODEs of stream lines for non-isotropic plasma endowed with Berwald-Moór metric of order four become: • the ODEs of the horizontal Berwald-Moór non-isotropic stream lines of order four:

—

2

ds

where pjTO =

+


p + pc

p ro (i -4S km ) '

m

^

d

^ld^L=

' ds

ds

h

^

p + pc

2

rk,

m

dp/dx ;

• the ODEs of the vertical Berwald-Moór non-isotropic stream lines of order four: Tkx = 0. Open Problem. Are there real physical interpretations for the geometric dynamics of non-isotropic plasma, produced by the jet rheonomic Berwald-Moór metric of order four and the nonlinear connection (7.9)?

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CHAPTER 8

THE JET LOCAL SINGLE-TIME FINSLER-LAGRANGE GEOMETRY INDUCED BY THE RHEONOMIC CHERNOV METRIC OF ORDER FOUR

In the background of Finsler spaces which exhibit the total equivalence of all the non-isotropic directions and which are relevant for Relativity, an important model is given by the Chernov metric of order four ([28], [15]), F:TM-*R,

F(y) = ^ y V y 3 + y'y2y4

+ yW

+ y2y3y4.

(8.1)

The larger class of Finsler metrics to which this metric belongs (we refer to the mroot metrics) have been previously studied by the Japanese geometers Matsumoto and Shimada ([52], [53], [90]). Considering the former geometrical and physical reasons, the present Chapter develops on the 1 -jet space J 1 (R, M 4 ) the Finsler-Lagrange geometry, together with its attached gravitational and electromagnetic field theory, for the natural rheonomic jet extension of the Chernov metric of order four,

Fm(t,y) = y/JFJij- Vylyjyf + y\y\y\ + yiyfyf + yfyiyf, where hn(t) is a Riemannian metric on Rand {t,xl ,x2 ,xs,x4 ,y\,yj,yf,yf) are the coordinates of the 1 -jet space J 1 (R, M4). Consequently, in the sequel, we apply the general geometrical results from Chapter 4 to the jet rheonomic Chernov metric of order four F[3]. Jet Single-Time Lagrange Geometry and Its Applications 1st Edition. By Vladimir Balan and Mircea Neagu. © 2011 John Wiley & Sons. Inc. Published 2011 John Wiley & Sons, Inc.

99

100

8.1

THE LOCAL GEOMETRY INDUCED BY THE RHEONOMIC CHERNOV METRIC

PRELIMINARY NOTATIONS AND FORMULAS

Let (R, hn(t)) be a Riemannian manifold, where K is the set of real numbers. The Christoffel symbol of the Riemannian metric hu(t) is » ¡ , - £ $ 1 ,

where

*« . ¿

> 0.

(,2,

Let also M 4 be a real manifold of dimension four, whose local coordinates are ( x 1 , ! 2 , ^ 3 , ! 4 ) . Let us consider the 1-jet space J X (R,M 4 ), whose local coordinates are (t, x 1 , x2, x3, x4, y\,yl, yf, y\). These transform by the rules (the Einstein convention of summation is assumed) ~

drp dt

~

t = t(t),

3? = χΡ(χ<>), ¡^ = ^L^.yl

p,q = l,4,

(8.3)

where dt/dt φ 0 and rank (dxp/dx") = 4. We further consider that the manifold M 4 is endowed with a tensor of kind (0,3), given by the local components Spqr(x), which is totally symmetric in the indices p, q, and r. We shall use the notations Siji = &Sijpyl,

SHI = 3SiPq(x)y^yQi,

s

i n = S^rViViVi-

(8.4)

We assume that the d-tensor S^i is non-degenerate [i.e., there exists the d-tensor Sjkl on J ^ R . M 4 ) , such that Si:nSjkl = Sk]. In this context, we can consider the third-root Finsler-like function [90], [12] (which is 1-positive homogenous in the variable y) F{t,x,y) = ^'Spqr{x)yly\y\ ■ ^ ψ )

= ^S in (x,y) · ^V\t),

(8.5)

where the Finsler function F has as the domain of definition all values (t, x, y) which satisfy the condition Sm(x,y) φ 0. Then the 3-positive homogeneity of the"y-function"5n 1 [which is a d-tensor on the 1-jet space J X (R, M 4 )] leads to the equalities c

*iii =

j .„ óyU/j - 2bm,

dSm i o ,i » ¿aiUi = ¿»¿»in, SijiViUi — öoiiii

p. fc — 65¿jfc,

dSm d Sm S^i = ——j- = ,,

c

S{jVyx = — o¿ji·

The fundamental metrical d-tensor produced by F is given by the formula

,.

,

9ij{t,x,y)=

hn{t)jPl*_ „

i a j

·

THE RHEONOMIC CHERNOV METRIC OF ORDER FOUR

101

By direct computations, the fundamental metrical d-tensor takes the form 1/3

9ij{*,y)

°111

(8.6)

-SiuS 'jll 35i 11

Siji

Moreover, since the d-tensor S^i is non-degenerate, the matrix g = (
g

where S{ = S^plSpn

— ¿>¿> m

and 3 6

m

S^l +

s{s^

(8.7)

3 (Sill-©111)

S"1SpiiSqU.

=

8.2 THE RHEONOMIC CHERNOV METRIC OF ORDER FOUR Beginning with this Section we will focus only on the rheonomic Chernov metric of order four, which is the Finsler-like metric (8.5) for the particular case {p, q, r} - distinct indices, Spqr '■= S[3]pqr = \

·*·

0,

otherwise.

Consequently, the rheonomic Chernov metric of order four is given by

F[3] (t,y) = y/h"(t) ■ i/yjyjyf + y¡yhí + ylv&í + vlvbi-

(8.8)

Moreover, using the preceding notations and formulas, we obtain the following relations:

Sin := S[3]m = vlvlvl + vlvlví + y\y\yt + vívívt Q a Sin - S[mi

Siji

:—

S[3]iji

dSmni dy\

s - 9 , %]ιιι [3]my\ - % ] i i n - - ^ — ^ , ,

d2S[3]ul

_ \ Smi-y\-yl,

dy\dy[

iφj,

0,

3,

where S[4]nn = ylyfyfyi and Smi = y\ +yf + yf + y\. Note that, for i φ j , the following equality holds true as well: J

•%]tii · S[3]ju = S[3]m \S\i]i -y\-y{)

+

[4jllll

(y\Y{n)

102

THE LOCAL GEOMETRY INDUCED BY THE RHEONOMIC CHERNOV METRIC

Because we have 0 φ det [Sijl)i find

kl

Si

1

:= sg} = <

j = n

-rt)

» i

»mi

= 4 [ 4 S [ 4 ] m i - S j ^ S ^ m ] := » m i , we

y{

2/i?/i +

,· L.

ViVi

n(»i+»S)

, 3 Φ k, j = k.

)

.1=1

Further, laborious computations lead to

(8.9) S i l l : = 6[3]111 = &β] S ß j p n S ß ] , ! ! =

-S[3]m·

Replacing now the above computed entities into the formulas (8.6) and (8.7), we get gij : = 0 [ 3 ] i j = 0-V3 °[3]111

ο/Ό

*

Λ

^4]1111

1

2 (S[i]i - yi - j/íj - -ή± V

-,-4/3 '[3)111 "9

'

S[3!in

——2

« ^ J,

2

(8.10)

(y\) (yí)

Q2 ώ

í =J,

[3]»11'

and o

· - 5[3] -

^[Sjlll

gjkl

+ 65[ ]ΐ] -víi/í

(8.11)

3

Consequently, using the general formulas (4.7), (4.8), and (4.9), we find the following geometrical result: Proposition 143. For the rheonomic Chernov metric of order four (8.8), the energy action functional E [3 ](í,x(í))

= /

Ja

Ff3]y/hiidt

= I \¡(y¡yívi + vlvhi + y\y\v\ + y\y\y\Y ■ h11 y/KTidt Ja

produces on the l-jet space J 1 (R, M 4 ) the canonical nonlinear connection Γ = (M ( ( f n = -n\iy\,

Arg. = θ) .

(8.12)

Because the canonical nonlinear connection (8.12) has the spatial components equal to zero, it follows that our subsequent geometrical theory becomes trivial, in

CARTAN CANONICAL LINEAR CONNECTION. d-TORSIONS AND d-CURVATURES

103

a way. For such a reason, in order to avoid the triviality of our theory and in order to have a certain kind of symmetry, we will use on the 1-jet space J r ( R , M 4 ), by an a priori definition, the following nonlinear connection (which does not curve the space): Γ[3] = (MU = -nl.yl

Ν$, = - ^ δ ή ,

(8.13)

where ¿j is the Kronecker symbol. Remark 144. The spatial components of the nonlinear connection (8.13), which are given on the local chart U by the functions iV

- \f(i)i -

2 V'

do not have a global character on the 1-jet space J X (R, M 4 ), but have only a local character. Consequently, taking into account the transformation rules (1.8) and (1.18), it follows that the spatial nonlinear connection N has in the local chartW the following components:

8.3

CARTAN CANONICAL LINEAR CONNECTION. d-TORSIONS AND d-CURVATURES

The importance of the nonlinear connection (8.13) comes from the possibility of construction of the dual local adapted bases of d-vector fields

\Tt = Wt + ^M

;

Ί^ = Ί^ + ΎΜ

;

M\CX{E)

(8 14)

·

and d-covector fields dt ; dxi ; Sy\ = dy\ - K^J/JÄ - ^γάχΛ

C X*(E),

(8.15)

where E = J X (R,M 4 ). Note that, under a change of coordinates (8.3), the elements of the adapted bases (8.14) and (8.15) transform as classical tensors. Consequently, all subsequent geometrical objects on the 1-jet space J 1 (R, M 4 ) (as Cartan canonical connection, torsion, curvature, etc.) will be described in local adapted components. Using the general result of the Theorem 76, by direct computations, we can give the following important geometrical result: Theorem 145. The Cartan canonical T ^-linear connection, produced by the rheonomic Chernov metric of order four (8.8), has the following adapted local components:

rr,,

- ÍK1 rk - n r¿ - " " r - ' W r ^ A

Cl [3] - l ( t n , G j ! - U, hjk - ~YCj(ky

Cj(k)\ ,

104

THE LOCAL GEOMETRY INDUCED BY THE RHEONOMIC CHERNOV METRIC

where C

*(i)

j\k)

35

~

S

?3?

S

+ Q-^2 υ °[3]111

[3}jkm

1 1 6 Si[3]111

[3}jnS[3}kny[

(8.16)

Vi [3]jkl-¿- + öjS[3]kll

S

S S

+

k [3]jll

Proof: Using the Chernov derivative operators (8.14) and (8.15), together with the relations (8.10) and (8.11), we apply the general formulas which give the adapted components of the Cartan canonical connection (see Theorem 76): _ %T rk <Jjl - —

¿g¡3]mj St ,

9[3\ ft>g[3]jm

n L·, jk

2 I

i) _ 5 ρ | [ ( dg[3]jm

c3(k)

dg\í\km

¿ff[3]fcm _ ¿g[3]jfc

Sxk

Sxn

δχί

_ dg[3]jk \

_ 9¡3¡ dg\3]jk

dyf ' dy{ where, by computations, we have (for more details, see [90] and [12]) 2 I

dy\

dyf

1/3 „ [3}UlS[3}jkm

dg[3}jk

. 4 + 27St3]lll5[3]jll'%]kllST3]mll

2S

dyf

)

" " Q ^ P I H I \^[3}3k\S[3]mll

+ S[3]kmlS[3]jn

+ S'[3]m;;l'%]fcll }

¿(1)

Remark 146. The following properties of the d-tensor C ,Λ hold true: >i(l)

J'(fe)

ii(l)

—

0

fe(i)'

«(!) „m ifm)»l

0.

Theorem 147. The Cartan canonical connection CT[3] of the rheonomic Chernov metric of order four (8.8) has three effective local torsion d-tensors: oik) (1) (1)<(J)

pfc(l) _ ^fe(l)

1 ^fc(l) o K l l ° i t(j) '

(fc)

i? (1)1¿

1 f άκ\χ dt

κ

ι ι κ π I o,·.

Proof: A general /i-normal Γ-linear connection on the 1-jet space J 1 (R, M 4 ) is characterized by eight effective d-tensors of torsion (see Theorem 55). For our Cartan canonical connection CT[3¡ these reduce to the following three (the other five cancel):

p(fe) (1) _ Umi) "

dN{

]

U1

\l)i dyj

_

Tk

(fc)

RyK

J¿'

6M™ (1)1

W SN(1)J

δχί

St

(i

pfc(l)

Hi)

=

^fc(l) i(j) ■

Theorem 148. The Cartan canonical connection CTß] of the rheonomic Chernov metric of order four (8.8) has three effective local curvature d-tensors: fíl

n

_

ijk "

i1

4

l l fΛ

K

*.! cc«(l)(l) Hl)(l) Kl

p i (1) _ * 1 o'(l)(l)

ii K ii' : 'i(j)(fc)'

ñCK) Ul)

o!(l)(l) _ " ' " ¿ ( j ) _ °¿(j)(fc) ~ 0yk

a

o^l(l) r)C* °»(fc) 9j/J

^ij(k) -

o^n^jXfc)'

, ^ m ( l ) ^ i ( l ) _ ^771(1)^(1) U « 0 ) °m(fc) i(fc) U m O T

+L/

APPLICATIONS OF THE RHEONOMIC CHERNOV METRIC OF ORDER FOUR

105

Proof: A general /i-normal Γ-linear connection on the 1-jet space J 1 (R, M4) is characterized by five effective d-tensors of curvature (see Theorem 58). For our Cartan canonical connection CTpj these reduce to the following three (the other two cancel): oi

l k

i~

_

dL

ij

ÖLik

6xk

TmTi

iJ

δχί

mk

ττητΐ ik

mp

pi r

(1) _ ^i¿ _ rl{\) rl(l) p (m) (1) ij(k) - Qyk ^i(k)\j+^i(m)^(imk) '

oi(l)(l) _ iU)(k)~

ö

»(j) _ dyk

»(fc) dyj

+

__ ^,771(1)^(1) rm(l)rl(l) °i(j) °m(fc) °i(t) °m(i)'

where _"^i(fc) , ^ m ( l ) r j _ r , i ( l ) r m _ r , ¿ ( l ) r m *(fe)|j ~" ¿ x j + ¿(fc) m3 ^mik^i] °i(m)^r

nl(l)

Remark 149. We have denoted by " / j , " "μ," and " | L the Cartan covariant derivatives with respect to the corresponding R—horizontal (temporal), M—horizontal, and vertical vector fields of the basis (8.14). 8.4

8.4.1

APPLICATIONS OF THE RHEONOMIC CHERNOV METRIC OF ORDER FOUR Geometrical gravitational theory

From a physical point of view, on the 1-jet space J 1 (R, M 4 ), the rheonomic Chernov metric of order four (8.8) produces the adapted metrical d-tensor dx> + hng[3]ij6y{

6y{,

(8.17)

where <7[3]¿j is given by (8.10). This is regarded again as a "non-isotropic gravitational potential." Also, the nonlinear connection Γρι [given by (8.13) and used in the construction of the distinguished 1-forms Sy\] is regarded as a sort of "interaction" between (i)-, (x)-, and (y)-fields. We postulate that the non-isotropic gravitational potential G[3j is governed by the geometrical Einstein equations N

Ric (Cr[3])

Sc (Cr.31)

_

Y^iZG^KT,

(8.18)

where Ric (CT[3¡) is the Ricci d-tensor associated to the Cartan canonical connection CT[3], Sc (CT[3]) is the scalar curvature, /C is the Einstein constant, and T is the intrinsic stress-energy d-tensor of matter.

106

THE LOCAL GEOMETRY INDUCED BY THE RHEONOMIC CHERNOV METRIC

In this way, working with the adapted basis of vector fields (8.14), we can find the local geometrical Einstein equations for the rheonomic Chernov metric of order four (8.8). Firstly, by direct computations, we find the following Theorem 150. The Ricci d-tensor of the Cartan canonical connection CTp] of the rheonomic Chernov metric of order four (8.8) has the following effective local Ricci d-tensor components: i?

n

ij

— pr

_ l i

p (1) _ i(j) -

r

— 4KHK11S(¿)Ü) '

p ( l ) . _ pr (1) (i)j — rij(r)

_ 1 - 2

1 „(1)(1) " (0(3) '

_

_nQP<31Crml [3] [3] \S[3]ijpS[3]qrm

r

?(!)(!) S

^ι s(i)(i)

ijr

U

■-

WÜ)

'-'[3]ipr'-'[3].7gmJ

~

1 1 „ 1 _1_ + 7 5 ς-—S[3]iji - τ^-ξ2

^ α ι ■%];,· n

[3]1

where § ( ) { ? = S^,-wr\

is the vertical Ricci d-tensor field.

Proof: Using the equality (8.16), by laborious direct computations, we obtain the following equalities (we assume implicit summation by r and m): ÖC

5?oÖS[3]%

l l

5

s

1

.

s

[311 ÖO

i(r)

Λ7 dj/l

°i(i)

„ Οώ [3]

°m(r) -

„

2

öl ~^~T;S— mirm 9 2/l

— o-

9,:,

[3]

Λ

-

a

[3)

1

o"?~ J i |3]lH

„

5

2

1

[3]ί3ΐ + ö " 5 2 οώ [3]111

2

5

[3]ί115[3];ηΐ,

1

a [3Wpa|3]mr<j + „ 0 2 [3]ill^[3ljll 3 °|3]111

5 " ñ c [3] { 5 [3]ir«S(3|;jll + 5 [ 3 ] ^ > < ) % ] ί 1 ΐ } ~ 7 T! S[3)i;íl, ¿ <3[3]111 I 1 b Λ[3]ιιι

^ÍTr) C u ) =

9S

\3]" S[3] S[3\irpS[3]mjg + - -35 °

5 Γ " n c [ 3] { ^ ¿ Γ , ^ ΐ ί ΐ ΐ ¿ Λ[3]111 I '

D

[3|lll

S|3] m S[ 3 ]j n

+ S [ 3 ] j r ? S [ 3 ] i l l } ~ 7 5 "= 12 ώ[3]ιιι

Finally, taking into account that we have

3(1)(D _

er(l)(l) _

U

^i(j)

_

U

^i(r)

rm(l)rr(l)

_

rm(l)rr(l)

S[3)iji.

APPLICATIONS OF THE RHEONOMIC CHERNOV METRIC OF ORDER FOUR

107

and using the equalities Q'[3]T~S[3]ijm -

eSjgj 5[3j

S[3]ijpS[3]mrq,

ßerml ~j &{3]irm =

—

dy[

""[3]

'"'[3]

S[3]irp&[3]jmqi

we obtain the required result. Remark 151. The vertical Ricci d-tensor S;(i)(i) J, J has the following property of sym (¿)(J) metr

;(l)d)

=

§(!)(!)

y : %KJ) - -urn ■ Proposition 152. The scalar curvature of the Cartan canonical connection CT[3] of the rheonomic Chernov metric of order four (8.8) is given by Sc (Cr [ 3 ] )

4/ln +ΚΠΚΠ

3ll

nP9c(l)(l) %ΠΡ)(
§ 11

where

Proof: The general formula for the scalar curvature of a Cartan connection is (see Proposition 90)

Sc (CTm) = 9^R„ +

hng^J^y

Describing the global geometrical Einstein equations (8.18) in the adapted basis of vector fields (8.14), wefindthe following important geometrical and physical result: Theorem 153. The local geometrical Einstein equations that govern the nonisotropic gravitational potential (8.17), produced by the rheonomic Chernov metric of order four (8.8) and the nonlinear connection (8.13), are given by n 6 i S311 ¿ii=T„, K

llKllg|(l)(l)

i ¿

oll

T

(8.19)

-(!)(!)

S(!)(l)

¿ W + f c i S " * " * ^'(i)U) . ο = τ (;(ti))l '

0 = TiU

Tu (1)

ZHi)

κ

ίι σ (ΐ)(ΐ) 2/c <*>ω

T

(1)

ZUgWii)

=

j - (i)

where

(8.20)

4fen + KJiKJi 8/C Remark 154. The local geometrical Einstein equations (8.19) and (8.20) impose as the stress-energy d-tensor of matter T to be symmetric. In other words, the stressenergy d-tensor of matter T must satisfy the local symmetry conditions :n

TAB = TBA,

VA,Be{l,i,§}.

108

THE LOCAL GEOMETRY INDUCED BY THE RHEONOMIC CHERNOV METRIC

8.4.2

Geometrical electromagnetic theory

For the rheonomic Chernov metric of order four (8.8) and the nonlinear connection (8.13), via the general formulas (4.10), we find the electromagnetic 2-form F := F [3] = 0. In conclusion, the locally-Minkowski rheonomic Chernov geometrical electromagnetic theory of order four is trivial. In our opinion, this fact suggests that the metric (8.8) has rather gravitational connotations than electromagnetic ones in its t-deformed (x-independent) version, which leads to the need of considering x-dependent conformal deformations of the structure (as, e.g., recently proposed by Garas'ko in [36]).

CHAPTER 9

JET FINSLERIAN GEOMETRY OF THE CONFORMAL MINKOWSKI METRIC

This Chapter develops the Finsler-like geometry on the 1-jet space for the jet conformal Minkowski (JCM) metric, which naturally extends the Minkowski metric in the Chernov-Pavlov framework. To this aim, there are determined the canonical nonlinear connection, distinguished (d-) Cartan linear connection, d-torsions, and d-curvatures. The field geometrical gravitational and electromagnetic d-models based on the JCM metric are discussed.

9.1 INTRODUCTION Let (R, /in (i)) be a Riemannian manifold, where R is the set of real numbers. The Christoffel symbol of the Riemannian metric hu(t) is i

11

fc^dÄn 2 dt

where

hu

= {h v

)-i>0. '

Let also M4 be a manifold of dimension four, whose local coordinates are x = (a; x ,x 2 ,x 3 ,a; 4 ). These manifolds produce the 1-jet space J J (R, M 4 ), whose local coordinates are (t; x; y), where y — (y\, y\, yf, yf). We recall that these transJet Single-Time Lagrange Geometry and Its Applications 1st Edition. By Vladimir Balan and Mircea Neagu. © 2011 John Wiley & Sons, Inc. Published 2011 John Wiley & Sons, Inc.

109

110

JET FINSLERIAN GEOMETRY OF THE CONFORMAL MINKOWSKI METRIC

form by the rules i=t(t),

ff>

= x"(x"),

f1

=

^ - ^ . y l

p,q=l,4,

(9.1)

where dt/dt Φ 0 and rank (dxp/dxq) = 4. In recent physical and geometrical studies ([7], [37], [79], [80]), an important role is played by the jet Finsler-like metric

F[2] (t, y) = y/hJHf) ■ y¡y\y\ + y\y\ + y\y\ + y\y\ + y\y\ + yfyf,

(9.2)

which produces the fundamental metrical d-tensor

Using again the a priori jet nonlinear connection

ΓΜ = ( < ! = -«ίι»ί, <

w

= -^q)

.

(9-3)

we deduce that the Carian Γ[2]-linear connection is

CT [2]

= (K\u ckn = o, vjk = o, c$> = o).

For the Cartan connection CTp] all torsion d-tensors vanish, except 1

(k)

ñ (i)ij

~ 2

and all curvature d-tensors are zero. Consequently, all Ricci d-tensors vanish and the scalar curvature cancels. Consequently, the geometrical Einstein equations (8.18), produced by the jet Finslerian metric (9.2) and the nonlinear connection (9.3), become trivial, namely, 0 = TAB,

VA,Be{l,i,§}.

At the same time, the electromagnetic 2-form associated to the jet Finslerian metric (9.2) and the nonlinear connection (9.3) has the trivial form F := F[2] = 0. In conclusion, both the metric-tensor based geometrical (gravitational and electromagnetic) theories are shown to be trivial for the case of the rheonomic jet locally Finslerian metric (9.2). Hence, for developing a non-trivial model, one may need to consider other closely related alternatives, such as the jet conformal Minkowski (JCM) metric F : J\R, M 4 ) -> R, defined by4 F(t,x,y) = e"{x) · y/VHfy · sjy\y\ + y\y\ + y\y\ + yjyf + yfä + yfyf, 4

(9.4)

In the following we shall reduce the domain of the constructed geometric objects in order to ensure their existence and, where this is required, their smoothness.

THE CANONICAL NONLINEAR CONNECTION OF THE MODEL

111

where σ(χ) is a smooth non-constant function on M 4 . Remark 155. It is easy to verify that (as emphasized in the recent studies [28] and [80]) the geometrical object

Gn (y) d= v\vl + y\y\ + vhi + vlvl + ylvt + vhi

(9-5)

is a quadratic form in y = (y{, yf, yf, y\), whose canonical form is the Minkowski metric. Namely, denoting x = (χι,χ2,χ3,χΑ),χ = (x1,x2,x3,x4),and I l/v/6 l/v/6 l/v/6 \ l/v/6

A=

-1/V3 2/v/3 0 -l/v/3

1 0 0 -1

-l/v/6 -l/v/6 3/v/6 -l/\/6

if we apply on the product manifold K x MA the invertible linear coordinate transformation then in the induced coordinates (t, x, y) on 7Χ(Μ, Μ 4 ), we have T ?/ = A ■ Ty, and the JCM Finslerian metric (9.4) has the particular form ■ ΤΛ> · v / ^ H Í ) - ^ } )

F(t, x, y) = e^

2

- (Ö?)2 - (Ö?)2 - (£Í) 2 ·

In the sequel, we apply the general geometrical results from Chapter 4 to the particular jet conformal Minkowski-metric (9.4). 9.2

THE CANONICAL NONLINEAR CONNECTION OF THE MODEL

Let's consider on J X (K, M 4 ) the JCM metric (9.4), whose domain of definition consists of all values (t; x; y) which satisfy the condition Gn(y) > 0, where G n is given by (9.5). If we use the notation 5 [ i ] i = í / í + y i + ! / ? + !/!, then the following relations are true: def dGu il

G

=

i

~dy~r=

def dGu dy{

[l]1 Vl

~ '

=

d2Gu dy[dy{

=ί_δ^

where <5¿j is the Kronecker symbol. Obviously, the homogeneity of degree 2 of the "y-function" C7n [which is in fact a d-tensor on J 1 (M, M 4 )] leads to the equalities Gliy[=2Gn,

Gijy\y{

= 2Gn-

112

JET FINSLERIAN GEOMETRY OF THE CONFORMAL MINKOWSKI METRIC

By direct computation, we get the following: Lemma 156. (a) The fundamental metrical d-tensor produced by the JCM Finslerian metric F is given by the formula Ö2F2 dy\dy{'

ftu(t) 2

.. . 9ij{t,x,y) which in our case leads to

e2a(x)

9Φ)

=—j-&

(9·6)

- δϋ) >

and the matrix g = (gij) admits the inverse g~1 = (<7J'fe)> whose entries are 2„-2σ(χ)

93k{x) = — 3 — (1 - 3Äjfc) (b) The divergence of the σ-diagonal vector field on M 4

D =o{x) +a{x) +σ(χ) +σ(χ)

" h i έ έ

has the expression

div Da = σχ + σ2 + σ3 + σ 4 ,

where σ, = da¡dx%. Hence, using the general results from Chapter 4, we yield the following: Proposition 157. For the conformal JCM metric (9.4), the energy action functional E(t,x(t))

=

/ F2(t,x,y)VhTidt=

j ε2σ(χ) ■ Gxl(y) ■

Ja I

=

-Ja 2σ(χ) I \ 2 .

/ e

Ja

h1\t)s/h~^)dt

1 3 ,

[y1yl + y^

1 4 ,

2 3 ,

2 4 ,

3 4 \ . 1 1 IT

+ yij/i + 2/i?/i + ViVi + ViVi) h

.,

Vhudt,

where y = dx/dt, produces on the 1-jet space J X (R, M 4 ) the canonical nonlinear connection r = ( M ( ( í ¡ 1 = -KÍ 1 yÍ, J V « . ) , (9.7) where

Nl% = °jy\+°my?t}

+ °~i - 0 d , v D" ( S [ i ] i - t f í ) ·

Proof: The Euler-Lagrange equations of the energy action functional E are

^f

+ 2H& (Í,xfc,rf) + 2 G g x (t,x\»f) =0,

^ =^ ,

(9.8)

CARTAN CANONICAL LINEAR CONNECTION. d-TORSIONS AND d-CURVATURES

113

where we have the local geometrical components

HllL

Ai)

defJ

(i)

def

=

Um ~

G (1)1

-Wn{t)y\, hng,ik

4

d2F2 x dxmdy\ :V'i

ÖF_ 2

+ ^WK\1(t) dy\

dF2

Q2F2 k

dtdyk

dx

2hnK1ngkmyYl

+

°~my?y\ +
9.3

_ ofjW (i)i - ΖΛ(ΐ)ΐ

MW M

_ _Kκl „A ιι2/ΐ'

njW iV (i)7

dG (i) (1)1 dy\

CARTAN CANONICAL LINEAR CONNECTION. d-TORSIONS AND d-CURVATURES

The canonical nonlinear connection (9.7) is essential in constructing the dual adapted bases of distinguished (d-) vector fields

{¿-! + "»τ^έ=έ-<<ΜΗ (β) (9·9) and distinguished covector fields [dt; dxl ; Sy\ = dy\ - n\iy[dt

+ N¡¡>)pdxp} C X*(E),

(9.10)

where E = J1(M.,Mi). Note that, under a change of coordinates (9.1), the elements of the adapted bases (9.9) and (9.10) transform as classical tensors. Consequently, all subsequent geometrical objects on the 1-jet space J 1 ( E , M 4 ), like Cartan canonical linear connection, torsion, curvature, etc., will be described in local adapted components. In this respect, using a general result from Chapter 4, by direct computations, we have the following: Proposition 158. The Cartan canonical T-linear connection, produced by the jet conformal Minkowski metric (9.4), has the following adapted local components: CT - ^ κ η , Gjl - 0, L*fe, Cj(k) - 0J ,

(9.11)

114

JET FINSLERIAN GEOMETRY OF THE CONFORMAL MINKOWSKI METRIC

where L)k = Sijak + Sikaj +

Ojk

(l-5jk)ai

div D„

Proof: Using the local derivative operators (9.9) and the general formulas which provide the adapted components of the Cartan canonical connection, we get _ gkm Sgmj

m

0,

Sg.j m 2 V 6xk

Ljk

-lm I dg.j m dgkm k + 2 \dy ' dy{

dgjk

δχί

Sxm

0.

dyt

Remark 159. It is straightforward to check the relation U-k =

' } " , which, con

W ., leads to sidering the homogeneity of degree 1 of the local functions N}A

dN,(1)3

dy?

V?=N{ d)i

*

L)myT

N,d)

(9.12)

Proposition 160. The Cartan canonical Γ-linear connection CT of the jet conformal Minkowski metric (9.4) has a single effective local torsion d-tensor, namely, p(0

_ nil

,.P

where, using the notations 32σ dxldxJ'

grada = (σ1,σ2,σ3,σ4), {divDa)l

d{divDa) dxi

\\grada\\

= σ\ + σ\ + σ | + σ|,

°~\i + &2i + 0~3i + Gu

we have 9t ijk

$j (o'ik - o~iak) - Sk (atj - CTjffj) + (1 - Sij) (aik - σ/σ/t) - (1 - 5ik) (σ ϋ - σ/σ^·)

(9.13)

+ - (divDa) (ak - Oj + Sikaj - 8ijak) \grada\\2 -

+-

-(divDaf

(Sk-Slj+Stk6l}-StjSlk)

(div Da)j - (div Da)k + Sij (div Da)k - 8ik (div Ώσ)^

Proof: A general /i-normal Γ-linear connection on the 1-jet space J 1 (K, M 4 ) is characterized by eight effective d-tensors of torsion. For our Cartan canonical connection (9.11), these reduce only to one (the other seven cancel): R .(«)

(l)jk

5N,(I)

(l)j

6xk

SN{1) 0JV

(l)fc δχί

GEOMETRICAL FIELD MODEL PRODUCED BY THE JET CONFORMAL MINKOWSKI METRIC

115

Using now the expressions of the derivatives δ/δχι, formula (9.12), and the y-independence Lljk — Lljk(x), we find p(') -(l)jk

_ ml ,,P - - n p¿fcfl>

n

where n>l

._

ϋ _ dLik

ik '

dxk

, rr rl

*J

dxJ

rfe

_ jr

jl

Xj

ikL'ry

Finally, laborious computations lead to expression (9.13) of the d-tensor SK',fe. Proposition 161. The Carian canonical T-linear connection CT of the jet conformal Minkowski metric (9.4) has a single effective local curvature d-tensor, namely, pi ijk

foi ~nijki

Jx

where 9t' -fc is given by (9.13). Proof: A general /i-normal Γ-linear connection on the 1-jet space ^(Μ,,Μ4) is characterized by five effective d-tensors of curvature. For our Cartan canonical connection (9.11), these reduce only to one (the other four cancel), namely, , ijk

n

def ~

SL

ij ¿xk

_¿L\k ¿xj

+

,Tr rl ¿'ij^rk

ττ jl . ^ W n W - ^tfc-^rj + ° i ( r ) n(l)jk

f£ü _ ÖJL· 4. fr Jl k Sx

-

~

9.4

9.4.1

δχί

'■?

dxk

dL

ik

dxi

i]

i rr jl ij

^jrjl

rk

^ik^rj

jrjl

rk

ik

rj

—

_«»i

ijk

'

GEOMETRICAL FIELD MODEL PRODUCED BY THE JET CONFORMAL MINKOWSKI METRIC Gravitational-like geometrical model

From a geometric-physical point of view, on the 1-jet space J 1 (M, M 4 ), the jet conformal Minkowski metric (9.4) produces the adapted metrical d-tensor & = hndt ®dt + gijdxi dxj + hllgij5yi1

5y{,

(9.14)

where gij is given by (9.6). This may be regarded as a "non-isotropic gravitational potential." In such a "physical" context, the nonlinear connection Γ (used in the construction of the distinguished 1-forms 5y\) prescribes, probably, a kind of "interaction" between (t)-, (x)-, and (y)-fields. We postulate that the non-isotropic gravitational potential G is governed by the geometrical Einstein equations Ric (CT) -

SC ( C r ) G o

=KT,

(9.15)

116

JET FINSLERIAN GEOMETRY OF THE CONFORMAL MINKOWSKI METRIC

where Ric (CT) is the Ricci d-tensor associated to the Cartan canonical connection CT (in Riemannian sense and using adapted bases), Se (CT) is the scalar curvature, K, is the Einstein constant, and T is the intrinsic stress-energy d-tensor of matter [55]. In this way, working with the adapted basis of vectorfields(9.9), we can find the local geometrical Einstein equations for the metric (9.4). First, by direct computations, we find the following: Lemma 162. The Ricci d-tensor ofthe Cartan canonical connection CT ofthe metric (9.4) has a single effective local Ricci d-tensor, namely, Rij

=

-2(aij-aiOj) 1 - 6ij

3Aa + G\\grada\\2 -2{divDaf

where

- 6

(9.16)

4

Δσ = ση + σ22 + σ 33 -i- σ44,

® = ^Ζ

σ

νι·

p,q=l

Proof: A general /i-normal Γ-linear connection on the 1-jet space J 1 (R, M 4 ) is characterized by six effective Ricci d-tensors. For our Cartan canonical connection (9.11), these reduce only to one (the other five cancel): D ixij

^ £ / om _ mm — n ¿ j m — -r*-ijm-

Then, a direct computation gives expression (9.16) of the Ricci d-tensor R^. Lemma 163. The scalar curvature Sc (CT) of the Cartan canonical connection CT of the jet conformal Minkowski metric (9.4) is given by R = 4ε-2σ 3Δσ + 3 \\grada\\2 - {divD„f

- &

(9.17)

Proof: The general formula of the scalar curvature of the Cartan connection yields Sc (CT) d=f g™Rpq := R, where R is given by (9.17). By describing the global geometrical Einstein equations (9.15) in the adapted basis of vector fields (9.9), wefindthe following important geometrical and physical result: Proposition 164. The local geometrical Einstein equations that govern the nonisotropic gravitational potential G [produced by the jet conformal Minkowski metric (9.4)] are given by p Rij ~ ~^9ij = K-Tij, (9.18)

GEOMETRICAL FIELD MODEL PRODUCED BY THE JET CONFORMAL MINKOWSKI METRIC

0 = Tii,

-Rhn=2ICTlu

0 = T«,

117

0 = 7$}, (9.19)

0=Φ

0=T «

0=7$,

-

β

^ = 2£Τ^>.

Remark 165. The Einstein geometrical equations (9.18) and (9.19) impose that the stress-energy d-tensor of matter T be symmetric. In other words, the stress-energy d-tensor of matter T must satisfy the local symmetry conditions

V A f l e j i , i, g } .

TAB=TBA,

gijTuh11.

Moreover, we must a priori have the equality T,\J,\s'hn =

By direct computations, the geometrical Einstein equations (9.18) and (9.19) imply the following identities of the stress-energy d-tensor:5

T-l

de

f hllT

T™ d^J ^nrrr

—

lx

= h in — - 2 Τ >

li

- ti l\i — Ö,

^-1(1) de/ ,11^(1) _ 7

-Ί — 5 -Ή = U,

i¿ — g n

n

r (m)(l)

i r ¿ - p \g de/ ,

^-(m) def ,

mr/r(l)

¿ ( 1 ) 1 = Ain3

__ n

i ( r ^ — U,

tin - -r-Oj I ,

nmrT(l)(l)

_

R

m

(i) - ft Λ ω ~ υ > 7(1)(¿) - ftH3 V)(<) _ " " ^ c " i ' T^d^hll9m%)]=0, T ^ f ^ f T ^ O .

(9.20)

Consequently, the following local identities for the stress-energy d-tensor of matter hold true:

"Ί/l + J l|m + J (l)li(m) ~ "Ί/Ι ~

2AC

¿Í '

-T-l i -T-m , -T-( m )|(l) _ --rro _ ( .mr p Am I A / 1 + "'¿Im + J(l)i l(m) - "'¿Im ~ fc \ 9 " ~~ ~2 l ) ' |m 7

1(1) _,_^η(1) j_T-(m)(l)|(l) _ ^ ( m ) ( l ) , ( l ) ( ¿ ) / l "+" l (¿)|m + ■'(1)(¿) l(m) - ■'(!)(») I(m)

^Γ,^.Ν "21)

(9

1 9R

2K, dy\'

where 'Summing over both indicesTO,r is assumed in (9.22), over r in (9.20), and over m in (9.21) and (9.23).

118

JET FINSLERIAN GEOMETRY OF THE CONFORMAL MINKOWSKI METRIC Tl

1/1

def ST\

l

~ ~~δΓ

, y n

r(m),(l) J

de/ ^ ( 1 ) 1 ( l ) l l(m) - - ^ m " XT™ fixm "+" Λ

I

-

rl 1

( l ) de/ (i)/l -

i\m

ZH-HV (i) ¿í

0 i

^-m(l) d e / 7

L

¿xm

U

>

m

ι _ δΤγ

χ

n

~ ~δΓ'

2

~~ Jx™

rm

l

~

χ de/ dT ¿ _ χ , A / 1 - - ^ - + Λ «11 - U>

-*r ^ ¿ m '

T-r(l)

+

def δΤχ"

1|m

τ

, ο-τ-Ι(Ι) i _ „ + ^ W Kl1 ~ '

(j)

«)|m -

l

~

r m

y

(l)¿ l(m) ~

ÖT(m) ¿fym ~


def Km) ~

_

'

artmXD UJ (l)(i) aym '

r(m)(l)|(l)

(1)W

U

/

n

(i) L ™ - ¿ ( r ) ¿im - U.

Ü W

(9-22)

Taking into account that we have the y-independence R — R(x), we obtain the following result: Corollary 166. The stress-energy d-tensor of matter T must verify the following conservation geometrical laws: T 1 _ Un ■Ί/1 - '

9.4.2

q-m, _ } _ „mr n Rri i\m - JQ 9

I

Λp

χπι g-öj

T(m)(l)|(l) J

(l)(¿)

_ „ \(m) - " ·

,q?1i ^ y - Z -V

Related electromagnetic model considerations

On our particular 1-jet space J 1 (R, M 4 ), the jet conformal Minkowski metric (9.4) (via the Lagrangian function L — F2) produces the electromagnetic 2-form which, due to (9.12), trivially vanishes F = 0. In conclusion, the jet conformal Minkowski extended electromagnetic geometrical model constructed on the 1-jet space J X (R, M 4 ) is trivial. Namely, in our jet geometrical approach, the jet conformal Minkowski electromagnetism, produced only by the metric (9.4) alone, leads to null electromagnetic geometrical components and to tautological Maxwell-like equations. In our opinion, this fact suggests that the jet conformal geometrical structure (9.4) of the 1-jet space J 1 (M, M 4 ) is suitable for modeling gravitation rather than electromagnetism.

PART II

APPLICATIONS OF THE JET SINGLE-TIME LAGRANGE GEOMETRY

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CHAPTER 10

GEOMETRICAL OBJECTS PRODUCED BY A NONLINEAR ODEs SYSTEM OF FIRST-ORDER AND A PAIR OF RIEMANNIAN METRICS

10.1

HISTORICAL ASPECTS

According to Olver and Udri§te ([77], [94]), a lot of applicative problems arising from Physics [79], [80], Biology [68], [75] or Economy [95] can be modeled on 1-jet spaces. In such a context, we point out that the Riemann-Lagrange geometry of the 1-jet space J 1 (R, M) (which was already developed in the preceding chapters) is a very applicative mathematical framework. For example, this geometry contains many fruitful ideas for the geometrical interpretation of the solutions of a given ODEs (or a PDEs system in [71]). In such a perspective, it is important to note that Udri§te proved in [94] that the orbits of a given vector field may be regarded as horizontal geodesies in a suitable Riemann-Lagrange geometrical structure, solving in this way an old open problem suggested by Poincaré [84], [85]: Find the geometrical structure which transforms the field lines of a given vector field into geodesies. In the sequel, we present the main geometrical ideas used by Udri§te in order to solve the open problem of Poincaré (for more details, the reader is invited to consult the works [94] and [95]). In this direction, let us consider a Riemannian manifold (Mn, ifij (x)) and let usfixan arbitrary vector field X = {X1 (x)) on the manifold M. Obviously, the vector field X produces the first-order ODEs system (the dynamical Jet Single-Time Lagrange Geometry and ¡ts Applications 1st Edition. By Vladimir Balan and Mircea Neagu. © 2011 John Wiley & Sons, Inc. Published 2011 John Wiley & Sons, Inc.

121

122

OBJECTS PRODUCED BY AN ODEs SYSTEM AND A PAIR OF RIEMANNIAN METRICS

system) ?—=X\x(t)), Vt = l,n. (10.1) at Differentiating the first-order ODEs system (10.1) and making a convenient arranging of the terms involved, in the Udriste framework one constructs (via the vector field X, the Riemannian metric φ^, and its Christoffel symbols 7Jfc) a second-order prolongation (the single-time geometric dynamical system) having the form

£ Η'ΤΓΤΓ - 3 ΊΓ + Λ « ^ χ ' , ν, = Ώ ,

(10.2,

where V is the Levi-Civita connection of the Riemannian manifold (M, φ) and Fj = VjX* -
-* R + ,

given by Ls(x,y) = 1-ψφ) {yl - Xl{x)) (y* - Xj{x)),

(10.3)

where (χ', yl) are the local coordinates on the tangent space TM. It is obvious now that the field lines of class C 2 of the vector field X are the global minimum points of the least squares energy action attached to Lg, so these field lines are solutions of the Euler-Lagrange equations produced by L$. Because the Euler-Lagrange equations of Ls are exactly equations (10.2), it follows that the solutions of class C 2 of the first-order ODEs system (10.1) are horizontal geodesies on the Riemann-Lagrange manifold [94] ( R x M , l + φ,

Ν ^ =

Ί

^ - η ) .

Remark 168. The authors of this book believe that the preceding least squares variational method for the geometrical study of the ODEs system (10.1) can be reduced to a natural extension of the following well known and simple idea coming from linear algebra: In any Euclidian vector space (V, (, }) the following equivalence holds true: v = 0V & \\v\\2 = 0 , (10.4) where ||υ|| 2 = (v, v). This fact will be applied in the next Sections.

CANONICAL NONLINEAR CONNECTIONS

123

In order to find the jet geometrical objects that characterize a given ODEs system (which governs any applicative dynamical system), let us consider the jet fiber bundle of order one J 1 (R, M) - » t x M associated to the Riemannian manifolds (R, hu (t)) and (M, φ^(χΙί)). It is important to note that we use the notation V = (/in, φ^) for our pair of Riemannian metrics and the inverse tensors of the preceding Riemannian metrics are denoted by h11 and ψ%Κ We recall that these tensors verify the formulas h11 = 1/Λ.ιι > 0 and ^ ¡ ¡ , = 1 φιτη'ψτηί = fy. Also, we use the notations «} 1 and 7jfe for the Christoffel symbols of the Riemannian metrics hn and φ^. Let us consider an unknown curve c = {xl{t)) and an arbitrary given d-tensor field

X=(x$(t,xk)) that define on the 1-jet space J J (R, M) the following ODEs system of order one: Vi =T~ñ

<=>

y[=X$(t,xk(t)), í/í-X ( ( ; ) ) (í,x f c (í))=0, where y\ — dxl /dt. Remark 169. Recently, using the theory of Lie systems, Cariñena and de Lucas ([25]) intensively studied various geometrical features for a class of systems of first-order ordinary differential equations, whose general solutions can be described in terms of certain finite families of particular solutions and a set of constants (e.g., systems of first-order ordinary linear differential equations). The connection between their results and the further developed theory, is a subject of ongoing research for the authors of the present monograph.

10.2 SOLUTIONS OF ODEs SYSTEMS OF ORDER ONE AS HARMONIC CURVES ON 1-JET SPACES. CANONICAL NONLINEAR CONNECTIONS In the sequel, let us show that the solutions of class C 2 of the ODEs system (10.5) can be naturally regarded as harmonic curves on 1-jet spaces. In this way, under our geometrical assumptions, we prove the following interesting qualitative-energetic result (see Theorem 98): Theorem 170. All solutions of class C2 of the ODEs system (10.5) are harmonic curves on the 1-jet space JX(]R, M)for the time-dependent semispray

124

OBJECTS PRODUCED BY AN ODEs SYSTEM AND A PAIR OF RIEMANNIAN METRICS

whose components are given by the formulas 1 l il l2 /,.i / li , Hfl '(i)i = - - T2K h '

_ I r(i) Gj*) (i)i ~= 2-7J,jfc l^l/f - Λ ι ι ^ ί ί , α ; * , ^ ) ,

where

α«ί/

ay(')

v(i)

_

(i)//i_

ay>

-(l) _

dt

ay(«)

v(i)

(i)

i

21

'

v(i)

_

(DIIJ ~

"Λ(1)

,

y(m)

+ Λ

ax¿

i

(ΐ)Ν-

Proof: Using the idea from Linear Algebra (10.4) (which may be called in this case they'ei least squares variational method), the initial first-order ODEs system (10.5) is equivalent with the ODEs system 52_{ft U (t)<M* f c ) (y[ - Xgit,**(*))) (y{ - Xl(>(t,xk(t)))}=

0.

(10.6) This is because the Riemannian metrics hn(t) and
JLS°ms

:= j ; Λ11 (*)¥>«(sVil/í + Σ

+ *(*.**) = °> ( 1 0 · 7 )

U

$)fr*k)v\

1

i,j = l

where U§(t,xk)

n

= -2h11 J ]

toxJ»

m=l

(10.8)

and n

#(i,Xfc) = hU 52 Vrajtfjxííj. r,s—\

(10.9)

Using the Einstein convention for summations, let us consider now the jet least squares time-dependent Lagrangian of electrodynamics type, which is given by JCS°DEs

:=

JL50 D E s v^ 1 T(í) = l | C - X | | 2 v / M í )

= {hn(t)Vij(xk) [y\ - Xg] [y{ - Xg] } v^(t) {ft n (*)<M* fc )i/ÍJ/í + U$(t,xk)y\

+ Φ(ί,χ*)} χ/Λΐί(ί),

where C = y\{d/dy\) ?t/j) and X == X,y>(9/dy|). Let us also consider the y'ef least squares energy action functional EJcs^:C2([a,blM)-^R+,

CANONICAL NONLINEAR CONNECTIONS

125

given by E jcs\

w,{c)

Í

JCS^dt

■

Ja

f

|C-X|| 2 V7ni(t)«ft>0.

Ja

It is obvious now that a smooth curve c € C2(\a,b],M), locally expressed by c(t) = ( r r 1 ^ ) , ^ 2 ^ ) , ...,xn(t)), is a solution of the ODEs system (10.7) if and only if the curve c cancels the time-dependent Lagrangian v /£<S° DEs . In other words, the curve c is a solution of the ODEs system (10.7) if and only if c is a global minimum point for the jet least squares energy action functional E^^ODES. Therefore, every curve c = (xl(t)) of class C2 is a solution of the initial ODEs system (10.5) if and only if it verifies the Euler-Lagrange equations ODEsi l

dx

d_ (d[JCS%DEsi dt \ dy\

(10.10)

= 0, Vi = l,n.

Taking into account the expression of the time-dependent least squares Lagrangian JCS^Es and some local differential computations in the Euler-Lagrange equations (10.10), we claim (cf. Theorem 98) that equations (10.10) can be rewritten in the form (1.12) of the second-order ODEs system of the harmonic curves of the time-dependent semi spray «SoDEsOP) = {^(1)1^(1)1)

(10.11)

'

whose temporal components are given by (i) H'(i)i

-

1

.i „.« --^K\IV-Í

and the spatial components are expressed by

'(1)1

1 2»l\M\

hu
U

(l)m.yi

+

dU(x) dt

u

{i)

K

n

3Φ dxl

where dU,( i ) dxi

fEs£ dxl

TW. and Φ [see formulas (10.8) In what follows, using the expressions which give WJ and (10.9)], together with some direct local computations, we find

126

OBJECTS PRODUCED BY AN ODEs SYSTEM AND A PAIR OF RIEMANNIAN METRICS

X (m)

nh11

r/d) _

_

y'

m

dU{l) (»)

dt 0χΖ ~~ ¿a

n

^« m V S -(i)//i'

~~

Vmr^-W Λ(1)||,,

where dX,(j> ,., _ "^(1) _ Y(i) ι 1 Α í )// ~ oí (1)Κ"'

v (¿) 1

&*, (0 __ "J*(l) (l)l|j" #xj ,·χ

γ(ΐ) A

y{m) i (1J lmj

(10.12)

are the local covariant derivatives of the Berwald Γ-linear connection BY produced by the pair of Riemannian metrics V = (/lili ¥>¿j)· In conclusion, all our preceding computations imply the required result. Definition 171. The time-dependent semispray SoDEs(P) =

(#(Í)1'G(1)I)

given by Theorem 170 is called the canonical relativistic time-depedent semispray produced by the first-order ODEs system (10.5) and the pair of Riemannian metrics V = (/in,Vij)·

Remark 172. In other words, Theorem 170 asserts that the C 2 solutions of the initial first-order ODEs system (10.5) are verifying the second-order ODEs system (1.12) of the harmonic curves for the time-dependent semispray S0DES(P)Remark 173. It is well known that, in Theoretical Physics, the Riemannian metrics are modeling the gravitational potentials of a space of events [92]. Thus, by a natural extension, we may assert that, in our context, the Riemannian metrics may model some gravitational potentials of the abstract microscopic phenomena intrinsically produced by the matter governed by the given ODEs system (10.5). Taking into account the geometrical connection between the time-dependent semisprays and the nonlinear connections on 1 -jet spaces (given by Proposition 26), we easily deduce the following important geometrical Corollary and Definition: Corollary 174. On the 1-jet space J1(R, M), the canonical nonlinear connection

which is produced by the ODEs system (10.5) and the pair of Riemannian metrics 'P = (hu^ij), has the components M

(Í!i = -»hl/i

and Ν^=Ί%ν\-Ή%ρ

(10.13)

ODES SYSTEMS, RIEMANNIAN METRICS AND JET GEOMETRICAL OBJECTS

where

10.3

(l) nnwj

--

,nirY^

v(»)

- 2

127

in

FROM FIRST-ORDER ODEs SYSTEMS AND RIEMANNIAN METRICS TO GEOMETRICAL OBJECTS ON 1-JET SPACES

The geometrization on the 1-jet fiber bundle ^(Μ,,Μ) of the relativistic timedependent Lagrangians was completely developed in Chapter 5: "The jet singletime electrodynamics." Moreover, some generalized geometrical gravitational and electromagnetic field theories were described over there, too. Consequently, in this Section, we will particularize the main geometrical and physical results from Chapter 5 for our particular time-dependent Lagrangian of electrodynamics type JCS^Es. We underline that the jet least squares Lagrangian JCS™Es from Theorem 170 is derived only from the given ODEs system (10.5) and the pair of Riemannian metrics V = (/in> V^)· As a consequence, we introduce the following concept: Definition 175. The pair

JCS^RRUl

JLS%DEs),

= (J\R,M),

endowed with the nonlinear connection Γ,^^-ρ) given by (10.13), is called the canonical relativistic time-dependent Lagrange space produced by the ODEs system (10.5) and the pair of Riemannian metrics V = (hn,
covariant derivatives of the d-tensor X\lMt, xk) are given by Λν(0

Yd)

A

(I)IIJ//I

_

_

aA

(DllJ _ Yd) Qt

i

(ΐ)ΙΙ^ 1 1 '

128

OBJECTS PRODUCED BY AN ODEs SYSTEM AND A PAIR OF RIEMANNIAN METRICS

v(i)

(l)llJ

_

dxk

^(i)lb'Hfc ~

, γ(τη)

i

W\\Jlmk

_ γ(ί)

m

^(i)llmTjfe·

(iii) The curvature d-tensor R of the canonical generalized Cartan connection CTsimh(v) of the space JCS^P^RRL™ is determined only by one adapted component, namely, pi ^ijk

nil •n-ijk'

that is exactly the components of the classical curvature tensor of the Riemannian metric ψ^. In the sequel, following the geometrical and abstract physical ideas from Chapter 5, by direct local computations, we can construct the electromagnetic distinguished 2-form of the space JCS™EsRRLi and describe its geometrical Maxwell equations [see formulas (5.2) and (5.3)]. Theorem 177. (i) The electromagnetic field F of the reiativistic time-dependent Lagrange space JCSjPEsRRL™ is expressed by the distinguished 2-form W=

F¡^Sy\AdxK

where Sy\ = dy[ + M^dt +

N^dx*

and F (i)

_ Λ11

(ii) The adapted components Ft?, of the electromagnetic field F are governed by the following geometrical Maxwell equations *(Oj//i - 1A^i) \ .

^{i,j,k}F(i)j\\k

= 0

h φ

™ L (DlbV/i " φ

A

(i)||r//i^'J / »

>

where A{i¿] represents an alternate sum and ]TV ■feimeans a cyclic sum. Remark 178. We do not describe the jet single-time gravitational theory of the reiativistic time-dependent Lagrange space J£SjPEsRRLi because its time-dependent gravitational field G and its attached geometrical Einstein equations are independent on the distinguished tensorial components X^l(t,xk) that define the ODEs system (10.5). In other words, the geometric time-dependent gravitational entities are depending only by the pair of Riemannian metrics V = (/in,
GEOMETRICAL OBJECTS PRODUCED ON 1 -JET SPACES BY FIRST-ORDER ODEs SYSTEMS

10.4

129

GEOMETRICAL OBJECTS PRODUCED ON 1-JET SPACES BY FIRST-ORDER ODEs SYSTEMS AND PAIRS OF EUCLIDIAN METRICS. JET YANG-MILLS ENERGY

In order to use the preceding geometrical results for the study of some ODEs systems of order one coming from diverse Applied Sciences, let us consider the particular Euclidian metrics as the pair V of Riemannian metrics: V := Δ = (hn = 1,
_ / 0, i^j, " I 1, i =3, is the classical Kronecker symbol. In this particular situation, we are placed on the 1-jet space J1(R, R") produced by the Euclidian manifolds (R, 1) and (R n , ay). Consequently, Theorem 170 asserts now that all solutions of C2-class of the ODEs system (10.5) may be regarded as the harmonic curves of the relativistic time-dependent semispray •SoDEs(A) = (^(l)l>C(l)lJ ' whose components are given by the formulas o-W

rt

(i)i-u-

_

ft

r>(«)

G

_

Ι / γ Μ

ΓγΜ

(i)i - _ 2 ΐ Λ ( ΐ ) Ι Ι < 1

,,rl

(χ) ~yA

I y(') A

„m ι y(i)

(i)llm»i

\

+A(i)//i/'

where

*,(ίί,„=^

and

-dX$

*«

(i)//i dt Wllí ~ dxi ' In other words, following the Proof of Theorem 170, we conclude that the solutions of C2-class of the ODEs system (10.5) are minimizing the jet least squares timedependent Lagrangian

£SöOD&(t,rcfc,I,i) = ¿ (y* -

X^(t,xk))\

4=1

which is obviously the time-dependent Lagrangian of electrodynamics type j7X«S°DEs for the particular pair of Euclidian metrics Δ = (1,5^). We shall further denote the relativistic time-dependent Lagrange space produced by the ODEs system (10.5) and the pair of Euclidian metrics Δ = (1,5¿j) by CSQ0DEsRRL^

= (^(Κ,Μ), X

CSQ). n

Definition 179. Any geometrical object on J (R, R ), which is produced by the jet least squares Lagrangian function £<SQODEs, via its attached second-order EulerLagrange equations, is called a geometrical object produced by the jet first-order ODEs system (10.5).

130

OBJECTS PRODUCED BY AN ODEs SYSTEM AND A PAIR OF RIEMANNIAN METRICS

By simple computations, the geometrical results of Theorems 176 and 177 lead us to the following Theorem 180. (i) The canonical nonlinear connection r 5 o i ) l , ( A)=(M ( ( 1 i j 1 ,7Vg.) S

of the space CSQ

RRL™ is given by the components

(i) M,(1)1

0

Λγ(')

dX (U) i)

dxi

dx%

(i)

and

N,(ib'

(ii) The canonical generalized Cartan connection CT,SODFS(A) of the relativistic % time-dependent Lagrange space CSQ RRU{ has all its adapted components equal to zero. (iii) The torsion d-tensor T of the canonical generalized Cartan connection CTsoms(A) °f the space CSQOOEsRRLi is determined by two adapted local dtensors: *

>*$

R (i) (l)lj

*

«

;

■

and

dxidt

dxWt

R,(i) (i)jfc

dxWxk

dxidxk

(iv) All adapted components of the curvature d-tensor R of the canonical geners alized Cartan connection CTsODI!!,(A) of the space CSQ RRLi vanish. OOEs (v) The electromagnetic 2-form F of the space CSQ RRL^ is expressed by W=

F¡l)j5yilAdx^

where Sy\

=dy{--

(i) dX, (1)

dxi

SYÜ) (i)

dA

i

dx

dx3

and

FM

- i

dX

(i) (1)

dxi

dX, ( i ) dxi

(vi) The electromagnetic components FL·, of the relativistic time-dependent Lagrange space CSQODEsRRLrl are governed by the following geometrical Maxwell equations

(i)j//i

4^{«vj}

p(D

• ^{i,j,k}F(i)j\\k

- ° '

'^ίί! dxidt

ñ2 Υ^

fl2yW

d2x({Í]

dxidt

dxidt

dx^

0 A(1)

GEOMETRICAL OBJECTS PRODUCED ON 1 -JET SPACES BY FIRST-ORDER ODEs SYSTEMS

131

where A^jj represents an alternate sum, J2u ,· k} means a cyclic sum, and we have 7(D

dt

(¿)J//1

'

(i)i\\k

gxk

■

In our geometrical and physical approach, we point out that the distinguished electromagnetic 2-forms F produced by the ODEs systems and pairs of Euclidian metrics may be regarded as o(n)-valued 1-forms on the time manifold R, setting F = F{i)dt e IXA^T'R) ® o(n)), where

0

(1)3

(i)

(2)3

(2)3

0

r(i)

(1)2 7 (1)

F r

(1)3

F ((1) 1

F(D

r(i)

(1)2

Ml)« r

{2)n

Γ

(3)η

e o(n).

7W

(n — \)n

V

Γ (1) (1)»

7W

(3)n

(2)n

- F ,(n—l)n

0

Note that o(n) is the set of skew-matrices as the Lie algebra L(0(n)) of the subgroup of orthogonal matrices 0(n) c GLn(R). As a conclusion, we can introduce the following geometrical and abstract physical concept: Deñnitíon 181. The Lagrangian function of Yang-Mills type, which is given by the formula n— 1

n— \

n

n

(i) ΘΧ,(i)

syuT^rn^) = Σ Σ [4ΐΓ = ϊ Σ Σ9xJ i = l j=i+l

i—1 j=i+l

dX

(i)

dx*

is called the jet geometric Yang-Mills energy produced by the ODEs system (10.5). Remark 182. If we use the matrix notations (i) dX, (1) \

dxi

• iV(j) = (iV,j!) • R(i)i = ( ^ m i ■ 1

- the Jacobian matrix, '

¿>j=l*ro

- the nonlinear connection matrix, , - the temporal torsion matrix,

132

OBJECTS PRODUCED BY AN ODEs SYSTEM AND A PAIR OF RIEMANNIAN METRICS

• -R(i)fe — (-^f ii,-fc ) • F(i) := F^

, V fc = 1, n, - the spatial torsion matrices, - the electromagnetic matrix,

= [FfJA

then the following matrix geometrical relations attached to the jet first-order ODEs system (10.5) hold true: 1-NW

-\[J(XW)-TJ(X{1))];

=

2 R

· (m = - | [%)];

4. F(D = -N{1); 5. SyMlDEs(t,x)

= 1 · Trace [ F ^ · TF^]

,

that is the jet geometric Yang-Mills energy [produced by the ODEs system (10.5)] coincides with the square of the norm of the skew-symmetric electromagnetic matrix F ' 1 ' in the Lie algebra o(n) = L(0(n)). In many problems from Applied Sciences we meet d-tensors X^l which are not dependent on the time coordinate* e K. In other words, in many applicative problems we are working with ODEs systems of order one given by d-tensors X on J 1 (R, K") whose components have the particular form Λ

(ΐ) -

Λ

(ΐ)(χ

I-

In these time-independent situations, it is obvious that many geometrical objects fc studied by us cancel. In fact, for time-independent d-tensors XL((i) j! =- AXd( j!(x ), we ) ( have the ODEs system of first-order (jet dynamical system) ^

= Xl\\(xk(t)),

Vt = M .

(10.14)

Using Theorem 180, we can assert that the Riemann-Lagrange geometrical objects produced by the ODEs system (10.14) and the pair of Euclidian metrics Δ are given in the following result: Corollary 183. (i) The canonical nonlinear connection Γ - ( M(l)

N{i) λ

GEOMETRICAL OBJECTS PRODUCED ON 1 -JET SPACES BY FIRST-ORDER ODEs SYSTEMS

133

produced by the ODEs system (10.14), is given by the components M¡'=0

1 i («) V)i - = - -

and

dxi

dx1

(ii) All adapted components ofthe canonical generalized Cartan connection CT, produced by the ODEs system (10.14), are zero. (iii) The torsion d-tensor T of the canonical generalized Cartan connection CT, produced by the ODEs system (10.14), is determined only by the adapted components nz?(i)

ft2 Y ( i )

_

WJk

~

^ * $ dxidxk

dxWxk

(iv) All adapted components of the curvature d-tensor R, produced by the ODEs system (10.14), are zero. (v) The electromagnetic 2-form F, produced by the ODEs system (10.14), is expressed by ¥■■

F¡gSy\Adx^

where 5y\ = dy\

dX

U)

(i)

dx'

dxi

3

dx

and

dX ((0 i)

(i)

F,

Ü) dX, (i)

dxi

dxi

r(D are governed by the following simpler geometrical (vi) The components F,V. Maxwell equations V F(1) =0 {i,j,k}

(vii) The jet geometric Yang-Mills energy, produced by the jet first-order ODEs system (10.14), has the expression n-l

n

n

n—1

n

^ ^ ) = Σ Σ Κ] = Ϊ Σ Σ 1

¿=1 i = ¿ + l

¿=1

J'=¿+1

dX^

L

(i)

ΘΧ^ k (i)

9x*

Remark 184. Using the matrix notations from Remark 182, the following matrix geometrical objects characterize the jet first-order ODEs system (10.14): LNW

-\[J(X(1))-TJ(X(1))];

=

2. A (1)fc = ¿ [ i \ r ( 1 ) ] , V * = M ; 3. FW = -N(1); 4. SyM^(x)

= ^ -Trace [F™ ■ TF^]

.

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CHAPTER 11

JET SINGLE-TIME LAGRANGE GEOMETRY APPLIED TO THE LORENZ ATMOSPHERIC ODEs SYSTEM

In this Chapter, using the general formulas from Corollary 183 and Remark 184, we apply the Riemann-Lagrange geometrical results from Chapter 10 to the Lorenz five-components atmospheric ODEs system introduced by Lorenz [49] and studied, via the Melnikov function method for Hamiltonian systems on Lie groups, by Birtea, Puta, Rajiu, and Tudoran [21].

11.1 JET RIEMANN-LAGRANGE GEOMETRY PRODUCED BY THE LORENZ SIMPLIFIED MODEL OF ROSSBY GRAVITY WAVE INTERACTION The first model equations for the atmosphere are that so called primitive equations. It seems that this model produces wave-like motions on different time scales: • on the one hand, this model produces the slow motions which have a period of order of days (these slow-waves are called Rossby waves); • on the other hand, this model produces fast motions which have a period of hours (these fast-waves are called gravity waves). Jet Single-Time Lagrange Geometry and Its Applications 1st Edition. By Vladimir Balan and Mircea Neagu. © 2011 John Wiley & Sons, Inc. Published 2011 John Wiley & Sons, Inc.

135

136

JET LAGRANGE GEOMETRY APPLIED TO THE LORENZ ATMOSPHERIC ODEs SYSTEM

The question of how to balance these two time scales led Lorenz [49] to consider a simplified version of the primitive equations model, which is given by the following nonlinear system of five differential equations [21]: dx1 ~dt dx2 dt dx3 ~dT dx4 dt dx5 dt

— -x2x3

+ εχ2χ5,

= xlx3 — εχ1χ5, =

-xlx2,

(11.1)

= x4 + εχ1χ2,

where the variables x4, x5 represent the fast gravity wave oscillations and the variables x1, x2, x3 are the slow Rossby wave oscillations, with a parameter ε which is related to the physical Rossby number. Remark 185. It is obvious that, from a physical point of view, the Lorenz atmospheric ODEs system (11.1) couples the Rossby waves with the gravity waves. Naturally, the Lorenz atmospheric ODEs system (11.1) can be regarded as a nonlinear ODEs system of order one on the 1-jet space J X (R, R 5 ), which is produced by the d-tensor field X = ( x W (x) j , where ¿ = 1,5 and x having the local components

X(^{x)

-x2x3

X^ix)

xxx3

* $; (( *- >> -

(xl,x2,x3,x4,xh)

+εχ2χ5, εχχχ5,

-x1x2,

(Π.2)

4>) ,(5)

X)¡(x)

=

x4+£x1x2.

Consequently, via the Corollary 183, we assert that the Riemann-Lagrange geometrical behavior on the 1-jet space J 1 (R, R 5 ) of the Lorenz atmospheric ODEs system (11.1) is described in the following result (see also Remark 184): Theorem 186. (i) The canonical nonlinear connection on J^RjR 5 ), produced by the Lorenz atmospheric ODEs system (11.1), has the local components

Ms,).

LORENZ ODEs SYSTEM AND ITS JET GEOMETRICAL OBJECTS Λ

137

(i)

where Ν)Λ- are the entries of the matrix 0

/

Ñ

0

x3 - ε χ 5

1

0

0

0

εχ1

- x 3 + εχ5

0

-x

0

X1

0

0

0

0

0

0

0

1

0

-1

<" - O W ^ V

0

—εχ

1

\

o )

(ii) All adapted components of the canonical generalized Cartan connection CT, produced by the Lorenz atmospheric ODEs system (11.1), are zero. (iii) A11 adapted components of the torsion d-tensor T ofthe canonical generalized Cartan connection CT, produced by the Lorenz atmospheric ODEs system (11.1), are zero, except p(3) __p(2) _ , -"■(1)21 ~ -"-(1)31 ~ x '

p(5) _ _ ό ( 2 ) _ -""(1)21 ~ -"-(1)51 ~~

A(2) __p(l) _ _·, Λ -"-(1)13" (1)23 ~ -1'

Λ

. '

fc

0(2) _ _ 6 ( 1 ) _ „ (1)15 — ■fl(l)25—&-

(iv) All adapted components of the curvature d-tensor R of the canonical generalized Cartan connection CT, produced by the Lorenz atmospheric ODEs system (11.1), cancel. (v) The geometric electromagnetic distinguished 2-form, produced by the Lorenz atmospheric ODEs system (11.1), has the expression ¥ =

F$jSy{Adx\

where k 6y\=dy\+N^)kdxy(0 ,

Vi = l , 5 ,

and the adapted components F¡J. are the entries of the matrix

'"'-(Ο^-Λ

(1)·

(vi) The jet geometric Yang-Mills energy, produced by the Lorenz atmospheric ODEs system (11.1), is given by the formula £yMLorem{x)

=

^ 5 _ ^ 2

+

^

2

+

^

2

+

^

Proof: The Lorenz atmospheric ODEs system (11.1) is a particular case of the jet first-order ODEs system (10.14) for n = 5 and X = [XJ'2(x)) given by V

*'

/i=l,5

relations (11.2). In conclusion, applying Corollary 183 (together with Remark 184)

138

JET LAGRANGE GEOMETRY APPLIED TO THE LORENZ ATMOSPHERIC ODEs SYSTEM

and using the Jacobian matrix / J (*(!)) =

—X

V

-x 3 + εχ 5

-x

o

εχ2

EX

0

xl

o

—εχ1

2

-x1

0

o

0

0

o

0

1

0

εχ

εχ"

1

\

we obtain what we were looking for. Remark 187. Let us remark that, although the jet geometric Yang-Mills energy EyMLorem, produced by the Lorenz atmospheric ODEs system (11.1), depends only by the coordinates x1, x 3 , and x5, it still couples the slow Rossby wave oscillations with the fast gravity wave oscillations. However, the coordinates x2 and x4 are missing in the expression of SyM. orenz. An open question is whether there exists a physical interpretation of this fact.

11.2

YANG-MILLS ENERGETIC HYPERSURFACES OF CONSTANT LEVEL PRODUCED BY THE LORENZ ATMOSPHERIC ODEs SYSTEM

In the preceding Riemann-Lagrange geometrical theory on the 1-jet space J 1 (R, M5) the Lorenz atmospheric ODEs system (11.1) "produces" a jet Yang-Mills energy given by the formula £yMUmm(x)

= (1 + ε 2 ) (x1)2 + (x 3 ) 2 + ε 2 (χ 5 ) 2 - 2εχ 3 χ 5 + 1,

5 where x = (x 1 ) . In what follows, we study the jet geometric YangMills energetic hypersurfaces of constant level, produced by the Lorenz atmospheric ODEs system (11.1), which are defined by the implicit equations

^Lorenz . (£

5

„ . 32 ^ . /-.

. o.\ / 1\2 η + (ι+ εη(χΐγ = ο-ι,

where C is a constant real number. Because Sp orenz is a quadric in the system of axes Ox 1 x 3 x 5 for every C e R , then, using the reduction to the canonical form of a quadric, we find the following geometrical results: 1. If C < 1, then we have E ^ f = 0. 2. If C — 1, then we have V Lorenz L 'C=Í

x1 = 0, ■ ex

JET LAGRANGE GEOMETRY APPLIED TO EVOLUTION ODES SYSTEMS FROM ECONOMY

that is, Σ ^ " ζ is a straight line in the system of axes

139

Ox1x3x5.

3. If C > 1, then we have Σ ^ Τ : (1 + ε 2 ) (χ1)2 + (x3)2 + ε2(χ5)2

- 2εχ3χ5 - C + 1 = 0,

that is, Σ^™21 is adegenerate non-empty quadric in the system of axes whose canonical form is

Σ ^ Γ : ( * 3 ) 2 + (* 5 ) 2 =

^ ·

Note that the rotation of the system of axes Ox1x3xb OX1X3X5 is given by the matrix relation

x1 \ 3

x

χ5

=

J

.

l

/ 0 VTTs2

JTTe^ [

e

0

1

0

Ox1xsx5,

into the system of axes

0

In conclusion, the degenerate non-empty quadric Σ^?!^* is in the system of axes Oxlx3x5 a slant circular cylinder of radius R

! C

-1 ί + ε2'

having as an axis of symmetry the straight line Σ^?^ 2 Open problem. Find physical interpretations for our preceding geometrical results regarding the study of the Lorenz atmospheric ODEs system (11.1).

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CHAPTER 12

JET SINGLE-TIME LAGRANGE GEOMETRY APPLIED TO EVOLUTION ODEs SYSTEMS FROM ECONOMY

Using again the general results from Chapter 10, in the next Sections, we apply our jet Riemann-Lagrange geometrical results to certain evolution equations that govern certain phenomena in Economy, extending in this way the geometrical studies initiated by Udriste, Ferrara, and Opri§ in the book [95]. In fact, we will study the jet RiemannLagrange geometry produced by the following two economic mathematical models: • the Kaldor nonlinear model of the business cycle (cf. Gandolfo [35]); • the Tobin-Benhabib-Miyao economic model regarding the role of currency on economic growth (cf. Tobin [93] and Benhabib-Miyao [22]).

12.1

JET RIEMANN-LAGRANGE GEOMETRY FOR KALDOR NONLINEAR CYCLICAL MODEL IN BUSINESS

The national revenue Y(t) and the capital stock K{t), where t € [a, b], are the state variables of the Kaldor nonlinear model of the business cycle. The kinetic Kaldor model of a commercial cycle belongs to the category of business cycles described by Jet Single-Time Lagrange Geometry and Its Applications I-st Edition. By Vladimir Balan and Mircea Neagu. © 20! I John Wiley & Sons, Inc. Published 2011 John Wiley & Sons, Inc.

141

142

JET LAGRANGE GEOMETRY APPLIED TO EVOLUTION ODEs SYSTEMS FROM ECONOMY

the Kaldor flow (for more details, see [35] and [95]) dY

s[I(Y,K)-S(Y,K)\, (12.1)

I(Y,K)-qK, where • / = I(Y, K) is a given differentiable investment function, which verifies some economic-mathematical Kaldor conditions; • S = S(Y, K) is a given differentiable saving function, which verifies some economic-mathematical Kaldor conditions; • s > 0 is an adjustement constant parameter which measures the reaction of the model with respect to the difference between the investment function and the saving function; • q € (0,1) is a constant representing the depreciation coefficient of capital. Remark 188. Details of the Kaldor economic-mathematical conditions imposed on the given functions I and S are found in [35] and [95]. We underline that, from the point of view of our jet Riemann-Lagrange geometry produced by the Kaldor evolution equations (geometry that we will describe in the sequel) the economicmathematical hypotheses of Kaldor can be neglected because all our necessary geometrical informations are concentrated in the Kaldor flow (12.1). The Riemann-Lagrange geometrical behavior on the 1-jet space J x ([a,6],R 2 ) of the Kaldor economic evolution model is described in the following result (cf. Corollary 183 and Remark 184): Theorem 189. (i) The canonical nonlinear connection on J 1 ([a, 6], K2), produced by the Kaldor flow (12.1), has the local components

where, if Ιγ, Ιχ, and SK are the partial derivatives of the functions I and S, then (i) es of the matrix ΝΪΑ. are the entries of the matrix

(

o

Nm

V -\[IY

- *VK - SK)]

\{W - s(iK - sK)\ \

2

0

)

(ii) All adapted components of the canonical generalized Cartan connection CT, produced by the Kaldor flow (12.1), are zero.

JET RIEMANN-LAGRANGE GEOMETRY FOR KALDOR NONLINEAR CYCLICAL MODEL IN BUSINESS

143

(in) All adapted components of the torsion d-tensor T of the canonical generalized Cartan connection CT, produced by the Kaldor flow (12.1), are zero, except ¿(ί)2ΐ = ~¿(i)ii = \\IYY

- *VYK -

-^(1)22 = --^(1)12 = 2^ΙΥΚ

~~ S(lKK

SYK)},

~

S K K

^

where Ιγγ, Ιγκ, Ικκ, SYK, and SKK are the second partial derivatives of the given functions I and S. (iv) All adapted components of the curvature d-tensor R of the canonical generalized Cartan connection CT, produced by the Kaldor flow (12.1), cancel. (v) The geometric electromagnetic distinguished 2-form, produced by the Kaldor flow (12.1), has the expression

where

5y\=dy\+Ñi§)kdxk,Vi=T71,

and the adapted components F¡.J. are the entries of the matrix UM

F^

'

0

K

hiY - s(iK - sK)}

= -Ña (i)

_i[jy_e(/K_5jf)] >

o

,

(vi) The economic geometric Yang-Mills energy, produced by the Kaldor flow (12.1), is given by the formula Ka]dor

£yM

(Y,K)

= \[IY-

s(IK -

SK)f.

Proof: Let us regard the Kaldor flow (12.1) as a particular case of the jet first-order ODEs system (10.14) on the 1 -jet space ^([a, b],R2), taking n = 2,

x1 = Y,

x2 = K,

and putting

X^(x\x2)

= s [l(x\x2)

- S(x\x2)}

,

xfife1,!2)

= I(x\x2)

Now, taking into account that we have the Jacobian matrix jIγ \ J{X{1))

( S [Ixl - Sxl]

=

V

S [IX2 -

^

I*-q

S\IY-SY]

IY

Sx2]

S[IK-SK} IK

-q

- qx2.

144

JET LAGRANGE GEOMETRY APPLIED TO EVOLUTION ODEs SYSTEMS FROM ECONOMY

and using Corollary 183 and Remark 184, we obtain what we were looking for. Open problem. The geometric Yang-Mills economic energetic curves of constant level produced by the Kaldorflow (12.1), which are different by the empty set, are the curves in the plane Y OK having the implicit equations T C : [IY - 8(IK - SK)}2 = 4C, where C > 0. Determine the relevance of the shapes of the plane curves Tc for real economic processes. 12.2 JET RIEMANN-LAGRANGE GEOMETRY FOR TOBIN-BENHABIB-MIYAO ECONOMIC EVOLUTION MODEL The Tobin mathematical model [93] regarding the role of money on economic growth was extended by Benhabib and Miyao [22] by incorporating the role of some expectation constant parameters. Thus, the Tobin-Benhabib-Miyao (TBM) economic model relies on the variables k(t) = the capital labor ratio, m(t) = the money stock per head, q(t) = the expected rate of inflation, whose evolution in time is given by the TBM flow [95] dk — = sf(k(t)) ^

- (1 - 8)[θ - q(t)]m(t) - nk(t)

= m(t) {θ-η-

q(t) - e[m(t) - l(k(t),q(t))}}

< 12 · 2 )

^=μεΗί)-ί(Λ(ί),«(«))]. where the f(k) and l(k, q) are some given differentiable real functions and s, Θ, n, μ, ε are expectation parameters: s = saving ratio, Θ = rate of money expansion, n = population growth rate, μ = speed of adjustement of expectations, ε = speed of adjustement of price level. Remark 190. From the point of view of economists, the actual rate of inflation in the TBM economic model is given by the formula [95]

p(t)=e[m(t)-l(k(t),q(t))}+q(t). In what follows, we apply our jet Riemann-Lagrange geometrical results to the TBM flow (12.2). In this context, we obtain (cf. Corollary 183 and Remark 184) the following: Theorem 191. (i) The canonical nonlinear connection on J l ([a, 6], R 3 ), produced by the TBM flow (12.2), has the local components (

»

■

<

)

·

JET RIEMANN-LAGRANGE GEOMETRY FOR TOBIN-BENHABIB-MIYAO ECONOMIC EVOLUTION MODEL

145

where, if Ik and lq are the partial derivatives of the function I, then ΝΪΛ, are the entries of the matrix 0

-(!-')('-«)—ernlk

' , ' +emlk \ — (1 — s)m — με/fc

0

( N,(D

^

{ 1

.

μεΐΗ\

8 ) η +

—m + emL — ue 0

m — emlq+με

(ii) All adapted components ofthe canonical generalized Cartan connection CT, produced by the TBM flow (12.2), are zero. (iii) The effective adapted components of the torsion d-tensor T of the canonical generalized Cartan connection CT, produced by the TBM flow (12.2), are the entries of the matrices £(i)i = - ^ (

0 kk -ßelkk

£ml

smlkk 0 -emlkq

ßslkk emlkq

R
£(i)3 = ~ ñ

0 - 1 + s + emlkq \ -ßehq i

1 - s - emlkq 0 -emlqq

ßelkq emlqq 0

where Ikk, hq, and lqq are the second partial derivatives of the function I and

(iv) All adapted components of the curvature d-tensor R of the canonical generalized Cartan connection CT, produced by the TBM flow (12.2), cancel. (v) The geometric electromagnetic distinguished 2-form, produced by the TBM flow (12.2), has the expression W=

Ftl)6y\Adx3,

where 5y[=dy[+Ñ$kdxk,Vi=tt and the adapted components F^J. are the entries of the matrix F™ = -JV (1) .

/

146

JET LAGRANGE GEOMETRY APPLIED TO EVOLUTION ODEs SYSTEMS FROM ECONOMY

(vi) The economic geometric Yang-Mills energy, produced by the TBM flow (12.2), is given by the formula 0>.MTBM(fc,m,
=

^{[(l-s)(e-q)+emlkf

+ ßelk}2

+ [(l-s)m

+ [m — emlq + με] >. Proof: We regard the TBM flow (12.2) as a particular case of the jet first-order ODEs system (10.14) on the 1-jet space ^({a, b], R 3 ), taking n = 3, x1 = k, x2 = m, x3 = q and putting X f í j í x 1 , ! 2 , ! 3 ) = sf{xl) X$(x\x2,x3) and

- (l _ S)[0 - x3}x2 -

nx\

= x2 {Θ - n - x3 - ε[χ2 - l(x\x3)}}

X({V{x\x2,x3)

= με{χ2 -

,

l(x\x3)

It follows that we have the Jacobian matrix -(1-8)(θ-χ3) -2εχ2 + Θ - χΛ - n+ +εΙ(χχ,χ3)

■7 ( * d ) )

με

I sf'(k) - n emlk \

-με/fc

-(1-sW-q) -2ετη + θ — q — n+ +sl{k,q) με

(1 - s)x2 JU

1

\

*Z-*JL· ^''7*'-*

-μεΙχ3 (l-s)m

\

-m + emlq -με19

J

where / ' is the derivative of the function / . In conclusion, using Corollary 183 and Remark 184, we find the required result. Open problem. The Yang-Mills economic energetic surfaces of constant level produced by the TBMflow (12.2), which are different by the empty set, have in the system of axis Okmq the implicit equations Σ σ : [(1 -s)(e-q)+

emlkf

+ [(1 - s)m + με^]2 + [m - emlq + με}2 = AC,

where C > 0. Find economic interpretations for the geometry· of the E c surfaces.

CHAPTER 13

SOME EVOLUTION EQUATIONS FROM THEORETICAL BIOLOGY AND THEIR SINGLE-TIME LAGRANGE GEOMETRIZATION ON 1-JET SPACES

Using the applicative results from Chapter 10 (particularly the results from Corollary 183 and Remark 184), the aim of this Chapter is to construct the Riemann-Lagrange differential geometry on 1-jet spaces (in the sense of nonlinear connections, generalized Cartan linear connections, d-torsions, d-curvatures, jet electromagnetic fields and jet electromagnetic Yang-Mills energies), starting from some given nonlinear evolution ODEs systems modeling biologic phenomena such as: • the model of the cancer cell population (see [38] and [91]); • the model of the infection by human immunodeficiency virus-type 1 (HIV-1) model (see [83]); • the model of cytosolic calcium oscillations in living cells (see [42]); • the model of calcium oscillations in living cells, through endoplasmic reticulum, mitochondria and cytosolic proteins (see [50]). For more theoretical-biologic details, see the works of Nicola [74], [75], and [68]. Jet Single-Time Lagrange Geometry and Its Applications 1st Edition. By Vladimir Balan and Mircea Neagu. © 2011 John Wiley & Sons, Inc. Published 2011 John Wiley & Sons, Inc.

147

148

SOME EVOLUTION EQUATIONS FROM THEORETICAL BIOLOGY

13.1

JET RIEMANN-LAGRANGE GEOMETRY FOR A CANCER CELL POPULATION MODEL IN BIOLOGY

The mathematical model of cancer cell population, which consists of a two-dimensional ODEs system having four parameters, was introduced in 2006 by Garner et al. (see [38]). The cancer cell populations consist of a combination of proliferating, quiescent, and dead cells that determine tumor growth or cancer spread. Moreover, recent research in cancer progression and treatment indicates that many forms of cancer arise from one abnormal cell or a small subpopulation of abnormal cells. These cells, which support cancer growth and spread, are called cancer stem cells (CSCs). Targeting these CSCs is crucial because they display many of the same characteristics as healthy stem cells, and they have the capacity of initiating new tumors after long periods of remission. The understanding of the cancer mechanism could have a significant impact on cancer treatment approaches as it emphasizes the importance of targeting diverse cell subpopulations at a specific stage of development. The non-dimensionalized model introduced by Garner et al. is based on a system of Solyanik et al. (see [91]), which starts from the following assumptions: 1. the cancer cell population consists of proliferating and quiescent (resting) cells; 2. the cells can lose their ability to divide under certain conditions and then transit from the proliferating to the resting state; 3. resting cells can either return to the proliferating state or die. The dynamical system has two state variables, namely, P is the number of proliferating cells and Q is the number of quiescent cells, and their evolution in time is described by the following differential equations {cancer cell population flow): (

dP

<

at

^ I at

1

M)

= P-P(P =

_rQ

1+

+

+ 0 p(p

fcP2'

Q)+F(P,Q), (13.1)

+ Q)_F(P,Q),

6'

ac'

c2 '

where • a is a dimensionless constant that measures the relative nutrient uptake by resting and proliferating cells; • b is the rate of cell division of the proliferating cells; • c depends on the intensity of consumption by proliferating cells and gives the magnitude of the rate of cell transition from the proliferating stage to the resting stage in per cell per day;

JET RIEMANN-LAGRANGE GEOMETRY FOR A CANCER CELL POPULATION MODEL IN BIOLOGY

149

• dis the rate of cell death of the resting cells (per day); • A represents the initial rate of increase in the intensity of cell transition from the quiescent to proliferating state at small P; • A/B represents the rate of decrease in the intensity of cell transition from the quiescent to proliferating state when P becomes larger. The Riemann-Lagrange geometrical behavior on the 1 -jet space J 1 (R, R 2 ) of the cancer cell population flow is described in the following result (cf. Corollary 183 and Remark 184): Theorem 192. (i) The canonical nonlinear connection on JX(R, R 2 ), produced by the cancer cell population flow (13.1), has the local components

f = (o,iv ( ( ;j.),¿,j = T72, where, if

hQ(l~kP2) hP jri 2 2 2^ and FvQ = FP = —^ (1 + kP ) 1 + kP2

are the first partial derivatives of the function F, then we have N{1) iV

(l)l

- N{2) -

JV

-0

(1)2 -

U

>

^(í)2 = - ^ $ 1 = \ K2a + l)P + aQ-(FP

+ FQ)}

1 /„ ^ n r, hQ(l-kP2) hP —¿- - ■—= ~ 2 (2a + l)P + aQ- —^ (1 + fcP2)2 1 + fcP2 (ii) All adapted components ofthe canonical generalized Cartan connection CF, produced by the cancer cell population flow (13.1), are zero. (iii) All adapted components ofthe torsion d-tensor T ofthe canonical generalized Cartan connection CT, produced by the cancer cell population flow (13.1), are zero, except 6(1) _ (1)21 —

Λ

Λ

(1)22 ~

0(2) ""(1)11

a+-(l~FPp-

-"(1)12

- (a - FPQ - FQQ) = - (a - FPQ),

FPQ) ,

where Fpp =

2hkPQ (3 - kP2)

(1 + kP2f

Fp

Q

=

h(l-

kP2)

/■■ , ,

ο 2 '

(1 + kPπ 2)λ2

are the second partial derivatives of the function F.

'

a n d

F

QQ

=

°

150

SOME EVOLUTION EQUATIONS FROM THEORETICAL BIOLOGY

(iv) All adapted components of the curvature d-tensor R of the canonical generalized Cartan connection CT, produced by the cancer cell population flow (13.1), cancel. (v) The geometric electromagnetic distinguished 2-form, produced by the cancer cell population flow (13.1), has the expression :Ρ™δυ\Γ\άΧ*, where

Sy\=dyÍ+Ñ¡?)kdxk,Vi

= l,2,

and the adapted components Pyj., i,j = 1,2, are given by pW r

_p(D r

(l)l ~

F

$i

_

0

( 2 ) 2 ~ U>

= - ^ m = \ K 2 ° + DP + aQ-(Fp

+ FQ)}

hQ (1 - kP2)

hP

(2a + 1) P + aQ -

2 2

1 + kP2

(1 + kP )

(vi) The biologic geometric Yang-Mills energy, produced by the cancer cell population flow (13.1), is given by the formula 2

hQ(l-kP2) hP £yM (P,Q) (2a + l)P + aQ- - ^ J- —-, (1 + kP2)2 1 + kP2 Proof: We regard the cancer cell population flow (13.1) as a particular case of the first-order ODEs system (10.14) on the 1-jet space J1(R, R 2 ), with i

cancer

n

= 2, x1 = P, x2 = Q,

and Xfgix1,!2)

-

x1-x1(x1+x2)

+

X((2](x\x2)

=

-rx2 +

axl(xl+x2)-F{x\x2).

F(x\x2),

Now, using Corollary 183, together with Remark 184, and taking into account that we have the Jacobian matrix 1 - 2 P - Q + FP 2aP + aQ-FP

' (*(!>) ( 1

-P + FQ -r + aP - FQ

kP2) + (1 + kP2)2 ' hQ(l-

1-2P-Q 2aP + aQ

VV

kP2)

hQ (1 -

2 2

{1+kP )

hP

-P- +1 + kP2 -r +

aP-

hP

1 + kP2 J

THE JET RIEMANN-LAGRANGE GEOMETRY OF THE INFECTION BY HIV-1 EVOLUTION MODEL

151

we obtain what we were looking for. Open problem. The Yang-Mills biologic energetic curves of constant level produced by the cancer cell population flow (13.1), which are different from the empty set, are, in the plane POQ, the curves of implicit equations

rc

(2a + 1) P + aQ

hQ (1 - kP2) 2 2

(1 + kP )

hP_ kP2

4C,

where C > 0. For instance, the zero Yang-Mills biologic energetic curve produced by the cancer cell population flow (13.1) is, in the plane POQ, the graph of a rational function: P ( l + fcP2)[/t-(2a + l ) ( l + fcP2)] r

°'

a{l +

kP2f-h(l-kP2)

As a possible opinion, we consider that if the cancer cell population flow does not generate any real Yang-Mills biologic energies, then it is to be expected that the variables P and Q vary along the rational curve FQ. Otherwise, if the cancer cell population flow generates a real Yang-Mills biologic energy, then it is possible that the shapes of the constant Yang-Mills biologic energetic curves Γ<7 to offer useful interpretations for biologists. 13.2 THE JET RIEMANN-LAGRANGE GEOMETRY OF THE INFECTION BY HUMAN IMMUNODEFICIENCY VIRUS (HIV-1) EVOLUTION MODEL The major target of HIV infection is a class of lymphocytes, or white blood cells, known as CD4+ T cells. These cells secrete growth and differentiations factors that are required by other cell populations in the immune system, and hence these cells are also called "helper T cells." After becoming infected, the CD4+ T cells can produce new HIV virus particles (or virions) so, in order to model HIV infection, it was introduced a population of uninfected target cells T, and productively infected cells T*. Over the past decade, a number of models have been developed to describe the immune system, its interaction with HIV, and the decline in CD4+ T cells. We propose for our geometrical investigation a model that incorporates viral production (for more biologic details, see Perelson and Nelson [83]). This mathematical model of infection by HIV-1 relies on the variables T(t) - the population ofuninfected target cells, T* (t) - the population ofproductively infected cells, and V(t) - the HIV-1 virus, whose evolution in time is given by the HIV-1 flow [83] ^=s+(p-d)T-^-kVT, at dT* = kTV-5T*, dt dV_ ηδΤ* - cV, ~dt

m, (13.2)

152

SOME EVOLUTION EQUATIONS FROM THEORETICAL BIOLOGY

where • s represents the rate at which new T cells are created from sources within the body, such as thy mus; • p is the maximum proliferation rate of T cells; • d is the death rate per T cells; • δ represents the death rate for infected cells T*; • m is the T cells population density at which proliferation shuts off; • k is the infection rate; • n represents the total number of virions produced by a cell during its lifetime; • c is the rate of clearance of virions. In what follows, we apply our jet Riemann-Lagrange geometrical results to the HIV-1 flow (13.2) regarded on the l-jet space J X (R, R 3 ). In this context, we obtain (cf. Corollary 183 and Remark 184): Theorem 193. (i) The canonical nonlinear connection on J 1 (R, R 3 ), produced by the HIV-1 flow (13.2), has the local components f =(0,iV ( ( ^),z,j=T73, where Ñ^l are the entries of the matrix χ

JV(1) = - -[ 2

0 kV \ kT

I

-kV 0 -kT + ηδ

-kT \ kT-ηδ . O /

(ii) All adapted components of the canonical generalized Cartan connection CF, produced by the HIV-1 flow (13.2), are zero. (iii) A11 adapted components of the torsion d-tensor T of the canonical generalized Cartan connection CT, produced by the HIV-1 flow (13.2), are zero, except the entries of the matrices /

£ ( i)i =

0 ° \ -k/2

and Ä(i)3 =

/

0 -k/2 \ 0

0 ° k/2 k/2 0 0

k/2 \ - fc /2 0 / 0 \ 0 , 0/

THE JET RIEMANN-LAGRANGE GEOMETRY OF THE INFECTION BY HIV-1 EVOLUTION MODEL

153

where (*!$)ik)iJ=TZ>*k£l1>V·

*W* =

(iv) All adapted components of the curvature d-tensor R of the canonical generalized Carian connection CT, produced by the HIV-1 flow (13.2), cancel. (v) The geometric electromagnetic distinguished 2-form, produced by the HIV1 flow (13.2), has the expression F =

F{¿)jSy\AdxJ,

where Sy\=dyÍ

Ñ^kdxk,yi=T^,

+

and the adapted components F^J. i,j = 1,3, are the entries of the matrix 0 kV kT

-kV 0 -kT + ηδ

-kT kT -ηδ 0

(vi) The biologic geometric Yang-Mills energy, produced by the HIV-1 flow (13.2), is given by the formula SyMmvA

(T, T*, V) = i [k2(V2 + T2) + (kT - ηδ)2] .

Proof: Consider the HIV-1 flow (13.2) as a particular case of the first-order ODEs system (10.14) on the 1-jet space Jl(R, R 3 ), with n = 3, x1 = T, x2 = T*, x3 = V and

x3) = s + (p- d)xl - —(x 1 ) 2 -

Xfihx1,!2, Xlf}(x\x2,x3)

= kx1x3 - δχ2,

Xll))(x\x2,x3)

=

kx3x\

n6x2-cx3.

It follows that we have the Jacobian matrix 0

-kT

kV

-δ

kT

0

ηδ

-c

Í p-d-^T-kV m J x

( w)

= \

\

)

In conclusion, using Corollary 183 and Remark 184, we find the required result. Open problem. The Yang-Mills biologic energetic surfaces of constant level, produced by the HIV-1 flow (13.2), have in the system of axis OTT*V the implicit equations E c : k2(V2 + T2) + (kT - ηδ)2 = AC,

154

SOME EVOLUTION EQUATIONS FROM THEORETICAL BIOLOGY

where C > 0. It is obvious that the surfaces Σ
-knS 0 η2δ2 - 4C

it follows that its invariants are Ac = k4 (η2δ2 - 8 C ) , δ = 2/c4 > 0, and I = 3fc2 > 0. As a consequence, we have the following situations: η2δ2 1. If 0 < C < ———, then we have the empty set Σ 2 2

2. If C =

ηδ

8

„2*2 = 0.

vs<-'< s

8

, then the surface E„_ I¡ ¿ ¿ 2 degenerates into the straight line s Σ^_„2 6 2 : < C-

—

T= — 2k

V = 0.

.Jo2

3. If C > ——, then the surface Σ ^ „a^ is a right elliptic cylinder of equation

where a < b are given by V8C - n2<52 , 1/8C - n2<52 a = ^ 1 o= ;= · 2k kV2 Obviously, it has as an axis of symmetry the straight line Σ,,_ „2S2. C — —g—

To what extent might the shapes of the above Yang-Mills energetic constant surfaces Σ ο provide valuable information for biologists? 13.3

FROM CALCIUM OSCILLATIONS ODEs SYSTEMS TO JET YANG-MILLS ENERGIES

Taking into account that we would like to develop the jet single-time Lagrange geometry of some ODEs systems that govern the calcium oscillations in a large variety of

FROM CALCIUM OSCILLATIONS ODEs SYSTEMS TO JET YANG-MILLS ENERGIES

155

living cells, we would like to expose a few biological properties of these oscillations. So, we recall that the oscillations of cytosolic calcium concentration, known as calcium oscillations, play a vital role in providing the intracellular signaling and many biological processes are controlled by the oscillatory changes of cytosolic calcium concentration. Since the 1980's, when the self-sustained calcium oscillations were found experimentally by Woods, Cuthbertson, and Cobbold [ 100], many experimental works have been published. Various models have been constructed to simulate calcium oscillations in living cells. In this Chapter, we will consider and geometrically study only two of these models. These mathematical models were proposed in the course of investigations of plausible mechanisms capable of generating complex calcium oscillations. The first one was proposed by Borghans, Dupont and Goldbeter [23] and was deeply mathematically analyzed by Houart, Dupont, and Goldbeter [42]. This first mathematical model, describing the cytosolic calcium oscillations, relies on the interplay between CICR* (calcium-induced calcium release) and the Ca2+-stimulated degradation of InsP3 (inositol triphosphate). Alternatively, the second mathematical model of calcium oscillations was introduced by Marhl, Haberichter, Brumen and Heinrich [50] and was also intensively studied from a mathematical point of view. This second model is based on the interplay between three calcium stores in the living cells: endoplasmic reticulum, mitochondria and cytosolic proteins. In the following Subsections, we study the expressions of the abstract biological Yang-Mills energies produced by the calcium oscillations in this large variety of cell types. For reviews of these geometrical-biologic topics, see the works of Nicola [74], [68], and references therein.

13.3.1 Intracellular calcium oscillations induced by self-modulation of the inositol 1,4, 5-triphosphate signal The mathematical model that describes calcium oscillations which is based on the mechanism of calcium-induced calcium release, takes into account the calciumstimulated degradation of inositol triphosphate (InsPs). This is because in some cell types, particularly in hepatocytes, calcium oscillations have been observed in response to stimulation by specific agonists. As these cells are not electrically excitable, it is likely that this calcium oscillations rely on the interplay between two intracellular mechanisms capable of destabilizing the steady state: An increase in InsP3 is expected to lead to an increase in the frequency of calcium spikes, but at the same time the InsP3-induced rise will also lead to increased InsP3 hydrolysis due to the calcium activation of the InsP3 3-kinase. The classical mathematical model for the study of cytosolic calcium oscillations and their associated degradations of InsP3 in endoplasmic reticulum contains three variables Z(t), Y(t) and A(t), where • Z is the concentration of free calcium in the cytosol; • Y is the concentration of free calcium in the internal pool;

156

SOME EVOLUTION EQUATIONS FROM THEORETICAL BIOLOGY

• A is the InsP·! concentration. The time evolution of these variables is governed by the following first-order differential equations of cytosolic calcium oscillations, denoted by us with [Ca2+ — InsPs] (for more biologic details, see Houart, Dupont, and Goldbeter [42]): ~dt ~

M3

Kf

+ Z™' K2+Y2'

+ kfY-kZ d

X.-v

át dA

K

2

M2

K 2 + Z2

+ V0 + ßV1, z<2

M2

K\ + A*~

+Z

2

bY f

Ap

zm

v

M3

Kf

+

γ2

Ai 2

2

Z™'K +Y 'K4A+A^

Zn

where • VQ refers to a constant input of calcium from the extracellular medium; • V\ is the maximum rate of stimulus-induced influx of calcium from the extracellular medium; • β is a constant parameter reflecting the degree of stimulation of the cell by an agonist and thus only varies between 0 and l; • the rates V2 = VM2 κξ+ζ2 and V3 = VMa Κ,Γ+Ζ„, ■ KiY+Y2 · K*\A4 ref er to the pumping of cytosolic calcium into the internal stores and to the release of calcium from these stores into the cytosol in a process activated by cytosolic calcium, respectively. The constants V\¡2 and VM3 represent the maximum values of the preceding rates; • the parameters K2, Κγ, Κζ, and KA are threshold constants for pumping, release, and activation of release by calcium and by InsPß; • kf is a rate constant measuring the passive, linear leak of Y into Z; • k relates on the assumed linear transport of cytosolic calcium into the extracellular medium; • VM4 is the maximum rate of stimulus-induced synthesis of InsPs; • V5 = VMD KI^1A,, · K"'+z» ls t n e r a t e 0 ^ phosphorylation of the InsP 3 by the 3-kinase, which is characterized by a maximum value VM5 and a half-saturation constant K$; • TO, n, and p are the Hill's coefficients related to the cooperative processes; • ε is the rate of phosphorylation of the InsP3 by the 5-phosphatase.

FROM CALCIUM OSCILLATIONS ODEs SYSTEMS TO JET YANG-MILLS ENERGIES

157

From a biological point of view, we recall that the preceding ODEs system is based on the mechanism of Ca2+-induced Ca 2+ release (CICR), that takes into account the Ca 2+ stimulates the degradations of the inositol 1,4,5 triphosphate InsP3 by a 3kinase. From a geometrical point of view, the ODEs system of calcium oscillations and the degradation of the inositol trisphosphate InsP3 may be regarded as an ODEs system on the particular 1-jet space J 1 (K, M3). Let us denote the coordinates of the spatial manifold R3 by x 1 = Z, x2 = Y, and x 3 = A. Consequently, the local coordinates τα>3\ are on the 1-jet space J ^ KD>, JK") Y, x 3 = A, y\ = Z,y2

(t, x1 = Z, x2

= Ϋ, y3 = A).

In this geometrical context, the ODEs system [Ca2+ — InsPa], as a particular ODEs system of the form (10.14), is determined by the following distinguished tensorial components Χ^λ(χι,χ2,χ3): A4 K\ + A4

Y2

X¡l](Z,Y,A)

VM,

m

2

Kf + Z

K$ + Y

VM2

K2 + Z2

+V0 + ßVi + k}Y - kZ, X¡2](Z,Y,A)

=

VM2

z2

K2 + Z2

- kfY - VM3

ßvMi-vM6 Κζ

X\;UZ,Y,A)=

Κψ + Zm

Zn

AP + AP

Y2 Kl + Y2

A4 K\ + A4'

εΑ.

K2 + Ζη

Consequently, using the general results from Chapter 10, together with some partial derivatives and computations, we find the following important geometrical result that characterizes the cytosolic calcium oscillations in hepatocytes and the degradation of InsP3 through endoplasmic reticulum (cf. Corollary 183): Theorem 194. The biologic electromagnetic field F produced by the ODEs system [Ca2+-InsP3J and the pair of Euclidian metrics Δ = (1,5^) has the components (1)2

H

kf -

2VM2K2

K2Z K +Y2 2

7 (D

(1)3

2VM3K4lZm Kf + Zm

7m-1

(K + Z ) + vM. Kf + Zm 2

(2)3

2VM3K\ ■

Υ K + Y2 2

A4 K\ + A4

KfY + m Km + Zm Y2 ' K$+Y2' r

Γ(1)

2 2

z

Kf + Zm

A3 (K\ + A4)2 2

Y K\+ Y2

+

A" n 2- Κξ + AP

A3 {K\ + A4)2'

n 7n-l VMtK2Z (K% + Zn)2 '

158

SOME EVOLUTION EQUATIONS FROM THEORETICAL BIOLOGY

Proof: We have the formulas 7 (D

(1)2

1 2

OF

dZ

^(2)3 -

2

'

«ίί!

*α)3 - 2

dA

dA

(D

Sil

dY

So, some direct computations imply the required result. We recall that the formulas from Theorem 170 show that the spatial components G ' L of the semispray ¿>[ca2+-insP3] = (θ,ο^λΛ

, produced by the ODEs system

2+

[Ca -InsP3] and the pair of Euclidian metrics Δ = (1, ί ^ ) , are given by dX

G (i) (1)1

(r) (1) l

dx

X

dx ( i ) y[] + dxn Vi

(r) (1)

Consequently, Theorem 170 implies the following interesting qualitative geometrical result with biological-energetic connotations: Theorem 195. The solutions of class C2 of the ODEs system [Ca2+-InsP3] may be regarded as harmonic curves of the semispray <S[ca2+-insP3] on ^e ^'Jet sPace J ^ R . R 3 ) . In other words, the C2 solutions (Z(t),Y(t),A(t)) of the first-order ODEs system [Ca2+-InsP3] satisfy also the second-order biological ODEs system:

2

dY dt2 d2A dt2

2Ι®.*±-ΣθζΧ$-Χ%=0, W 3 dt

(D2 dt

dt2

k=l

M) dZ ^2dt

+

d

■2F, 4(D3 dt

(1) dA IF, 2 i3

3

£-Σ**ϊ

< > dt 1dY

+ 2*? ' <2)3 dt

(fc) )

x$=o,

(13.3)

fc=l 3

Σ^ί

(fc)

fc=l

(O

x,(1)

Proof: We use the general second-order differential equations of the harmonic curves of a semispray, which are given by (1.12). We take also into account that we have x1 = Z, x2 = Y, and x3 = A, together with the particular semispray *%a 2 +-InsP 3 ]·

Remark 196. The importance of the second-order system (13.3) is that its equations are equivalent with the Euler-Lagrange equations of the least squares Lagrangian on the 1-jet space JX(R, R 3 ), given by CSQlc^-msv3]

=

(z-

χ&>(Z, Y, A))* + (Ϋ - x[¡] (Z, Y, A)) *

+ (i-X ( (3 ) ) (Z,Y,A)) 2 .

FROM CALCIUM OSCILLATIONS ODEs SYSTEMS TO JET YANG-MILLS ENERGIES

159

Therefore, the solutions (Z(t),Y(t),A(t)) of class C 2 of the first-order ODEs sys2+ tem [Ca -InsP3] are global minimum points for the jet least squares biological Lagrangian£5Q [Ca2+ - InsP3 ]. Particularizing the general definition of the geometric jet Yang-Mills energy of a general first-order ODEs system to our present biological phenomena, we deduce that the biological Yang-Mills energy produced by the intracellular calcium oscillations in some non-excitable cell types has the form (cf. Definition 181) >(D ' r

SyM[Ca2+.lnsP3](Z,Y,A)

2

(l)2

>(!) ' (D3

+

2

>(D "

+ Λ2)3.

It is important to note that we have three main sets of parameter values for [Ca2+-InsP3], including bursting, chaos and quasiperiodicity situations, which are listed in the following table (for more biologic details, see Houart, Dupont, and Goldbeter [42]): |

Parameters

|

ß

1 1 1

n

| Bursting | Chaos | Quasiperiodicity | |

0.46

2

m

p

|

0.65

|

4

0.51

|

1 1 4 1 i

1 1 2 1 1 1 i 1

4

|

|

0.1

|

| 0.3194 |

0.3

|

2

1 1 1

|

Κ2{μΜ)

|

0.1

|

Κ5(μΜ)

|

1

|

ΚΑ(μΜ)

|

0.1

|

0.1

|

0.2

|

|

Κά(μΜ)

|

0.6

|

1

|

0.5

|

|

Κγ (μΜ)

1

0.2

1 0.3

|

0.2

|

|

Kz (μΜ)

|

0.3

|

|

0.5

|

|

1

his' )

| 0.1667 | 0.1667 |

0.1667

|

|

kf (s^·1)

| 0.0167 | 0.0167 |

0.0167

|

| | |

1

eCs" )

0.1

2

0.6

| 0.0167 | 0.2167 |

0.0017

|

-1

| 0.0333 | 0.0333 |

0.0333

|

_1

| 0.0333 | 0.0333 |

0.0333

|

VbiMMs ) Vi(/iMs ) -1

| VM2 ((iMs ) |

|

0.1

|

0.1

|

1

| VM3 (^Ms" ) | 0.3333 |

0.5

|

0.3333

|

| V M Í C / Í M S ^ 1 ) 1 0.0417 |

0.05

|

0.0833

|

0.5

1

1 VM6(//MS^) I

0.1

0.5

1 0.8333 |

160

SOME EVOLUTION EQUATIONS FROM THEORETICAL BIOLOGY

These parameter values correspond to the various types of complex oscillatory behavior observed in the model defined by equations [Ca 2+ -InsP3] and obviously produce particular and distinct geometric biological Yang-Mills energies: Theorem 197. The following formulas for the geometric biological energies of YangMills type are true: (i) Biological Yang-Mills energy o/bursting cytosolic calcium oscillations in the model involving Ca + activated InsP3 degradation. Y SyMZ*?, , , = |Í 0.00835 - (o.oi , °;?01 J L + 0.0081 °· 3 3 3 3 +· Z3 [tv+-m.,i>3l ■ + Z2)2 Z 4 0.04 + Y2 2

0.04 Z 0.04 + Y2

0.0162 Y 0.0081 + Z 4

A Í + A)2

0.18 ■ Z Ί2 2 2 (0.36+ Z ) j

f 0.6666 · 10" 4 · Z4 0.0081 + Z 4

Yz 0.04 + Y2

0.44435 · 10~ 8 · Z8 Y4 4 2 (0.0081 + Z ) (0.04 + Y2)2

A<

0.0001 + A4

(0.0001 + A4) 2 A6 (0.0001 + A4)4 '

(ii) Biological Yang-Mills energy of chaos cytosolic calcium oscillations in the model involving C a 2 + activated InsP3 degradation. 0.001 · Z ^■^*? + . Ι „,Γ3] = { 0 · 0 0 8 3 5 · (0.01 + Z2)2 0.09 ■ Z 0.09 + Y2

0.36 · Y 0.36 + Z2

X1 2

0.5 Z 0.36 + Z2

' 0.0001 · Z2 /0.C 36 + Z 2

2A 0.8333 Z 3 ! 2 . Η Γ 8 · Ζ 4 + 0.3194 +. A. · (1 ,. +. Zr»„ 4 2 ) J) + (0.36 + Z2)2

Y 0.09 + Y2

0.09 + Y2

0.0001 + A4

(0.0001 + A4) 2

Y4 (0.09 + Y2)2

A6 (0.0001 + A4)4 '

(iii) Biological Yang-Mills energy of quasipenodicity cytosolic oscillations in the model involving Ca + activated InsP3 degradation.

e y M

~ g £ *

0.04 · Z 0.09 + y 2 2A2 + 0.09 + A 2

= jo.00835 -

0.001 Z 0.3333 Z ^ ¿ ) 2 + (0.01+ Z 2 ) 2 0.25 + Z2

( 0

0.25 · Y Ί 0.25 + Z2 J

+

0.03125 Z 3 Y Γ¿ 4 2 (0.0625 + Z ) J

Y 0.04 + Y2

calcium

A4 0.0016 + A4

f 0.00106656 ■ Z 2 Y2 2 \ 0.25+ Z 0.04+· y 2

A3 (0.0016 + A4)2

0.0000113 o.i Z4 y4 2 2 (0.25 + Z ) (0.04 + Y2)2

A6 (0.0016 + A 4 ) 4

FROM CALCIUM OSCILLATIONS ODEs SYSTEMS TO JET YANG-MILLS ENERGIES

161

13.3.2 Calcium oscillations in a model involving endoplasmic reticulum, mitochondria, and cytosolic proteins The next mathematical model (that we will geometrically study in the sequel) represents a possible mechanism for complex calcium oscillations based on the interplay between three calcium stores in the biological living cells: the endoplasmic reticulum (ER), mitochondria, and cytosolic proteins. The majority of calcium released from the ER is first very quickly sequestred by mitochondria. Afterward, a much slower release of calcium from the mitochondria serves as the calcium supply for the intermediate calcium exchanges between the ER and the cytosolic proteins. We would like to point out that the oscillations of cytosolic calcium concentration play a vital role in providing the intracellular signaling. Moreover, many cellular processes, like cell secretion or egg fertilization, for instance, are controlled by the oscillatory regime of the cytosolic calcium concentration. In this second mathematical model, we have three variables Cacyt(t), C a e # ( i ) , a n d C a m ( i ) , where • Cacyt means the free cytosolic calcium concentration; • CaER means the free calcium concentration in the ER; • CaTO means the free calcium concentration in the mitochondria. The preceding variables of calcium are governed by the following first-order ODEs system of the calcium oscillations through endoplasmic reticulum, mitochondria and the cytosolic proteins, denoted by us as [Ca2+-ER-cyt.pr-m] (for more details, see Marhl, Haberichter, Brumen, and Heinrich [50]): dCa,cyt dt

C&cyt

K1 + Ca c!/t

(CaER — Cacyt) + kieak(CaER — Cacyt)

i I fcout

fcpump^acyt

a Cal cyt K\ + Ca2cyt

+

Ca;cyt + k. Catot — Cacyt — -5—CaER K2 + Cacyt PER dfracpmßmCam) — k+Cacyt( Pr ( 0 t -Ca t o t + Cacyt

+ ^CaER

+

PER

dCaER _ PER dt PER

~

rCc f^ch

Ca;cyt (CaER - Ca,cyt) K1 + Ca c¡/(

CaCyt)

Ca8

ßn Prr

^Caml,

DCa, pump^a Cyt

-kleak(CaER

dCav dt

Pm

■K*.

^Acyt

Ca;cyt

K\

Ca;cyt + km ' Cacyt

Ca n

162

SOME EVOLUTION EQUATIONS FROM THEORETICAL BIOLOGY

where • Pr t o t is the total concentration of cytosolic proteins; • Ca tot represents the total cellular Ca

+

concentration;

• K\ represents the half-saturation for Ca2+. • K2 represents the half-saturation for Ca 2+ of uniporters in the mitochondrial membrane; • Vpump = kpurnpCacyt is the adenosine triphosphate (vlTP)-dependent calcium uptake from the cytosol into the ER; Ca 2 • Vch = kch—2—cy 2 (CaER — Ca cyt ) is the calcium efflux from the ER Kx + CaC3/t through channels following the calcium-induced calcium release mechanism; • Vieak = heak(CaER — Cac?/t) represents the calcium leak flux from the ER into the cytosol; • Vin = kimnK —5—°y a ' s *!+G&t porters; * ■

-

C

a

:

• Vout — I kout —2—^~2 ,2

K1 +Ca cyt

tne act ve

'

calcium uptake by mitochondrial uni-

\~ km j Ca m is a very small non-specific leak flux;

J

• fe_ and k+ denote the off and the on rate constants of the calcium binding; • PER and pm represent the volume ratio between the ER and the cytosol or between the mitochondria and the cytosol, respectively; • PER and ßm are constant factors for relating the concentrations of free calcium in the ER and the mitochondria to the respective total concentrations; • kpump is the rate constant of the ATP; • kch represents the maximal permeability of the calcium channels in the ER membrane; • heak is the rate constant for calcium leak flux through the ER membrane; • kin represents the maximal permeability of the uniporters in the mitochondrial membrane; • kout represents the maximal rate for calcium flux through pores; • km stands for the non-specific leak flux;

FROM CALCIUM OSCILLATIONS ODEs SYSTEMS TO JET YANG-MILLS ENERGIES

163

Remark 198. From a biological point of view, in addition to the endoplasmic reticulum as the main intracellular calcium store used in the first mathematical model, in this second model, the mitochondrial and cytosolic Ca2+-binding proteins are also taken into account. We recall that this model was proposed in [50] especially for the study of the physiological role of mitochondria and the cytosolic proteins in generating complex Ca 2+ oscillations. For more biological details, please see the works [68] and [74] and references therein. From a differential geometric point of view, we underline that the first-order ODEs system of calcium oscillations through endoplasmic reticulum, mitochondria and cytosolic proteins may be regarded as an ODEs system on the particular 1-jet space J 1 (R, R 3 ). Denoting the coordinates of the manifold R 3 by x1 = CaCJ/t7 x2 = CaER and x3 — Ca m , it follows that the local coordinates on the 1 -jet space J 1 (R, R 3 ) are (t, x1 = C&Cyt, x2 = Ca B ñ , x3 = Ca m , y\ = Cacyt, y\ = CaER, y\ = Cam). We also remark that the ODEs system of order one [Ca 2+ -ER-cyt .pr-m] is deter1 mined by the following distinguished tensorial components X^Ux , x2, x3): (i)v l X}'(Ca (i) (LacjiiLaBÄ.Lamj — „2 Cyt,Ca.ER,Cam) =

\2 ( C a ^ -- CaC!/t) + heak{CaER - Cacvt)

Cacyt

«VumpCacyí Kout ~Tp2 "T^ ¡' « 22 ^'pump^acyt i~v |I Kout /Vj -\-Cacyt

+fc_ ( Catot - Cacyt - -~CaER PER

\ „

' ™m J Cam. /

,

Ca,.

"'ϊτι _^g g K 2 + CaCJ/t

- -r^Ca,, Pm

PER^„ R +, Pn -k+Cacyt I Pr¿0¡ ~Catot + Cacyt + ——Ca -¿—C-an E -—Ca E R + -5PER

X^}(CaCyt,CaER,Cam) = 1 ;

Pn

fcpump^acyt

t^leakx^aER

Ca^yij

PER

CaCcyt (CaE.R 2 , ^ 2 AT? + Ca ~cyt yt

kch „

-

Cacyt)

J

K¿n*^aCyf

I

y^ücyt

X,<ji)(Cac!/í,CaBi?.,Cam) = — \k°utK¡ + Ca%t+km]Ca" Pm K!+Ca%t As in the preceding biological case, it is obvious that again some partial derivatives and computations imply some geometrical results which, in our opinion, are possible to characterize the microscopic energetic changes produced by the calcium oscillations in the model involving endoplasmic reticulum, mitochondria, and cytosolic proteins (cf. Corollary 183): Theorem 199. The adapted components of the biologic electromagnetic field F produced by the ODEs system [Ca2+ -ER-cyt .pr-m] and the pair of Euclidian metrics Δ = (l,Sij) are given by the following expressions:

164

SOME EVOLUTION EQUATIONS FROM THEORETICAL BIOLOGY (2) dXf(1)

dX, ((ii))

(1) F;(1)2

dCaER x

pump

PER

dCacyt

2

{K¡+Ca

cyt)

(k- + k+Cacyt)

JER

τ-(ΐ)

(1)3

Ca;cyt 2

#i + Ca^ t

(CaER -Cacyt)

+ kch

i

kleak

PER PER

Ca;cyt + kleak K\ + Ca;2cyt

}·

dX, ((ii))

(3) dX, (1)

öCa,,

9Ca,cyt

δΚίηΚο

ñ { kch

1 ( 2 1

CalCJft

(Kf+Ca^)

2

W

Ca^ i i 2 + C a 2cyt

2 ¿rCout-K-l

T Kn

ßr, Pn

^--ä m V^ä C y£

( x 2 + ca2yt)

-(fc_ +fc + Ca c y t ) >,

F(1)

^ (2)3 -

- ί O

(2) 9X (1)

9X,(1)

(3)

9Ca„

dCaER

= 0.

-.(i) Taking again into account that the spatial components GLIj of the semispray

*^[Ca 2 +-ER-cyt.pr-m]

=

(Ο,β^μΐ i

produced by the ODEs system [Ca2+ — ER — cyt.pr — m] and the pair of Euclidian metrics Δ = (1,5 t j), are given by

(i)

(D1 ~

1 I dX(T)

(m)

9x,l ((!» i)

ax,k ((i m) ) '

2 1 öa;*

W

da;"

9a;'

2/i

we naturally establish the following qualitative geometrical result with biologicalenergetic meaning: Theorem 200. The C2 solutions of the ODEs system [Ca2+ -ER-cyt.pr-m] may be regarded as harmonic curves of the semispray ¿>[ca2+-ER-cyt .pr-m] on tne l~Jet space J 1 (K,K 3 ). In other words, the solutions (Ca C ! / t (t),Ca B ñ (t),Ca m (í)) of the firstorder ODEs system [Ca 2+ -ER-cyt.pr-m] verify the second-order biological ODEs system

FROM CALCIUM OSCILLATIONS ODEs SYSTEMS TO JET YANG-MILLS ENERGIES

d2c^cyt dt2 d2CaER dt2

d 2 Ca m di 2

op(i)

O F (i)

dCaER W 2 dt

dCam W dt

(i) (¿Cacyt

^-"-(1)2

^

d i

_ m(i) (¿Ca, dCacyf

(fc)

^

(k)

165

{k)

2^Λ(ΐ)·ύ^>Λ(ΐ)

fc=l (fe)

_

fe=l 3

γ^

(fe)

y(fe)_n

fc=l

(13.4)

Proof: We use the general second-order differential equations of the harmonic curves of a semispray, which are given by (1.12). At the same time, we take into account that we have x1 = CaC3/t, x2 = CZER, and x 3 = Ca™, together with the particular semispray S[Ca2+.BR.cytpr.m]. Remark 201. The second-order differential equations (13.4) are obviously equivalent with the Euler-Lagrange equations of the jet least squares Lagrangian cs^~ER-cyt.Pr-m]

=

^

_ χ (1)) 2

+

( ¿ ^ _ χ(2)^

,2

+ {Cam-X[f)y In other words, the C 2 solutions (Cacyt(t), CaER(t), Cam(r,)) of the first-order ODEs system [Ca2+ -ER-cyt.pr-m} are global minimum points for the jet least squares Lagrangian £5β[<*+-ΒΒ-«*ί.ΡΓ-™] In the sequel, particularizing the general definition of the geometric Yang-Mills energy of a general ODEs system to our present biological phenomena, we deduce that the biological Yang-Mills energy produced by the calcium oscillations in this model (that takes into account endoplasmic reticulum, mitochondria, and cytosolic proteins) has the form (cf. Definition 181) 2

£y-M[Ca2+-ER-cyt.pr-m\(Cacyt,C&ER,Cam)

= ^(1)2

+

i2

^(1)3

because FLL = 0. Also, it is important to note that we have three sets of parameter values (corresponding to three types of complex Ca 2+ oscillations in which the parameter values correspond to the various types of complex oscillatory whose behavior was observed in the model [Ca 2+ -ER-cyt.pr-m}), including bursting, chaos, and birhythmicity situations, which are listed in the following table (for more biologic details, see Marhl, Haberichter, Brumen and Heinrich [50]):

166

SOME EVOLUTION EQUATIONS FROM THEORETICAL BIOLOGY

Parameters

Bursting

Chaos

Birhythmicity

|

Ca to( (μΜ)

|

90

|

90

|

90

|

|

Prtot (μΜ)

|

120

|

120

|

120

|

|

PER

|

o.oi

|

o.oi

|

o.oi

|

|

pm

|

o.oi

|

o.oi

|

o.oi

|

|

ßER

| 0.0025 |

0.0025

|

0.0025

|

|

ßm

| 0.0025 |

0.0025

|

0.0025

|

|

Ki (μΜ)

|

5

|

5

|

5

|

|

K2 (μΜ)

|

0.8

|

0.8

|

0.8

¡

1968-2456

|

| | |

1

|

4100

1

|

20

|

20

|

20

|

kuakis- )

|

0.05

|

0.05

|

0.05

|

1

|

300

|

300

|

300

|

|

125

|

125

|

125

|

0.00625

|

0.00625

|

kchis' ) kpumpis' ) 1

| kin{ßM)(s- ) | |

fcouiCs"1) -1

fcm(s )

| 0.00625 | 1

| k+ (μΜ)(&~ ) |

| 2780-2980,3598-3636 |

k- (s _ 1 )

|

0.1

|

0.1

|

0.1

|

|

0.01

|

0.01

|

0.01

|

Obviously the preceding parameter values produce particular and distinct geometric biological Yang-Mills energies for calcium oscillations phenomena through endoplasmic reticulum, mitochondria, and cytosolic proteins. Theorem 202. The following formulas for the geometric biological energies of YangMills type are true: (i) Biological Yang-Mills energy of bursting calcium oscillations in the model involving endoplasmic reticulum, mitochondria, and cytosolic proteins.

syM

bursting [Ca=+ -ER-cyt.pr-m]

_1 Í 3075 · Ca; cyt ~ 4 | 25 + Ca^,t

+ (Ca E ñ - Ca cyt )

\ 51250 · Cacyt - 0.4 · Cacyt - 5.0025 / '(25 + Ca2cyt)2 100.663296 · Cal,,. cyt (0.16777216+ Ca«yi)2

125 · Ca?, cyt 25

+ C<, 4

2

1 [ 1562.5 ■ Ca m ■ Cacyf 4\ (25 + Ca^ ( )2

0.4 · Cac„t - 0.03375

FROM CALCIUM OSCILLATIONS ODEs SYSTEMS TO JET YANG-MILLS ENERGIES

167

(ii) Biological Yang-Mills energy of chaos calcium oscillations in the model involving endoplasmic reticulum, mitochondria, and cytosolic proteins. f1,

ί^

A/< chaos

JS/l

[C^+ -ER-cyt.pr-m]

■ 1 2 · 5 · kch ■?**

-

1

J ° · 7 5 · kch ■ Cacyt

~ 4 |

25 + Ca 2

V<-aßfl ~

^cyi)

- 0.4 - Ca,* - 5.OO25}2 + i { 1 5 6 2 - 5 - C a * · < * *

100.663296 · Cal,,, cyt (0.16777216+ Ca« yt )

2

+

125 · Cal a cyt ^ 7 - 0.4 · Ca cyt - 0.03375 25 + Ca cyt }

oc |

where kch e [2780,2980] U [3598,3636]; (iii) Biological Yang-Mills energy of birhythmicity calcium oscillations in the model involving endoplasmic reticulum, mitochondria, and cytosolic proteins. Ο Λ , A ^birhythmicity y [cJ+-ER-cyt.pr-m}

£ M

1 I 0.75 · kch · Cacyt = 4 j + (C*ER " Ca c y t ) 2g + C a 2 ^

■ 12 · 5 · kch f^f-2 - 0.4 · Cacvtcvt - 5.002512 + \ { 1 5 6 2 ' 5 ' ^ 2 ' (25 + Ca c %)

/

4 \

(25 + Ca yt )2

^

100.663296 · Ca7cyt 125 · Ca2 ^ - 2 ++ o c ' T - 0.4 ■ Qzcyt - 0.03375 \ , (0.16777216+ Ca* yt ) 25+Ca:cyt where fcc/l e [1968,2456]. 13.3.3

Yang-Mills energetic surfaces of constant level. Theoretical biological interpretations

In the opinion of the authors of this book, from a biological point of view, the appearance in our geometrical studies of a biological electromagnetic field F (unknown until now), directly and naturally provided by a nonlinear ODEs system of first-order that governs some biological phenomena, may probably have interesting connections with the intrinsic biological phenomena studied. In this way, we believe that our geometric biological field F may be regarded as follows: • Either this biological field F vanishes in order to realize a stability of the biological phenomena studied. This should be probably because such an "electromagnetic" field must not exist in such biological phenomena. • Or this biological field F does not vanish, having a natural and microscopic character in the biological phenomena studied. In other words, this microscopic

168

SOME EVOLUTION EQUATIONS FROM THEORETICAL BIOLOGY

biological field F may be probably regarded as being provided not necessarily by the ODEs systems involved in the studies, but by the pair of Euclidian metrics Δ = ( l , i y ) , which have the well-known physical meaning of microscopic gravitational potentials produced intrinsically by the biological matter. In this case, we believe that the classical geometric study (i.e., fundamental forms and main curvatures) of constant level Yang-Mills energy surfaces provided by the biological ODEs systems involved in this Chapter, aided by relevant computer imaging simulation, represents a promising research topic for Theoretical Biology. For example, the geometric features of the 20-th Taylor approximation of the rational constant level Yang-Mills energy surfaces for the energies of Theorem 197 and of Theorem 202 respectively, which are represented below, exhibit specific geometric features which both reflect properties of the associated ODEs systems, and provide biological relevance for the modeled phenomena.

Fig. 1. Taylor approximation of rational constant level Yang-Mills energy surfaces for intracellular calcium oscillations (bursting, chaos, and quasiperiodicity cases).

Fig. 2. Taylor approximation of rational constant level Yang-Mills energy surfaces for calcium oscillations through endoplasmic reticulum, mitochondria, and cytosolic proteins (bursting, chaos, and birhythmicity cases). Open problem. Determine the biological meaning of our previously described geometrical Yang-Mills energies.

CHAPTER 14

JET GEOMETRICAL OBJECTS PRODUCED BY LINEAR ODEs SYSTEMS AND HIGHER-ORDER ODEs

Using the applicative geometrical theory from Chapter 10, in this Chapter we construct the Riemann-Lagrange geometry on 1-jet spaces (in the sense of d-connections, dtorsions, d-curvatures, electromagnetic d-fields, and geometric electromagnetic YangMills energies), produced by a given linear ODEs system of first-order or by a given higher-order ODE. The case of a non-homogenous linear ODE of higher-order is also disscused. 14.1

JET RIEMANN-LAGRANGE GEOMETRY PRODUCED BY A NON-HOMOGENOUS LINEAR ODEs SYSTEM OF ORDER ONE

In this Section we apply our preceding jet Riemann-Lagrange geometrical results for a non-homogenous linear ODEs system of order one. In this direction, let us consider the following non-homogenous linear ODEs system of first-order, locally described, in a convenient chart on J 1 (M, K n ), by the differential equations

Jet Single- Time Lagrange Geometry and Its Applications I-st Edition. By Vladimir Balan and Mircea Neagu. © 20! I John Wiley & Sons, Inc. Published 2011 John Wiley & Sons, Inc.

169

170

JET OBJECTS PRODUCED BY LINEAR ODEs SYSTEMS AND HIGHER-ORDER ODEs

where the local components * = flȣ-aa)*'Vfc and

Ai) _dx^dt

, =

_— ' '

1 n

-y)

Remark 203. We suppose that the product manifold R x R " c J ^ R , ^ " ) is endowed a priori with the pair of Euclidian metrics Δ = (l,<$y), with respect to the coordinates

(t,xl).

It is obvious that the non-homogenous linear ODEs system (14.1) is a particular case of the jet first-order nonlinear ODEs system (10.5) for n fc=l

In order to present the main jet Riemann-Lagrange geometrical objects that characterize the non-homogenous linear ODEs system (14.1), we use the matrix notation V

v

/J

/t,j=l,n

In this context, applying our preceding jet geometrical Riemann-Lagrange theory to the non-homogenous linear ODEs system (14.1) and the pair of Euclidian metrics Δ = (1, Sij), we get (cf. Theorem 180 and Remark 182) the following: Theorem 204. (i) The canonical nonlinear connection on J 1 (R, Mn), produced by the non-homogenous linear ODEs system (14.1), has the local components

(»·*;;!,) where JVÍ! . are the entries of the skew-symmetric matrix

(ii) All adapted components of the canonical generalized Cartan connection CT, produced by the non-homogenous linear ODEs system (14.1), are zero. (iii) The effective adapted components ñ i . of the torsion d-tensor T of the canonical generalized Cartan connection CT, produced by the non-homogenous linear ODEs system (14.1), are the entries of the skew-symmetric matrices

¿(Di = « ) « )

= 2 [i(i) - TÁi)] -

RIEMANN-LAGRANGE GEOMETRY PRODUCED BY A NON-HOMOGENOUS LINEAR ODEs SYSTEM

171

where AD

= j t [An)] ■

(iv) All adapted components of the curvature d-tensor R of the canonical generalized Cartan connection CF, produced by the non-homogenous linear ODEs system (14.1), cancel. (v) The geometric electromagnetic distinguished 2-form, produced by the nonhomogenous linear ODEs system (14.1), is given by

where Sy[ =

W

dy[--[ °(l)fe

_

(fc)

a

(l)i

dxk, Vt = l,

and the adapted components F^J. are the entries of the skew-symmetric matrix

that is, we have

pw _ I ¡„a) _ JJ) ' WJ ~ 2 L dW W\ ' (vi) The jet geometric Yang-Mills energy, produced by the non-homogenous linear ODEs system (14.1), is given by the formula

i=l

j—i+1

Proof: Using relations (14.2), we easily deduce that we have the Jacobian matrix J(X(1))

=A(1),

whereX(1):=(x;;j)i=_. Consequently, applying Theorem 180 to the non-homogenous linear ODEs system (14.1), together with Remark 182, we obtain the required results. Remark 205. The entire jet Riemann-Lagrange geometry produced by the nonhomogenous linear ODEs system (14.1) does not depend on the non-homogeneity terms/ ( ( jj(i). Remark 206. The jet geometric Yang-Mills energy, produced by the non-homogenous linear ODEs system (14.1), cancels if and only if the matrix Ac\) is a symmetric one. In this case, the entire jet Riemann-Lagrange geometry produced by the non-homogenous linear ODEs system (14.1) becomes trivial, so it does not offer geometrical information about the initial system (14.1). However, it is important to

172

JET OBJECTS PRODUCED BY LINEAR ODEs SYSTEMS AND HIGHER-ORDER ODEs

note that, in this particular situation, we have the symmetry of the matrix ^4(1), which implies that the matrix A^ is diagonalizable. Remark 207. All jet torsion adapted components, produced by a non-homogenous (i)

linear ODEs system with constant coefficients U/Λ,, are zero. 14.2 JET RIEMANN-LAGRANGE GEOMETRY PRODUCED BY A HIGHER-ORDER ODE Let us consider the higher-order ODE expressed by y(n)(t)

= f(t,y(t),y'(t),...,y{n-'1)(t)),

where y(t) is the unknown function, y^k\t) function y(t) for each k e {0, l,...,n} depending on the distinct variables t, y(t), It is well known the fact that, using the x1 = y, x2

n>2,

(14.3)

is the derivative of order k of the unknown and / is a given differentiable function y'(t),..., j / n _ 1 ) ( i ) . notations

y , ···, x

=y

(n-l)

the higher-order ODE (14.3) is equivalent with the nonlinear ODEs system of order one dxx 2 ~dt=X ' dx2 , at (14.4) dx n - l dt dxn — j\t^x -,χ , · · · , χ )· ~~dT But, the first-order nonlinear ODEs system (14.4) can be regarded, in a convenient local chart, as a particular case of the jet nonlinear ODEs system of order one (10.5), setting

X((¡ht,x)=x2,X¡¡Ut,x)=x\. X?(t,x)

=

'

f(t,x\x2,...,xn),

(1)

V'->x) ~

X

"

(14.5)

where we suppose that the geometrical object X — (X,\l (t, x) j behaves like a d-tensoron J^JR.R"). Remark 208. We assume again that the product manifold R x R n c J ^ R , R n ) is endowed a priori with the pair of Euclidian metrics Δ = (1, £¿¿), with respect to the coordinates (t,xl).

JET RIEMANN-LAGRANGE GEOMETRY PRODUCED BY A HIGHER-ORDER ODE

173

Definition 209. Any geometrical object on J X (R, R"), which is produced by the first-order nonlinear ODEs system (14.4), is called a geometrical object produced by the higher-order ODE (14.3). In this context, the Riemann-Lagrange geometrical behavior on the 1-jet space J ^ R , R n ) of the higher-order ODE (14.3) is described in the following result (cf. Theorem 180 and Remark 182): Theorem 210. (i) The canonical nonlinear connection on J1 ( by the higher-order ODE (14.3), has the local components

ln), produced

MW' where Ñ^l

are the entries of the skew-symmetric matrix Ñ^

= I Ñ,

(lb'

df_

I

dx1

χ

dx2 dx3

df

gxn-2

V

0

0

0

df dx1

df_ dx2

dx3

df

-1

df

βχη-2

dx™-1

-1 +

df dxn-1

0

(ii) All adapted components ofthe canonical generalized Cartan connection CT, produced by the higher-order ODE (14.3), are zero. (iii) The effective adapted components of the torsion d-tensor T of the canonical generalized Cartan connection CT, produced by the higher-order ODE (14.3), are the entries of the skew-symmetric matrices (

0

0

0

0

0

0

d2f

1

dtdx d2f dtdx2

R (Di

d2f dtdx

1

d2f

d2f dtdx

2

dtdxn-1

d2f dtdx"-1 0

\

174

JET OBJECTS PRODUCED BY LINEAR ODEs SYSTEMS AND HIGHER-ORDER ODEs

and 0

0

0

0

0

0

0

0

d2f dxkdx1 d2f dxkdx2

\

R(i(l)fc

d2f \ dxkdx1

d2f dx^dx"-1

d2f dx dxn~

d2f dxkdx2

0

k

where k G {1,2, ...,n}. (iv) All adapted components of the curvature d-tensor R of the canonical generalized Cartan connection CT, produced by the higher-order ODE (14.3), cancel. (x)The geometric electromagnetic distinguished 2-form, produced by the higherorder ODE (14.3), has the form W = F?W where

'ΜΛάχΐ,

oyÍ=dy\+Ñ¡;\kdxk,Vi

= l

and the adapted components PL', are the entries of the skew-symmetric matrix

(vi) The jet geometric Yang-Mills energy, produced by the higher-order ODE (14.3), is given by the formula syMHUUb(t,x)

dxn-l

^

Z^\dxj

Proof: By partial derivation, relations (14.5) lead to the Jacobian matrix

( °0

1

0

0

o \

0

1

0

0

0

0

0

0

1

ÉL

df_ dx2

2L

·>(*(!))

V Οχλ

dx

3

df dx™"1

dij

dxn >

JET GEOMETRY PRODUCED BY A NON-HOMOGENOUS LINEAR ODE OF HIGHER-ORDER

175

where X a ) := (x ( ( j¡). In conclusion, Theorem 180, together with Remark 182, applied to first-order nonlinear ODEs system (14.4), give what we were looking for. 14.3

RIEMANN-LAGRANGE GEOMETRY PRODUCED BY A NON-HOMOGENOUS LINEAR ODE OF HIGHER-ORDER

If we consider the non-homogenous linear ODE of order n € N, n > 2, expressed by

a0(t)y(n) + ai(t)y(n-V

+ ... + an^(t)y'

+ an(t)y = b(t),

(14.6)

where b(t) and a¿ (f), V i = 0, n, are given differentiable real functions and a0 (t) φ 0, V t G [a, b], then we recover the higher-order ODE (14.3) for the particular function b(t)

an{t) a0(t)

f(t,x) = a (t) 0

x

!

α„_ι(ί) 7-T— a0(t)

·x

2

-

···

-

ai(t) a0(t)

(14.7)

where we recall that we have y

x\y'

x\...Jn-l)

Consequently, we can derive the jet Riemann-Lagrange geometry attached to the non-homogenous linear higher-order ODE (14.6) (cf. Theorem 210). Corollary 211. (i) The canonical nonlinear connection on JX(R, R n ), produced by the non-homogenous linear higher-order ODE (14.6), has the local components

f = (o,%), ¡CH») . are the entries of the skew-symmetric matrix where Ν,Λ

N,( i )

W.'L)

i,j=l,n

i o

1

0

0

0

0

1

0

0

α0 Oo O-n-2

-1

0

0

0

0

0

0

1

03

0

0

-1

0

ι +ί

On

O'n-X

O-n-2

03

_l_a1

Oo

a0

Oo

Oo

Oo

Oo

θ2

a0

a0

0

176

JET OBJECTS PRODUCED BY LINEAR ODEs SYSTEMS AND HIGHER-ORDER ODEs

(ii) All adapted components of the canonical generalized Cartan connection CT, produced by the non-homogenous linear higher-order ODE (14.6), are zero. (iii) All adapted components of the torsion d-tensor T ofthe canonical generalized Cartan connection CT, produced by the non-homogenous linear higher-order ODE (14.6), are zero, except the temporal components ñÜ) _ ñ(n) ( l ) l n - -R(l)li

_ -

W

a

'n-i+la0

-an-i+ia'o ^2

'

,,

V i

_ -

γ

=l.""1'

where we denoted by " ' " the derivatives of the functions α&(ί). (iv) All adapted components of the curvature d-tensor R of the canonical generalized Cartan connection CT, produced by the non-homogenous linear higher-order ODE (14.6), cancel. (v) The geometric electromagnetic distinguished 2-form, produced by the nonhomogenous linear higher-order ODE (14.6), has the expression W = F¡¡))jSy\AdxK where dyÍ=dy\+Ñ$kdxk,Vi

= l,n,

and the adapted components F^J. are the entries of the skew-symmetric matrix

^(1) = ( 4 ) i ) . - = -^· (vi) The jet geometric Yang-Mills electromagnetic energy, produced by the non-homogenous linear higher-order ODE (14.6), has the form n

0>M

NHLHODE

(i) = -

a0

„2

j^2a%

Proof: We apply the Theorem 210 for the particular function (14.7) and we use the relations df

dxi

_

Ctra-j + l

ao

-, V j = l,n.

Remark 212. The entire jet Riemann-Lagrange geometry produced by the nonhomogenous linear higher-order ODE (14.6) is independent by the term of nonhomogeneity b(t). In the opinion of the authors, this fact emphasizes that the most important role in the study of the ODE (14.6) is played by its attached homogenous linear higher-order ODE.

JET GEOMETRY PRODUCED BY A NON-HOMOGENOUS LINEAR ODE OF HIGHER-ORDER

177

I EXAMPLE 14.1 The law of motion without friction of a material point of mass m > 0, which is placed on a spring having the constant of elasticity k > 0, is given by the homogenous linear ODE of order two (harmonic oscillator) d2y

o

¿ Γ + ω 2 2 / = 0,

(14.8)

where the coordinate y measures the distance from the center of mass and ω2 = k/m. It follows that we have n = 2,

oo(i) = 1,

ai(t) = 0

and

2 ,

that is, the second-order ODE (14.8) provides the following jet geometric YangMills electromagnetic energy of harmonic oscillator: c-y i/Haraionic Oscillator

Z ( l _i_

4V

,2\2

/ '

Open problem. Is there a real physical interpretation for the jet geometric Yang-Mills electromagnetic energy attached to the harmonic oscillator?

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CHAPTER 15

JET SINGLE-TIME GEOMETRICAL EXTENSION OF THE KCC-IN VARI ANTS

In this Chapter we generalize on the 1-jet space J 1 ( E , M), where Mn is an arbitrary smooth manifold of dimension n, the basics of the KCC-theory for second-order systems of differential equations (see Kosambi [44], Cartan [26] and Chern [27]). In this respect, let us consider on J 1 (R, M) a second-order system of differential equations (SODEs) of local form ~+F((Í]1(t,xk,ykl)

= 0, ¿ = M ,

where yk = dxk/dt and the local components F^L(t,xk,yk) change of coordinates (1.2), by the rules

(15.1) transform, under a

p(r) _F(j) fdt\2dx^_d^dy{_dx^d^_~j (1)1 ~ ^Kdt) dxi dt dt dxi dx^n'

(

,,. '

Remark 213. The second-order system of differential equations (15.1) is invariant under a change of coordinates (1.2). Using a temporal Riemannian metric hn(t) on R and taking into account the transformation rules (1.7) and (1.10), we can write the SODEs (15.1) in the following Jet Single-Time Lagrange Geometry and Its Applications 1st Edition. By Vladimir Balan and Mircea Neagu. © 2011 John Wiley & Sons, Inc. Published 2011 John Wiley & Sons, Inc.

179

180

JET SINGLE-TIME GEOMETRICAL EXTENSION OF THE KCC-INVARIANTS

form: «} 1 í/í+2G^ 1 (í,x f c , 2 /f) = 0, ¿ = M ,

άχ1

dt2

where G<<> T

1 ^ > , . 1 . 1KnVl (i)i ~ 2 W1 + 2

are the components of a spatial semispray on J 1 (M, M). Moreover, the coefficients of the spatial semispray G?L· produce the spatial components N¡V,. of a nonlinear connection Γ on the I -jet space J 1 (R, M) by putting

_dGtm

(t)

_idFtin

i iH

In order to find the basic jet differential geometrical invariants of the system (15.1) (see the works [2], [4], [5], [19], [87]) under the jet coordinate transformations (1.2), we define the h-KCC-covariant derivative of a d-tensor of kind T,■ j ; (t, xk, y\) on the 1-jet space J1(R, M), via

DT { ( ;¡

&rW BTW i 8FW (*> i 1 (i)iT(r>

-

i V,- 1 τ<*>

where the Einstein summation convention is used throughout. Remark 214. The h-KCC-covariant derivative components —¿¡P- transform, under a change of coordinates (1.2), as a d-tensor of type 7/Λ'x. In such a geometrical context, if we use the notation y\ — dxl/dt, then the system (15.1) can be written in the following distinguished tensorial form: h Dy

l

_p(i)

-

(f

Tfc

,M

, ΛΓ(«) ,.r _

i 8F{i)

"

1 ,i

i

_ p « ) 1 ++ i l l i i l i ^1 _ I K i n y„ii ^(D 2 aj/ϊ ^ 2 '

Definition 215. The distinguished tensor

ε

(ΐ)ΐ

-

^(i)i

+

2

dy\

Vl

2

nVl

is called the^rsi h-KCC-invariant on the 1 -jet space J 1 (R, M) of the SODEs (15.1), which is interpreted as an external force [2], [19].

JET SINGLE-TIME GEOMETRICAL EXTENSION OF THE KCC-INVARIANTS

181

■ EXAMPLE 15.1 It can be seen easily that for the particular first-order jet time-dependent dynamical system dt -xwV'x where X,lJ(t,x) has the form

>^

dt* -

dt

+

(153)

dx™ Vi'

is a given d-tensor on J ^ M , M ) , the first /i-KCC-invariant

E

W ~

+

dt

2 dxr

Vl

2KnVv

In the sequel, let us vary the trajectories xz(t) of the system (15.1) by the nearby trajectories (#'(£, s)) s € (_ e i E ), where x'(£, 0) = xl{t). Then, considering the variation d-tensor field dx1 5=0

we get the variational equations

In order to find other jet geometrical invariants for the system (15.1), we also introduce the h-KCC-covariant derivative of a d-tensor of kind ξ%(ί,) on the 1-jet space J ^ R , M), via

dt -

dt

+ 7 N

W¿

- dt

+

2 dyf

ξ

+ Κηξ

2 -

h

Remark 216. The h-KCC-covariant derivative components ^¡- transform, under a change of coordinates (1.2), as a d-tensor T^l. In this geometrical context, the variational equations (15.4) can be written in the following distinguished tensorial form: h

/ h

dt\

dt ' ~

™ 1ιξ

where h. j11

dF(i) σ -Τ(1)1

öxJ

1 d2F(i) 1 " >

2 atayj

1 d2F{i) 1 σ ^r ( 1 ) 1 , Γ

2dx dy{

i d2F{i) 1 ° ^(1)1 „(r)

*dy\dy{

'

182

JET SINGLE-TIME GEOMETRICAL EXTENSION OF THE KCC-INVARIANTS h.

Definition 217. The d-tensor P j u is called the second h-KCC-invariant on the 1 -jet space J 1 (R, M) of the system (15.1), or the jet h-deviation curvature d-tensor. EXAMPLE 15.2 If we consider the second-order system of differential equations of the harmonic curves associated to the pair of Riemannian metrics (hn(t), φ^(χ)), the system which is given by (see Example 1.8) d2xl

, . , dxl

,· , . άχί dxk

where K,\1(t) and 7*-fe (x) are the Christoffel symbols of the Riemannian metrics hu(t) and φ^(χ), then the second /i-KCC-invariant has the form

where _

mi

d

d

"lpq'PQ

=

l'PI P] <~r «™g dxi ~ ipq

,r

i

_

r

,r

i

3 'pj
Ά„Ί

h

D

+*Ui/?i/?r = o,

where

h

C ^dí = ^dt++7 °*W lAr-

EXAMPLE 15.3 For the particular first-order jet time-dependent dynamical system (15.3), the jet /i-deviation curvature d-tensor is given by H _!#*£> " n - 2 dtdxi

+

! ^ [ 2dxWxryi

+

1 3 X $ < ! A~dx^ dxi

+

1 ^ 2 dt

Öj

Definition 218. The distinguished tensors h.

1

*5*i = 3

/

ft. dpi

jn dyki

h

dPjn dy{

jkm

dyf '

1 4ΚιιΚΐ1^·

JET SINGLE-TIME GEOMETRICAL EXTENSION OF THE KCC-INVARIANTS

183

and Dn

jkm

=

(i)i

dy{dy1dyf

are called the third, fourth, and fifth h-KCC-invariant on the 1-jet vector bundle Jl(W, M) of the system (15.1). Remark 219. Taking into account the transformation rules (15.2) of the components F(\)v w e immediately deduce that the components D%Xm behave like a d-tensor. I EXAMPLE 15.4 For the first-order jet time-dependent dynamical system (15.3) the third, fourth and fifth /i-KCC-invariants are zero. Theorem 220 (of characterization of the jet /i-KCC-invariants). All the five hKCC-invariants of the system (15.1) cancelen J 1 (M, M) if and only if there exists a flat symmetric linear connection Hfc(a;) on M such that (15.5)

F^^r^WM-K^m. Proof: "<=" By a direct calculation, we obtain e $ i = 0, Hn = -Kqjypiy¡

= 0 and D?kl = 0,

where T^lpqj = 0 are the components of the curvature of the flat symmetric linear connection Tl,k(x) on M. "=s>" By integration, the relation g3p(i) dy{dy1dy\ subsequently leads to

where the local functions Γ**.(ί, x) are symmetrical in the indices j and k. The equality ε' ( 1^) 1 = 0 on J ^ R , M) infers |/(1)1^U,

Consequently, we have

9í

?íu

M

(l)j

-

K

llöj-

, ·

— ^ ί - - 2Γ}ρν? - « i ^ -

184

JET SINGLE-TIME GEOMETRICAL EXTENSION OF THE KCC-INVARIANTS

and

p(i)

_ Γ« ,.Ρ,.9 _

Kl

,.»

The condition P j n = 0 on J^M, M) implies the equalities r*-fc - T)k{x) and

where P9J

Qxj

Qxq

^

PqL rj

L

pjL rq-

It is important to note that, taking into account the transformation laws (1.2), (15.2), (1.7), and Example 1.5, we deduce that the local coefficients r*-fe (x) locally transform like a symmetric linear connection on M. Consequently, fcp«„ represent the curvature of this symmetric linear connection. h.

On the other hand, the equality iC fcl = 0 leads us to 7^Lfc = 0, which infers that the symmetric linear connection n f c ( i ) on M is flat.

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INDEX

Γ-linear connection, 20, 21, 34, 37 h-normal of Cartan type, 38, 39 Benvald canonical, 35 Cartan canonical, 81, 103 Cartan canonical h-normal, 48 Cartan canonical ~ , 69 change of adapted components of ~ , 21 h-normal, 33, 37 jet ~ , 19 local adapted components of ~ ,21 1 -jet fiber bundle, xi 1-positive homogeneity condition, xiii adapted bases, 60 anisotropy, 5 autonomous Lagrange space of electrodynamics, 58, 61, 62 Berwald connection, 22 Berwald-Moór derivative operators, 70, 81 Berwald-Moór metric rheonomic of order four, 78, 79, 89 rheonomic of order three, 66, 67 Bianchi identities, 39, 52

canonical connection Cartan ~ , 52-54 generalized Cartan ~ , 60, 61, 63, 137, 142, 145, 149, 152, 170, 173, 176 nonlinear ~ , 50 canonical nonlinear connection, 53, 62, 68, 80, 102, 136, 142, 144, 149, 152, 170, 173, 175 of the autonomous jet single-time Lagrange space of electrodynamics, 60 canonical nonlinear connection of the relativistic time-dependent Lagrange space, 47 canonical projections, 20 canonical semispray, 60 of the autonomous jet single-time Lagrange space of electrodynamics, 60 of the relativistic time-dependent Lagrange space, 47 change of coordinates, xiv Chernov derivative operators, 104 Chernov metric, 99 of order four, 99 rheonomic of order four, 101

Jet Single-Time Lagrange Geometry and Its Applications 1st Edition. By Vladimir Balan and Mircea Neagu. © 2011 John Wiley & Sons, Inc. Published 2011 John Wiley & Sons, Inc.

191

192

INDEX

Christoffel process, 49 Christoffel symbols, 8, 34, 59 connection Berwald, 49 Cartan, 49 conservation laws, 75, 87 covariant derivative, 23, 34 M-horizontal, 61 curvature d-tensor, 60, 63, 71, 82, 104, 137, 143, 145, 150, 153, 174, 176 d-tensor identities, 51 of the relativistic time-dependent Lagrange space, 49 curvature tensor, 27, 36 d-tensor stress-energy ~ , 55, 63, 72, 75, 84, 86, 88, 107 d-tensor field, 4, 5, 172 ^-normalization ~ , 6, 15 d-tensor identities, 52 d-torsions, 60 d-vector field /i-canonical Liouville ~ , 6 canonical Liouville, 50 canonical Liouville ~ , 6 deflection d-tensors, 50 metrical ~ , xiv derivative operators Chernov, 104 direct sum, 20 distinguished tensor (d-tensor), 4 dual bases, 20 economic-mathematical Kaldor conditions, 142 Ehresmann-Koszul axioms, 21 electromagnetic components, 51 d-tensors, 62 field, 62 potential, 58 energy action functional, 45, 47, 59, 68, 80, 102 Euler-Lagrange equations, 11, 46, 47 external force, 180 extremals, 47 fast gravity wave oscillations, 136 Finsler-like geometry, 66, 78 metric, 101 third-root ~ function, 67 Finsler-like function third-root ~ , 100

time-dependent ~ , 78 first-order jet time-dependent dynamical system, 181 first-order nonlinear ODEs system, 173 flow of cancer cell population, 148 of the HIV-1 model, 152 foliation, xiv general relativistic time-dependent Lagrange space, 61-63 geometrical electromagnetic distinguished 2-form, 137, 143, 145, 150, 153, 171, 176 geometrical conservation laws, 55, 62, 63, 86 geometrical Einstein equations, xiv, 53, 54, 62, 63, 72, 73, 84, 85, 105, 107 geometrical gravitational theory, 63 geometrical Maxwell equations, xiv, 30, 39, 51,62 geometrical Yang-Mills energy biologic, 150 global Einstein equations, 55 gravitational potentials, 58, 87 h-deviation curvature d-tensor, 182 harmonic oscillator, 177 harmonic curve, 11, 17, 59 autoparallel ~ , 17 equations, 47 harmonic curves associated to a pair of Riemannian metrics, 182 of the relativistic time-dependent semispray, 59 of the semispray, 47 harmonic map, 11 higher-order ODE, 172 horizontal distribution, 12 horizontal vector field, 12 temporal ~ , 14 isotropic space-time, 65 jet autonomous single-time Lagrange space of electrodynamics, 63 jet differential geometrical invariants, 180 jet first-order nonlinear ODEs system, 170 jet geometric Yang-Mills electromagnetic energy, 176 of harmonic oscillator, 177 jet geometrical Riemann-Lagrange theory, 170 jet Lagrangian function, 44

INDEX jet relativistic time-dependent Lagrangian of electrodynamics, 58 jet Riemann-Lagrange geometrical objects, 170 jet Riemann-Lagrange geometry, 171, 175, 176 jet single-time electromagnetic field, 51 jet single-time gravitational field, 53 jet single-time gravitational potential, xiv jet single-time Lagrange space, 44 jet transformation group, xii Kaldor flow, 142 nonlinear model, 142 KCC-covariant derivative, 180, 181 KCC-invariant fifth- , 183 first h- ~ ,180 first- , 180 fourth ~ , 183 second ~ , 182 third- , 183 KCC-theory, 179 Kronecker h-regular Lagrangian, 44 h-regularity, 46 Lagrangian, xiv relativistic time-dependent ~ , 11 time-dependent ~ , 43 linear connection Cartan canonical, 49 local covariant derivatives, 61 local curvature tensors, 61 local derivative operators, 23 local symmetry conditions, 86 Lorenz atmospheric ODEs system, 136, 138 mathematical model HIV evolution - ,151 Kaldor ~ , 142 Solyanik - , 148 Tobin ~ , 144 metrical d-tensor, 84 fundamental vertical - , 62 fundamental ~ , 67, 79, 100 spatial ~ , 59 vertical fundamental - , 44 metrical deflection d-tensor identities, 52 d-tensors, 50, 61 multi-dimensional time, 77 non-autonomous

193

jet single-time Lagrange space of electrodynamics, 45 Lagrangian function of electrodynamics, 58 non-homogenous linear first-order ODEs system, 169 linear ODE, 175 linear ODEs system, 170, 171 non-isotropic gravitational potential, 84 nonlinear connection, xiii, 11, 19 canonical ~ , xiii components of - , 13 spatial components of a ~ , 180 nonmetrical deflection d-tensors, 31, 38 physical Rossby number, 136 Poisson form of Euler-Lagrange equations, 46, 47 Poisson brackets, 25 potential function, 58 process Christoffel ~ , 49 relativistic time-dependent Lagrange space, 44 rheonomic Berwald-Moór metric of order four, 79, 89 electrodynamics, 58 Ricci d-tensor, 53, 72, 73, 84, 106 of the Cartan connection, 53 vertical - , 107 Ricci identities, 31, 38 Ricci tensor, 63 classical ~ , 54 Riemann-Lagrange geometry, 59 Riemannian metric, 34, 43 temporal ~ , 180 scalar curvature, 54, 63, 72, 73, 85, 107 semi-Riemannian metric, 58, 61, 63 semispray, xiii, 7 relativistic time-dependent ~ , 9, 16, 17 spatial, 47, 180 spatial ~ , xiii, 7 temporal ~ , xiii, 7 single-time gravitational ft-potential, 62 SODE, 10 tangent bundle, xii time-dependent electrodynamics, 58 Finsler-like function, 78 semispray, 47, 60

194

INDEX

torsion, 61 torsion d-tensor, 70, 82, 104, 137, 143, 145, 149, 152, 170, 173, 176 torsion of the relativistic time-dependent Lagrange space, 49 torsion tensor, 24, 25, 35 transformation of jet local coordinates, 4 transformation rules, 46 transverse coordinates, xiv variation d-tensor field, 181 variational equations, 181 wave oscillations slow Rossby ~ , 138 waves gravity ~ , 135 Rossby- , 135 Yang-Mills energy biologic geometric ~ , 153 curves of constant level of biologic ~ , 151 curves of constant level of economic ~ , 144 economic geometric ~ , 143, 146 hypersurfaces of constant level of ~ , 138 jet geometric ~ , 137, 138 j e t - , 171, 172 surfaces of constant level of biologic ~ , 153 surfaces of constant level of economic ~ , 146