Integrable problems of celestial mechanics in spaces of constant curvature

Journal of Mathematical Sciences, Vol. 125, No. 4, 2005

INTEGRABLE PROBLEMS OF CELESTIAL MECHANICS IN SPACES OF CONSTANT CURVATURE T. G. Vozmishcheva

UDC 517.933

Abstract. The technique of topological analysis of integrable problems developed by A. T. Fomenko is applied for studying certain problems of celestial mechanics.

CONTENTS Chapter 1. Main Concepts and Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. Hamiltonian Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Lie Groups and Algebras. Noether Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Integrability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. Momentum Mapping. Bifurcation Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . 5. Fomenko–Zieshang Topological Invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter 2. Generalization of the Kepler Problem to Spaces of Constant Curvature . . . . . . 6. Historical Essay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7. Flat Kepler Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8. Dynamics in Spaces of Constant Curvature. Generalized Bertrand Theorem . . . . . . . . 9. Generalized Kepler Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10. Bifurcation diagrams. Geometry of the Phase Space of the Generalized Kepler Problem . 11. Regularization of the Kepler Problem on the Sphere . . . . . . . . . . . . . . . . . . . . . 12. Lie Algebra of First Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13. Neumann Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter 3. Two-Center Problem on the Sphere . . . . . . . . . . . . . . . . . . . . . . . . . . 14. Plane Two-Fixed-Centers Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15. Description of the System on the Sphere. Reduction . . . . . . . . . . . . . . . . . . . . . 16. Integrals of the System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17. Regularization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18. Bifurcation Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19. Topological Analysis of the Two-center Problem on the 2-Sphere . . . . . . . . . . . . . . 20. Motions on the Configuration Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter 4. Two-Center Problem in the Lobachevskii Space . . . . . . . . . . . . . . . . . . . 21. Description of the System. Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22. Integrals of the System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23. Bifurcation Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24. Classification of Motions on the Configuration Space. Limit Motions . . . . . . . . . . . . 25. Description of Noncompact Bifurcations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26. Passage to the Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27. Comparative Analysis of the Topology of Liouville Foliations of the Two-Center Problem Chapter 5. Motion in the Newtonian and Homogeneous Field in the Lobachevskii Space . . . 28. Reduction. Analog of Homogeneous Field . . . . . . . . . . . . . . . . . . . . . . . . . . . 29. Pseudosphero-Parabolic Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

422 422 429 433 440 441 444 444 448 460 465 465 470 470 471 472 473 479 480 484 486 489 498 502 502 508 511 512 514 518 520 521 521 524

Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 4, Dynamical Systems, 2003. c 2005 Springer Science+Business Media, Inc. 1072–3374/05/1254–0419


419

30. Integrals of the System . . . . . . . . . . . . . . . . . . . 31. Bifurcation Diagrams . . . . . . . . . . . . . . . . . . . 32. Description of Noncompact Bifurcations and Motions on References . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . the Configuration Space . . . . . . . . . . . . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

525 526 528 530

Introduction The problem of integrability and nonintegrability of dynamical problems occupies a central place in mathematics and mechanics. Integrable cases are of great interest since by examining their examples we can study the general principles of behavior of solutions of equations of motion. The classical approach to studying dynamical problems presupposes the search for explicit formulas of solutions of the equations of motion and then their analysis. This approach has stimulated the development of new directions in mathematics such as the algebraic integration and the theory of elliptic functions and theta-functions. In spite of this, the qualitative approach to studying dynamical systems is very urgent. It was Poincar´e who laid the base of the qualitative theory of differential equations. Developing the qualitative methods, Poincar´e has studied the problems of celestial mechanics and cosmology in which it is very important to understand the behavior of trajectories of motion, i.e., solutions of differential equations at infinite time. Precisely, as initiated by Poincar´e, sets of equations (in connection with the study of problems of celestial mechanics) whose right-hand sides do not depend on the independent variable time explicitly, i.e., dynamical systems, are studied. At present, qualitative methods for studying dynamical systems, in particular Hamiltonian systems, have been developed intensively. According to the modern viewpoint, one of the main qualitative characteristics of an integrable Hamiltonian system is the structure of its Liouville foliation, i.e., the foliation of phase space into invariant Liouville tori (cylinders) and singular integral submanifolds. A. T. Fomenko has suggested a new approach to studying this structure, which allows one to construct a complete topological invariant (Fomenko–Zieshang invariant). Using this new approach, in the qualitative theory of integrable Hamiltonian systems, a classification of such foliations on three-dimensional surfaces is constructed. In this paper, we also apply the technique of topological analysis of integrable problems. In celestial mechanics, the three-body problem is most famous; this problem can be formulated as follows. Three material points are mutually attracted by the Newton law, according to which an attracting force acts between every two of these points; it is directly proportional to the masses of these points and inversely proportional to the square of the distance between them; the points are located in any initial position and can move in the space freely. At present, the three-body problem is not solved in the general form. The problem of motion of a material point in the Euclidean space under the action of the fields of two fixed Newton centers (a particular case of the three-body problem) was considered by Euler for the first time. Precisely, Euler reduced the problem of two fixed centers to quadratures. Generally speaking, almost all achievements of modern celestial mechanics are based on Euler’s work, which is very fruitful, although S. V. Vavilov, speaking about him, commented that there is a lack of physical intuition in Euler’s mathematical genius and therefore, that he was a mathematician but not a physicist. Unfortunately, at that time, the results obtained by Euler had a purely theoretical significance, since two fixed gravitational centers is a system that does not occur in nature. Problems of the type Sun– Jupiter–Saturn could not serve as an adequate model since they did not satisfy the conditions of motion in the field of two fixed centers. In the work of E. P. Aksenov, E. A. Grebennikov, and V. G. Demin [22], a surprising field of application of the Euler problem was found, the motion of artificial Earth satellites in the field of a nonspherical planet. But Earth is, in fact, a nonspherical planet that can be considered as not a singular center but two imaginary centers. Moreover, we can also study the motion of a spacecraft in the gravitational field 420

of two planets, neglecting their displacement during the flight time of the spacecraft. Therefore, a field of application was found, and the problem was reborn. One more aspect of the two-center problem is the motion of a material point under the action of the Newton attraction force of a fixed center and one more force constant in magnitude and direction. This problem is the limit case of the two-center problem. In passing to the limit, the second center goes to infinity in the direction of the traction force (moreover, its must grow so as to ensure the constancy of the traction, i.e., proportional to the square of the distance). This problem was studied and integrated by Lagrange for the first time. The qualitative analysis of the Lagrange problem was carried out by V. V. Beletskii in [6] for the plane case. In the case of space flights, it is important to take into account the homogeneous field (under the action of which the spacecraft obtains a constant acceleration), as well as the field generated by the gravitational center, the planet. There arises the question on the extension of the properties of dynamical systems of celestial mechanics to the case of a curved space, in particular, the problem of the influence of curvature on the integrability of dynamical systems is of interest. It is natural to consider the case of spaces of constant (negative or positive) curvature. There are few works devoted to the study of dynamical systems in spaces of constant curvature, although this problem was posed by N. I. Lobachevskii for the first time when he studied the generalization of the attraction law for a space of constant curvature. The statement of the problem of dynamics itself in a space of constant curvature is often rather nontrivial. One of the main problems is related to the description of the potential generating a gravitating center. There are various approaches to generalization of the classical problems to the curvilinear case. Moreover, the systems considered in celestial mechanics have singularities and, formally speaking, are not Liouville integrable (the vector field generated by a Hamiltonian is not complete). In this work, we describe a certain regularization of systems after which the vector field becomes smooth and behaves regularly (integrably). Note that there are few integrable cases, and most of them are named by their discoverers. We present to the reader integrable problems of celestial mechanics on a sphere and on the Lobachevskii space. The problems under study are a natural generalization of the classical flat problems of celestial mechanics. However, the integrability of analogous problems on the sphere and on the Lobachevskii plane was found comparatively recently. Therefore, the topological properties of these problems have been little studied. In this work, we obtain a number of interesting results. In particular, some of the obtained topological invariants did not occur in the integrable cases studied earlier. The topology of their isoenergetic surfaces is also very different from what occurred earlier. In the work, we found certain new topological effects in problems of dynamics in spaces of constant curvature. At present attention to problems of such kind continue to grow. The proposed survey consists of 5 chapters. In the first chapter, we present main concepts, definitions, and theorems devoted to the integrability and qualitative analysis of dynamical systems, topological properties of integrable Hamiltonian systems, and the description of topological invariants and bifurcations. In the second chapter, we present the generalization of the Kepler problem to spaces of constant curvature. In this chapter, we present the results of studying the Kepler problem in a flat space, a sphere, and the Lobachevskii space. In the third chapter, we briefly present the results of studying the flat two-center problem obtained earlier. We study the motion of a point on a sphere (with metrics of constant positive curvature). We carry out the reduction of the system considered to the case of two degrees of freedom, prove the integrability, and write integrals of motions. Using qualitative methods, we carry out the bifurcation and topological analysis of the system considered and calculate the Fomenko–Zieshang invariants, which completely describe the topology of Liouville foliations of isoenergetic surfaces Q3 . We describe all types of motions (the regular motions and limit motions corresponding to bifurcations of Liouville tori) on the configurational spaces. We consider the relations between the Fomenko–Zieshang invariants (labelled molecules) and various types of motions. In the fourth and fifth chapters, we present the result of studying integrable problems of celestial mechanics in the Lobachevskii space. 421

In the fourth chapter, we study the two-center problem in the Lobachevskii space. We reduce the system considered to the case of two degrees of freedom, prove the integrability, and write the integrals of motion. We give a classification of regions of possible motion on the configurational space (we describe all types of motions: regular and limit motions). The following problem often arises in physics and mechanics. Let two Hamiltonian systems be given. It is required to show whether they are equivalent in the topological sense. In most cases, a practically unique method for solving this problem consists in calculating the corresponding Fomenko–Zieshang invariants. In Chapter 4, we prove the theorem on the Liouville equivalence of the two-center problem on a sphere, pseudosphere, and the plane under the condition of a constrained motion (in the case of a sphere, the constrained motion corresponds to motion on the upper hemisphere). We pass to the limit as λ → 0 (λ is the curvature of the space considered), and, as a result, we can make the conclusion that the integrable problems, i.e., the Kepler problem and the two-center problem, transform into one another in the spaces of constant curvature with change in two parameters: the curvature of the space and the distance between the centers. In Chapter 5, we study the generalization of the Lagrange problem to the case of the Lobachevskii space. This problem is the limit case of the two-center problem. In passing to the limit, the second center goes to infinity in the direction of the traction force. We obtain an analog of the homogeneous field in the Lobachevskii space and write the integrals of motion. We carry out the bifurcation and topological analysis. We describe pseudosphero-parabolic coordinates arising in passing to the limit as one of the centers goes to infinity. The Hamiltonian of the system has the Liouville form with respect to the new coordinates.

Chapter 1 MAIN CONCEPTS AND THEOREMS 1.

Hamiltonian Systems

Consider a vector field on a smooth manifold. Let x1 , . . . , xn be local coordinates; then the vector field can be written as dxi = ξ i (x1 , . . . , xn ), i = 1, . . . , n, (1) dt where ξ i (x1 , . . . , xn ) are smooth functions, being components of the field. Therefore, each vector field is interpreted as a set of ordinary differential equations on the manifold. The converse is also true: each set of ordinary differential equations describes a vector field on the corresponding manifold. In classical mechanics, the motion of a system can be described by using ordinary differential equations. Among mechanical systems, an important class of systems described by Hamiltonian equations is distinguished. These systems are represented on symplectic manifolds. A symplectic manifold. Definition
1.1. A symplectic structure on a smooth manifold M is a closed nondegenerate differential ωij (x)dxi ∧ dxj , i.e., the following conditions hold: 2-form ω = i<j

(1) dω =


i<j


 ∂ωij ∂ωjk ∂ωki dxk ∧ dxi ∧ dxj = 0 (closedness); dωij (x)dx ∧ dx = + + ∂xk ∂xi ∂xj i

j

k
(2) det Ω(x) = 0 ∀x, where Ω(x) = (ωij (x)) is the matrix of the form (nondegeneracy). Definition 1.2. A manifold equipped with a symplectic structure (M, ω) is said to be symplectic. 422

In contrast to a Riemannian structure, which exists on any manifold, a symplectic structure cannot be given on every manifold. For the existence of a symplectic structure on a manifold some necessary conditions should hold. Assertion 1.1. A symplectic manifold is of even dimension. Assertion 1.2. A symplectic manifold is orientable. The proofs of these assertions are presented in [8]. Note that the above conditions (even dimension and orientability) are not sufficient for the existence of a symplectic structure. In the symplectic geometry, the following Darboux theorem holds. Theorem 1.1 (Darboux). For any point x0 in an arbitrary symplectic manifold (M 2n , ω), there exists an open neighborhood U (x0 ) with local regular coordinates p1 , . . . , pn , q 1 , . . . , q n , in which the symplectic n
dpi ∧ dq i . form ω takes the canonical form ω = i=1

Correspondingly, the matrix of the canonical form is written as   0 E Ω= . −E 0 The coordinates in which the form becomes canonical are said to be canonical . We now present some examples of symplectic manifolds. 1. On any two-dimensional orientable Riemannian manifold, the area form can be introduced as a symplectic structure. 2. The linear symplectic space M 2n = R2n (p1 , . . . , pn , q 1 , . . . , q n ). The symplectic structure is independent of a point and is written as   0 E 1 n Ω= . ω = dp1 ∧ dq + · · · + dpn ∧ dq , −E 0 3. Let M n be an arbitrary smooth manifold and let T ∗ M be its cotangent bundle. The cotangent bundle T ∗ M of the smooth manifold M is the set of pairs of the form {(x, ξ)| x ∈ M, ξ ∈ Tx∗ M }, where the covector ξ is a linear functional. Assertion 1.3. On the cotangent bundle T ∗ M of an arbitrary smooth manifold M , a natural symplectic structure can be defined. Proof. Consider the projection p : T ∗ M → M , p(x, ξ) = x, i.e., the whole fiber is projected on the point x. Further, on T ∗ M , construct a certain 1-form α (α is a linear functional on the set of tangent vectors T ∗ M ), called the action form. Let a be a tangent vector to the manifold T ∗ M at a certain point (x, ξ), where x ∈ M , ξ ∈ Tx∗ M , and a ∈ T (T ∗ M ). Define the value of the form α on the vector a by α(a) = ξ(dp(a)), where dp : T(x,ξ) (T ∗ M ) → Tx M . The form ω = dα is a symplectic structure on the manifold T ∗ M . In local coordinates, the velocity vector and the covector are written as   i dx dξi , , ξ = (ξ1 , . . . , ξn ). a= dt dt Correspondingly, the action form is written as α(a) = ξ(dp(a)) =

n
i=1

ξi

dxi . dt 423

As a symplectic structure on T ∗ M , take the form ω = dα =

n


dξi ∧ dxi . Obviously, it satisfies all the

i=1

necessary conditions

 2

dω = d α = 0,

Ω=

 0 E . −E 0

Hamiltonian vector fields. Let H be a smooth function on a symplectic manifold (M 2n , w). For this function, define the vector of skew-symmetric gradient sgrad H from the identity w = (v, sgrad H), where v is an arbitrary tangent vector and v(H) is the derivative of H in direction v. In local coordinates x1 , . . . , x2n , we obtain the following expression: (sgrad H)i = ω ij

∂H , ∂xj

where ω ij are entries of the matrix inverse to Ω (the summation is carried out with respect to repeated superscripts and subscripts).  Definition 1.3. Vector fields of the form sgrad H are called Hamiltonian vector fields. The function H is called the Hamiltonian of the vector field sgrad H. In local symplectic coordinates q 1 , . . . , q n , p1 , . . . , pn , which always exist in a neighborhood of any point of a manifold, the Hamiltonian system is written as follows:  i ∂H dq    dt = ∂p , i i = 1, , . . . , n,  ∂H ∂p  i  =− i, ∂t ∂q H(q, p) = H(q 1 , . . . , q n , p1 , . . . , pn ), i.e., in appropriate coordinates, the components of the Hamiltonian vector fields are   ∂H ∂H ∂H ∂H ,..., ,− ,...,− n . sgrad H = ∂p1 ∂pn ∂q 1 ∂q Definition 1.4. A function f on a manifold is a first integral of a vector field v if it is constant on all integral trajectories of the system, i.e., f (γ(t)) = const, where γ(t) is an integral trajectory of our system. Definition 1.5. An integral trajectory of the system is a smooth curve whose velocity vector at each point of it coincides with the vector of the field v at this point: dγ(t) = v(γ(t)). dt The equivalent definition is as follows. Definition 1.6. A function f is a first integral of the vector field of the system if its derivative in the direction of the field v vanishes: Lv f = 0. (Correspondingly, for a Hamiltonian system with Hamiltonian H, we have Lsgrad H f ≡ 0.) 424

Fig. 1 f = const H = const v

Fig. 2 This notation L is chosen in honor of Sophus Lie: n
∂f vi i , Lv f = ∂x i=1

where the derivatives are taken at the point of application of the vector; here, xi are coordinates in a neighborhood of this point and vi are components of the velocity vector in this coordinate system. Theorem 1.2. The Hamiltonian function is a first integral of the Hamiltonian vector field. Proof.


 n  n 
∂H ∂H ∂H ∂H ∂H i ∂H x˙ = 0, Lsgrad H H = + p˙i i = − ∂xi ∂p ∂pi ∂xi ∂xi ∂pi i=1

i=1

which is what was required to prove. If a function f is an integral of a field v, then this field is tangent to level surfaces of f . Therefore, each Hamiltonian vector field is always tangent to a level surface of its Hamiltonian, i.e., the hypersurfaces given by the equation H = const are invariant with respect to the field v, and integral trajectories of the field fill in these surfaces and do not leave them (Fig. 2). Let a Hamiltonian vector field v have an additional integral f , and, moreover, let the integrals H and f be functionally independent. Definition 1.7. Two functions are said to be functionally independent on a manifold if their gradients grad H and grad f are linearly independent almost everywhere. Then the field v is tangent to the surfaces H = const, as well as to the surfaces f = const (both surfaces are invariant with respect to the field v), which, obviously, are of dimension 2n − 1. Therefore, the field v is tangent to (2n − 2)-surfaces. These surfaces are intersections of the surfaces H = const and f = const. Therefore, we can reduce the dimension of the set of ordinary differential equations. Hamiltonian phase flows. Definition 1.8. A one-parameter group of diffeomorphisms of a manifold M is a smooth mapping F : R× M → M such that the following conditions hold: 1. ϕ0 (x) = x, x ∈ M , 2. ϕt1 +t2 = ϕt1 ◦ ϕt2 , where ϕt (x) = F (t, x). 425

ϕt (x) ξ(x)

Fig. 3 The mapping ϕt : M → M defines the smooth vector field ξ(x) on M by  d  ξ(x) =  ϕt (x) ∈ Tx M, dt t=0

ϕt (x)

and is an orbit of a point x. The converse is also true. Given a vector field ξ(x), we can construct a one-parameter group of diffeomorphisms of M (at least, locally). Let ξ i (x1 , . . . , xn ) be a C 1 -smooth vector field. Consider an arbitrary point x on the manifold and draw an integral curve γ(t) through it such that γ(0) = x. Then we can reconstruct the one-parameter group ϕt setting ϕt (x) = γ(t). Here ϕt is the shift along the integral trajectory γ(t) of the vector field ξ(t) by time t. Definition 1.9. A vector field ξ(x) is said to be complete if each integral curve can be continued from −∞ up to ∞. Assertion 1.4. If the completeness condition holds, then there exists a one-to-one correspondence between one-parameter groups of diffeomorphisms ϕt and complete vector fields. Poisson brackets. Definition 1.10. Let (M 2n , ω) be a symplectic manifold and let C ∞ (M 2n ) be the space of all smooth functions on this symplectic manifold. Let f and g be two smooth functions, f , g ∈ C ∞ (M 2n ). By definition, we set {f, g} = ω(sgrad f, sgrad g), {f, g} ∈ C ∞ (M 2n ). In local coordinates, Poisson brackets become {f, g} = ωαβ (sgrad f )α (sgrad g)β ∂f ∂g ∂f ∂g ∂f ∂g = ωαβ ω αi i ω βj j = ω βj δβi i j = ω ij i j ∂x ∂x ∂x ∂x ∂x ∂x
∂f (we have used that (sgrad f )i = wij j ). ∂x j

Assertion 1.5. {f, g} = (sgrad f )g = −(sgrad g)f . Proof. {f, g} = w(sgrad f, sgrad g) = (sgrad f )g. The assertion is proved. We now formulate the properties of the Poisson bracket in the form of theorems. Theorem 1.3. The Poisson bracket satisfies the following properties: 1. skew-symmetry: {f, g} = −{g, f }; 2. bilinearity of the field of real numbers: {αf + βh, g} = α{f, g} + β{h, g} 426

α, β ∈ R;

3. the Poisson bracket {·, ·} satisfies the Jacobi identity {f, {g, h}} + {h, {f, g}} + {g, {h, f }} ≡ 0; 4. the Leibnitz formula {h, f g} = {h, f }g + f {h, g} holds. Proof. The bilinearity and skew-symmetry of the Poisson bracket are obvious. Let us prove the Jacobi identity. The following Cartan formula is well known: dw(ξ, η, ζ) = ξw(η, ζ) − w([ξ, η], ζ) + (cyclic permutation), where w is an arbitrary 2-form and ξ, η, and ζ are arbitrary vector fields. Let ξ = sgrad f , η = sgrad g, ζ = sgrad h, and ω be a symplectic structure. Since a symplectic structure is closed by definition, we have sgrad f (ω(sgrad g, sgrad h)) − ω([sgrad f, sgrad g], sgrad h) + (cyclic permutation) = 0. By the definition of the Poisson bracket, we can rewrite this expression as sgrad f ({g, h}) − [sgrad f, sgrad g](h) + (cyclic permutation) = 0. Rewriting this once more, we obtain the Jacobi identity {f, {g, h}} − sgrad f (sgrad g(h)) + sgrad g(sgrad f (h)) +(cyclic permutation) = {g, {f, h}} + (cyclic permutation) = 0. We now obtain the Leibnitz rule {h, f h} = (sgrad h)(f h) = ((sgrad h)f )g + f (sgrad h)g = {h, f }g + f {h, g}. Remark 1. Property 3, i.e., the Jacobi identity for the Poisson bracket {·, ·}, is equivalent to the fact that the 2-form ω is closed, i.e., dω = 0. Remark 2. In constructing Hamiltonian mechanics, instead of a symplectic structure, one takes the Poisson bracket as the initial structure. In this case, the Poisson bracket is not assumed to be necessarily nondegenerate. Definition 1.11. If the Poisson bracket {f, g} of two functions f and g vanishes on a symplectic manifold (M, ω), then we say that the functions f and g are in involution. Assertion 1.6. A function f is a first integral of a Hamiltonian vector field iff {f, H} ≡ 0. Proof. Recall that a function f is a first integral of a Hamiltonian vector field if v(f ) ≡ 0. We have {f, H} = (sgrad H)f = v(f ) ≡ 0, which is what was required to prove. Corollary. A Hamiltonian system v = sgrad H always has a first integral ; this is the Hamiltonian H. In fact, we have formulated Theorem 1.2. Proof. Let us prove Theorem 1.2 in terms of Poisson brackets. Indeed, {H, H} = ω(sgrad H, sgrad H) = 0. Assertion 1.7. Let f and g be two first integrals of a Hamiltonian vector field sgrad H; then their Poisson bracket is also a first integral. 427

Proof. The proof follows from the Jacobi identity. Indeed, using Assertion 1.6, we obtain {H, {f, g}} = {f, {H, g}} − {g, {H, f }} = 0. Assertion 1.8. Any linear combination of first integrals λf + µg and their product f g is also a first integral. Proof. The bilinearity property of the Poisson bracket implies {λf + µg, H} = λ{f, H} + µ{g, H} ≡ 0, since f and g are first integrals. The Leibnitz formula implies {f g, H} = f {g, H} + {f, H}g ≡ 0, which is what was required to prove. Theorem 1.4 (Noether). Let H be the Hamiltonian of a dynamical system. Let σ t be the one-parameter group generated by the Hamiltonian f . Let H be invariant with respect to σ t . Then f is a first integral of the Hamiltonian system sgrad H. Proof. By the assumption of the theorem, H is a first integral for the flow of f , i.e., {f, H} = 0. The converse is also true: f is a first integral for the flow of H. Definition 1.12. A manifold M of dimension n (not necessarily even) is said to be Poisson if the following linear operation {·, ·} defined on the set of smooth functions {·, ·} : C ∞ (M ) × C ∞ (M ) → C ∞ (M ) is given:
∂f ∂g Aij i j , {f, g} = ∂x ∂x i,j

where Aij is a skew-symmetric matrix, not necessarily nonsingular, or, in other words, a skew-symmetric tensor field, i.e., a 2-form on the cotangent bundle of the manifold smoothly depending on a point. This operation satisfies the Jacobi identity. The operation {·, ·} defines the Poisson structure, and the relation on the components of the tensor Aij is exactly equivalent to the Jacobi identity. Remark 3. The terms “Poisson bracket” and “Poisson structure” are often used as synonyms. Remark 4. The Poisson structure can be degenerate. Remark 5. Any symplectic manifold is Poisson, but the converse is not true (for example, an odddimensional manifold is not symplectic). Remark 6. The condition of writing a form in the canonical form can be equivalently written in terms of Poisson brackets: {pi , pj } = 0,

{pi , q j } = δij ,

{q i , q j } = 0,

i, j = 1, . . . , n.

Remark 7. On the cotangent bundle T ∗ Rn , in terms of the Poisson bracket, the Hamiltonian equations are written as follows: dp dq = {q, H}, = {p, H}. (2) dt dt Let (M, ω) be a symplectic manifold and let C ∞ (M ) be the space of smooth functions on M . Let the Poisson bracket {f, g}, f, g ∈ C ∞ (M ) be defined on this space. Theorem 1.5. The Hamiltonian system sgrad H preserves the symplectic structure, i.e., Lsgrad H ω = 0. 428

(3)

Proof. We want to prove that the form vanishes on any two vectors. Consider arbitrary functions f and g. Since the form is nondegenerate, without loss of generality we can assume that the vectors sgrad g and sgrad f are arbitrary. We have Lsgrad H [ω(sgrad f, sgrad g)] = Lsgrad H {g, f } = {H, {g, f }}. We use the Leibnitz rule and write Lsgrad H [ω(sgrad f, sgrad g)] = (Lsgrad H ω)(sgrad f, sgrad g) +ω(Lsgrad H sgrad f, sgrad g) + ω(sgrad f, Lsgrad H sgrad g) (here, we have used the property Lsgrad H sgrad f = sgrad {H, f }). Then using the Jacobi identity, we obtain the desired result. Corollary (Liouville theorem). A Hamiltonian system sgrad H preserves a smooth measure (volume form): Lsgrad H ω ∧ . . . ∧ ω = 0. Definition 1.13. The rank of the Poisson bracket at a fixed point x ∈ M is the rank of the tensor Aij (x). The rank of the Poisson bracket on the whole manifold M is the number R = max rank Aij (x). x∈M

Definition 1.14. A function f : M → R is called a central function or a Casimir function of the Poisson bracket {·, ·} if {f, g} = 0 for any smooth function g. If the Poisson bracket is nondegenerate, i.e., its rank is equal to dim M , then only constants are central functions. With each smooth function H on a Poisson manifold M , we can associate the vector field v = sgrad H given by v(g) = {H, g} for any smooth function g, or, in local coordinates, ∂f v i = (sgrad H)i = Aij j . ∂x Vector fields of the form v = sgrad H are said to be Hamiltonian vector fields. The following theorem helps us to understand the structure of Poisson manifolds and their relation to symplectic manifolds. A Poisson manifold falls into symplectic fibers each of which is locally a symplectic submanifold whose dimension is equal to the rank of the Poisson bracket at any its point. The tangent space of a symplectic fiber at a point x ∈ M is composed of vectors of the form sgrad H(x). We now present the definition of the symplectic fiber. Two points x, y ∈ M of a Poisson manifold are said to be equivalent if there exists a piecewise-smooth curve connecting them each segment of which is a trajectory of a Hamiltonian vector field. Then the symplectic fiber passing through the point x is the equivalence class of this point. Clearly, if the Poisson bracket is nondegenerate, then the Poisson manifold is symplectic. 2.

Lie Groups and Algebras. Noether Theorem

In studying Hamiltonian systems, we inevitably encounter such a phenomenon as symmetries. The existence of integrals of motion is one of the main properties of a Hamiltonian system with symmetry. The theory of Lie groups and algebras is a working tool for studying symmetries. We now give a formal definition of a Lie algebra. Definition 2.1. Let G be a linear space over the fields of real or complex numbers. The space G is called a Lie algebra if in this space there is an operation [X, Y ] satisfying the following properties: 1. [αX + βY, Z] = α[X, Z] + β[Y, Z] for α, β ∈ R(C) (bilinearity); 2. [X, Y ] = −[Y, X] (skew-symmetry); 429

3. [X, [Y, Z]] + [Y, [Z, X]] + [Z, [X, Y ]] = 0 (Jacobi identity). Definition 2.2. Let G be a finite-dimensional Lie algebra, and let e1 , . . . , en be its basis. Then [ei , ej ] = ckij ek . The tuple ckij is a tensor called a structure tensor. In terms of the structure tensor, Property 2 is rewritten in the form ckij = −ckji , and Property 3 in the form cbpk cpij + cbpi cpjk + cbpj cpki = 0. Remark. The Poisson bracket transforms the space of smooth functions on a manifold M into an infinitedimensional Lie algebra. The algebra of matrices M (n, Rn ) of order n in which the Lie product is given by [X, Y ] = XY − Y X is the simplest example of a Lie algebra. Let ξ and η be two vector fields on a manifold M . A vector field can be understood as a differential operator defined on scalar fields ϕ. Consider the commutator [ξ, η] = ξη − ηξ (here, ξη is the composition of the operators ξ and η). Assertion 2.1. Let (x1 , . . . , xn ) be a local coordinate system on a manifold M n with respect to which the coordinates of vector fields ξ = (ξ 1 , . . . , ξ n ) and η = (η 1 , . . . , η n ) on it are calculated ; then [ξ, η] is a vector ∂η i ∂ξ i field with the components [ξ, η]i = ξ k k − η k k . ∂x ∂x Proof. Calculate the commutator       j j ∂ϕ i ∂ j ∂ϕ i ∂ j ∂ϕ i ∂η i ∂ξ η −η ξ = ξ −η . [ξ, η]ϕ = ξηϕ − ηξϕ = ξ ∂xi ∂xj ∂xi ∂xj ∂xi ∂xi ∂xj Here, we have taken into account that ξiηj

2 ∂2ϕ j i ∂ ϕ − ξ η =0 ∂xi ∂xj ∂xj ∂xi

by the relation ∂2ϕ ∂2ϕ = . ∂xi ∂xj ∂xj ∂xi The assertion is proved. Definition 2.3. The vector field [ξ, η] is called the commutator of the vector fields ξ and η. Obviously, the space V (M ) of vector fields on a smooth manifold M composes a Lie algebra with respect to the commutator [ξ, η] of vector fields. The Lie algebra of vector fields is infinite-dimensional. Definition 2.4. For an arbitrary Lie algebra G, define the linear mapping ad ξ : G → G, ξ ∈ G, by the relation ad ξ (η) = [ξ, η]. The mapping ad ξ is a derivation of the Lie algebra G: ad ξ ([η, ζ]) = [ad ξ (η), ζ] + [η, ad ξ (ζ)]. Definition 2.5. A subspace H in a Lie algebra G is called a subalgebra if [ξ, η] ∈ H for any ξ, η ∈ H. Definition 2.6. A subalgebra K in a Lie algebra G is called an ideal if [ξ, η] ∈ K for any ξ ∈ K, η ∈ G. Definition 2.7. A Lie algebra G is said to be commutative (or Abelian) if [ξ, η] = 0 for any its elements. If G is a commutative Lie algebra, then any its subspace is a subalgebra and an ideal. The subspace Z(G) = {ξ ∈ G|ad ξ = 0} is called the center of a Lie algebra G; Z(G) is an ideal in G. Definition 2.8. A homomorphism of a Lie algebra G1 into a Lie algebra G2 is a linear mapping f : G1 → G2 such that f [ξ, η] = [f (ξ), f (η)]. The subspace Ker f = {ξ ∈ G1 |f (ξ) = 0} is called the kernel of the homomorphism f . 430

Assertion 2.2. The kernel of any homomorphism is an ideal. Let G be an arbitrary Lie algebra. Define a chain of ideals G ⊃ G(1) ⊃ · · · ⊃ G(i) ⊃ · · · by G(i) = (G(i−1) )(1) , where G(1) is the subalgebra, which is the linear span of commutators [ξ, η], ξ, η ∈ G. For a finite-dimensional Lie algebra G, this chain stabilizes, i.e., there exists a number i such that G(i) = G(i+1) = . . .. Definition 2.9. If a chain of derivation Lie algebras, G ⊃ G(1) ⊃ · · · ⊃ G(i) ⊃ · · · , is stabilized at zero, then G is called a solvable Lie algebra. Definition 2.10. The radical of a Lie algebra G is a maximal solvable ideal in G. Definition 2.11. A Lie algebra is said to be semisimple if its Killing form is nondegenerate. Definition 2.12. The value of the Killing form on two elements of a Lie algebra is defined by (ξ, η) = Tr ad ξ ad η . Assertion 2.3. If the radical J of a Lie algebra G is equal to zero, then G is a semisimple Lie algebra. Definition 2.13. A Lie algebra G is said to be reductive if G = H ⊕ G1 , where H is an Abelian Lie algebra, G1 is a semisimple Lie algebra, and [ξ, η] = 0 for ξ ∈ H and η ∈ G. Definition 2.14. A manifold G is called a Lie group if it is a group, and, moreover, the mappings ϕ and ψ defining the group structure are smooth: ϕ : G → G, where ϕ(g) = g −1 , ψ : G × G → G, where ψ(g, h) = gh. For each element g ∈ G of a Lie group G, two diffeomorphisms Rg : G → G and Lg : G → G, called the left and right shifts, respectively, are defined by Rg (x) = xg and Lg (x) = gx, respectively, where x ∈ G. Definition 2.15. A vector field ξ on a Lie group G is said to be left-invariant (right-invariant) if (La )∗ ξ = ξ for any a ∈ G ((Ra )∗ ξ = ξ). Each left-invariant vector field is uniquely defined by its value at the identity e ∈ G of a Lie group G. The space of left-invariant vector fields on a Lie group is a finite-dimensional linear subspace in the space of all vector fields. Its dimension is equal to the dimension of the Lie group. If ξ and η are left-invariant vector fields, then their commutator [ξ, η] is a left-invariant vector field. Definition 2.16. The Lie algebra G of left-invariant vector fields on a Lie group G is called the Lie algebra of the group G. Assertion 2.4. The operator sgrad defines a homomorphism of the Lie algebra of smooth functions on a manifold into the Lie algebra of vector fields, i.e., for any two functions f and g on a symplectic manifold, sgrad {f, g} = [sgrad f, sgrad g]. Proof. Write the Jacobi identity {h, {f, g}} + {g, {h, f }} + {f, {g, h}} = 0 in the form −(sgrad {f, g})h − (sgrad g)(sgrad f )h + (sgrad f )(sgrad g)h = 0. Then we obtain (sgrad {f, g})h = −(sgrad g)(sgrad f )h + (sgrad f )(sgrad g)h = [sgrad f, sgrad g]h, which is what was required to prove. Corollary. Hamiltonian vector fields compose a subalgebra in the Lie algebra of all vector fields. 431

Definition 2.17. A smooth mapping f : R1 → G is called a one-parameter subgroup in a Lie group G if the following conditions hold: 1. f (t1 + t2 ) = f (t1 )f (t2 ) for all t1 , t2 ∈ R1 ; 2. f (0) = e, where e is the identity of the Lie group G. We say that a Lie group G acts on a manifold M if there exists a smooth mapping h : G × M → M such that the following conditions hold: ˆ1 · gˆ2 , g1 , g2 ∈ G; 1. g 1 g2 = g 2. eˆ = id , where gˆ(x) = h(g, x) and id (x) = x is the identity mapping. There is a natural correspondence between the following objects: 1. one-parameter subgroups in a Lie group G; 2. tangent vectors ξ ∈ Te G at the identity of the Lie group G; 3. left-invariant vector fields on the Lie group G. All one-parameter subgroups of a Lie group G can be collected into one universal mapping exp : G → G under which exp tξ : R1 → G → G defines a one-parameter subgroup with the velocity vector ξ at the identity e ∈ G. The mapping exp is called the exponential mapping of the Lie group G (G is the Lie algebra of the Lie group G). ˆ ξ ∈ G, ξˆ ∈ V (M ) of Let a Lie group G act on a manifold M . Then the tangent homomorphism ξ → ξ, the Lie algebra G of the Lie group G into the Lie algebra V (M ) on the manifold M is defined by  d 
ˆ ξ(x) =  exp(tξ)(x). dt t=0

The Lie group G acts on G by the conjugation gˆ(x) = gxg −1 , x ∈ G. The operator gˆ(x) transforms the identity element of the group G into itself. This action is said to be adjoint and is denoted by ig (x) = gˆ(x). It generates the linear action of the Lie group G on the Lie algebra G denoted by Ad . This representation is called the adjoined representation of the Lie group G:  d  Ad g x =  g exp(tx)g −1 . dt t=0

Let G∗ be the space dual to G, i.e., the space of all linear mappings α : G → R. The co-adjoined representation Ad ∗ of the Lie group G is defined by Ad ∗g ξ(x) = ξ(Ad g−1 x), where g ∈ G, ξ ∈ G∗ , x ∈ G. Let M be a manifold, F(M ) be the space of functions on M endowed with the Poisson structure, and let a Lie group G act on M . Then the Lie algebra G of the group G also acts on M (as the Lie algebra of ˆ vector fields on M ). That is, to an element ξ ∈ G, we put in correspondence the vector field ξ(x) on M . ˆ Assume additionally that the action is such that to each field ξ(x), it is possible to put in correspondence the function fξ on M such that ˆ ξ(x) = sgrad fξ (x),

fξ+η = fξ + fη .

Such an action of the group G is said to be Hamiltonian if f[ξ,η] = {fξ , fη }, (i.e., the correspondence ξ → fξ is a homomorphism of the Lie algebra G and the Lie algebra of functions on M ). We now formulate the Noether theorem in the following form. Theorem 2.1 (Noether). If a Hamiltonian H is invariant with respect to a group G whose action on M is Hamiltonian, then the vector field sgrad H has the algebra of first integrals isomorphic to the algebra G. These first integrals are called Noether integrals and are fξ . 432

3.

Integrability

Which set of differential equations can be considered as integrable (solvable)? The answer depends on what we mean by integrability. The classical approach is well known: a set of differential equations is integrable if its solution can be obtained by finitely many algebraic operations (including the solution of sets of algebraic equations) and quadratures, i.e., assuming that primitives of known functions are given. Such sets are said to be integrable in quadratures. Let a dynamical system given on a manifold M be described by a smooth vector field v(x) = (v1 , . . . , vn ): x˙ i = vi (x1 , . . . , xn ).

(4)

Its phase flow is a one-parameter family of diffeomorphisms, σ t : M → M . We now present a sufficiently simple, but important, integrability theorem. Theorem 3.1. If the set of n differential equations x˙ = v(x)

(5)

admits n − 1 independent first integrals almost everywhere, then it is integrable in quadratures. Proof. To prove this, we make the change of variables taking n − 1 integrals as new coordinates x2 , . . . , xn . Then system of equations (5) becomes x˙ 2 = · · · = x˙ n = 0.

x˙ 1 = v1 (x1 , x2 , . . . , xn ),

Therefore, xk = ck = const (k ≤ 2), and the coordinate x1 can be found by inverting the qudrature

dx1 . t= v1 (x1 , c)

In accordance with the theorem of strengthening of trajectories, in a small neighborhood of a nonequilibrium point of the phase flow of v (at which v is different from zero), there always exist n−1 independent integrals. However, only in exceptional cases can these local integrals be extended to functions on the whole phase space. We present one more important theorem on integrability whose proof is based on the previous theorem. Theorem 3.2 (Euler–Jacobi). If a set of n differential equations admits an invariant measure and n − 2 independent first integrals almost everywhere, then it is integrable in quadratures. 3.1. Separation of variables. Hamilton–Jacobi equation. Some integrable Hamiltonian systems can be explicitly solved by using a simple and effective separation variable method. According to Jacobi, the problem of solving the canonical Hamiltonian equations q˙ = p˙ = −

∂H , ∂q

∂H , ∂p (p, q) ∈ R2n ,

reduces to finding the complete integral of the Hamilton–Jacobi equation [2, 25]   ∂V ∂V +H , q, t = 0. ∂t ∂q

(6)

The complete integral is an n-parameter family of solutions V (t, q, x) satisfying the nondegeneracy condition  2  ∂ V   det   ∂q∂x  = 0. 433

If the Hamiltonian H(p, q) does not explicitly depend on time, then by using the change V (q, t) = −Kt + W (q), Eq. (6) reduces to   ∂W , q = K(x). (7) H ∂q The function W (q, x) is the complete integral of (7). It can be taken as a generating function of the canonical transformation (p, q) → (y, x): y=

∂W , ∂x

p=

∂W ; ∂q

moreover, the following condition holds:

 2   2  ∂ W  ∂ V      det  = det  ∂q∂x  = 0. ∂q∂x 

In new canonical coordinates x, y, the function H(p, q) = K(x) is independent of y, and, therefore, the Hamiltonian equations are immediately integrated: ∂K . ∂x To define the function K uniquely, we use additional conditions. As usual, we choose K(x1 , . . . , xn ) = xn : the trajectories of the system with such a Hamiltonian are straight lines in R2 = (x, y). If the Hamilton–Jacobi equation admits a complete integral of the form x = x0 ,

y = y0 + ω(x0 )t,

W (q, x) =

n


ω(x) =

Wk (qk , x1 , . . . , xn ),

k=1

then we say that the variables q1 , . . . , qn are separated . The following assertion holds. Assertion 3.1. Assume that in certain symplectic coordinates (p, q) = (p1 , . . . , pn , q1 , . . . , qn ), the Hamiltonian function H(p, q) has one of the following forms: (1) H = fn (fn−1 (. . . f2 (f1 (p1 , q1 ), p2 , q2 ), . . . , pn−1 , qn−1 ), pn , qn ),

(2) H = fs (ps , qs ) gs (ps , qs ) . Then the functions (1) F1 = f1 (p1 , q1 ), F2 = f2 (f1 (p1 , q1 ), p2 , q2 ), . . . , Fn = H, (2) F0 = H, Fs = fs (ps , qs ) − Hgs (ps , qs ), 1 ≤ s ≤ n, compose a complete tuple of integrals in involution of the Hamiltonian system with the Hamiltonian H. Liouville systems. Definition 3.1. A dynamical system is called a Liouville system if there exist coordinates qj in which the Hamiltonian is written as the sum of the potential energy U and the kinetic energy T : H = T + U, where

n n 1
1
q˙j2 = aj p2j , T = C 2 aj 2C j=1

U=

j=1

n 1
uj , C j=1

The functions aj , cj , and uj depend only on the variables qj . For this system, q˙j = 434

aj pj . C

C=

n
j=1

cj .

Such systems can be easily integrated. It is easy to verify that 1 Ij = aj p2j + uj − Hcj , j = 1, 2, . . . , n, 2 are integrals of motion. These integrals are called Liouville integrals. It should be noted that only (n − 1) integrals are independent, since n
Ij = 0. j=1

Therefore, taking into account the Hamiltonian H, we have n quadratic integrals of motion. Obviously, these integrals are in involution: {Ij , Ik } = 0, and, therefore, the system is completely integrable. The equations of motion can be integrated, for example, in the following way. From the relations Ij = αj = const we obtain the set of differential equations for qj : dqj dt , = aj pj C(q1 , . . . , qn )

p2j =

2(αj + Ecj − uj ) . αj

Here, E is the value of the complete energy. Therefore, dqj dt

= , C(q1 , . . . , qn ) 2aj (αj + Ecj − uj )

j = 1, . . . , n.

We now pass to a new local time τ according to dτ =

dt . C(q1 , . . . , qn )

Then we obtain the system dqj

= dτ. 2aj (αj + Ecj − uj ) As a result, using quadratures, we find qj = fj (τ ). Then, using the qudrature

τ t = C(q1 (τ  ), . . . , qn (τ  )) dτ  , we express τ in terms of t. That is, the solution of the problem reduces to the solution of sequential one-dimensional problems. Systems with two degrees of freedom. Consider a system with two degrees of freedom. The configurational space in this problem is M 2 (x1 , x2 ). The Lagrangian has the form 1 ϕ1 (x1 ) + ϕ2 (x2 L = (λ1 (x1 ) + λ2 (x2 ))(α1 (x1 )x˙ 21 + α2 (x2 )x˙ 22 ) − , 2 λ1 (x1 ) + λ2 (x2 ) where λ1 (x1 ) + λ2 (x2 ) = 0, α1 (x1 ) > 0, α2 (x2 ) > 0. The system under consideration has the energy integral ϕ1 + ϕ2 1 H = (λ1 + λ2 )(α1 x˙ 21 + α2 x˙ 22 ) + 2 λ1 + λ2 and the additional integral (Liouville integral) ϕ1 λ2 − ϕ 2 λ1 1 . L = (λ2 α1 x˙ 21 − λ1 α2 x˙ 22 ) + 2 λ1 + λ2 435

Solving the system of first integrals with respect to the generalized velocities x˙1 and x˙2 , we obtain from the conditions H = h and L = l that 1 [λ1 (x1 ) + λ2 (x2 )]a1 (x1 )x˙1 2 + Φ1 (x1 , h, l) = 0, 2 (8) 1 [λ1 (x1 ) + λ2 (x2 )]a2 (x2 )x˙2 2 + Φ2 (x2 , h, l) = 0, 2 where Φ1 (x1 , h, l) = ϕ1 (x1 ) − hλ1 (x1 ) − l, (9) Φ2 (x2 , h, l) = ϕ2 (x2 ) − hλ2 (x2 ) + l. We see from Eqs. (8) that the regions of possible motion are defined by the condition Φ1 (x1 , h, l) ≤ 0, Let

Φ2 (x2 , h, l) ≤ 0.

Ω1 = {x1 : Φ1 (x1 , h, l) ≤ 0} ⊂ R, Ω2 = {x2 : Φ2 (x2 , h, l) ≤ 0} ⊂ R.

(10)

Then we have (11) Ωh,l = Ω1 × Ω2 . Clearly, the boundary [24] of regions of possible motion (RPM) in this problem is determined (in the case of two degrees of freedom) by the condition rank

∂(h, l) < 2, ∂(x˙ 1 , x˙ 2 )

i.e., we have



 x˙ 1 x˙ 2 [λ1 (x1 ) + λ2 (x2 )]a1 (x1 )a2 (x2 ) det < 2. λ2 (x2 )x˙ 1 −λ1 (x1 )x˙ 2 Obviously, this condition holds in the following two cases. If x˙ 1 = 0, then (8) implies Φ1 (x1 , h, l) = 0, and if x˙ 2 = 0, then Φ2 (x2 , h, l) = 0. It follows from (10) and (11) that the type of motion changes only in passing through those values of the first integrals h and l for which at least one of the functions Φ1 (x1 , h, l) and Φ2 (x2 , h, l) has a multiple root. In this case, the vectors grad h and grad l are dependent. If sets (10) are bounded, then each connected component of RPM for (h, l) ∈ Σ is a rectangle inside which we have four admissible velocities, on its sides we have two, and at the vertices of the rectangle we have only one (zero velocity). The trajectories in the rectangle are similar to Lissajous figures. In the general case, the change of the type of RPM can also occur for (h, l) that are not critical values of the momentum mapping (see Sec. 5). St¨ ackel systems. Liouville systems are a particular case of the St¨ ackel systems. These systems were discovered by St¨ ackel in 1891. Theorem 3.3. Let Φ be the determinant of the matrix ϕij (qj ), 1 ≤ i, j ≤ n, and let Φij be the algebraic complement of the entry ϕij . Assume that in the canonical coordinates p1 , . . . , pn , q1 , . . . , qn , the Hamiltonian function has the form n
Φ1s (q)fs (ps , qs ) . (12) H(p, q) = Φ(q) s=1

Then the Hamiltonian equations have n independent integrals in involution, n
Φks fs , k = 1, . . . , n, Fk = Φ s=1

436

and can be explicitly integrated. Let K(x) = x1 ; write Eq. (7): 
 
∂W = 0. xk ϕkm (qm ) − fm , qm Φ1m ∂qm k

Its complete integral can be found as the sum
Wm (qm , x1 , . . . , xn ), W (q, x) = m

where Wm (as a function of qm ) satisfies the equation  
∂W , qm = xk ϕkm (qm ). fm ∂qm k

It remains to solve this ordinary differential equation. If the system is Liouville (particular case), then the Hamiltonian function becomes   n
p2j 1
+ Cj , Bj 2 Aj j=1



where the functions Aj , Bj , and Cj depend only on the coordinate qj and Bj , Cj are different from zero. For n = 2, each St¨ ackel system can be reduced to the Liouville form. Both types of systems often occur in applications to celestial mechanics. Certain facts from the elliptic function theory. We present definitions of Jacobi elliptic functions, since they will be used in what follows in integrating equations of motion. The integral

ϕ dx

(13) u = F (ϕ, k) = 1 − k 2 sin2 x 0

is called an elliptic integral of the first kind. The value k is called the modulus of the elliptic integral. It is assumed as usual that k satisfies the inequality 0 ≤ k < 1. The quantity K(k) = F

π 2



π/2

,k = 0

dx

1 − k 2 sin2 x

(14)

is called the complete elliptic integral of the first kind. It can be represented in the form of the following convergent series in powers of k:   1 2 π 1 + k + ··· . (15) K(k) = 2 4 The function, being the result of inversion of an elliptic integral of the first kind, is called the amplitude and is denoted by ϕ = amu. (16) The functions z = sn(u, k) (elliptic sine) and z = cn(u, k) (elliptic cosine) are defined by z = sn(u, k) = sin ϕ = sin amu,

(17)

z = cn(u, k) = cos ϕ = cos amu.

(18)

Since sin ϕ and cos ϕ are of period 2π with respect to ϕ, according to (13) and (14), the elliptic sine and elliptic cosine are of period 4K(k) with respect to u. 437

dnu

1

−1

snu 2K K cnu

u 3K 4K

Fig. 4

The function of delta-amplitude z = dn(u, k) is defined by   dϕ z = dn(u, k) = = 1 − k 2 sin2 ϕ = 1 − k 2 sin2 (u, k). du

(19)

The function of delta-amplitude is of period 2K(k) with respect to u. The functions ϕ = amu, z = sn(u, k), z = cn(u, k), and z = dn(u, k) are analytic in k, and as k → 0, they tend to the functions ϕ = u, z = sin u, z = cos u, and z = 1, respectively. The elliptic Jacobi functions satisfy the identities sn2 u + cn2 u = 1,

dn2 u + k 2 sn2 u = 1.

(20)

The following differentiation formulas hold for elliptic functions: d snu = cnu dnu, du d dnu = −k 2 sncnu. du

d cnu = −snu dnu, du

The graphs of elliptic functions are presented in Fig. 4. 3.2. Liouville integrability. Let x˙ = sgrad H be a Hamiltonian system on a symplectic manifold (M 2n , ω). Definition 3.2 (see [8]). The system of Hamiltonian equations x˙ = sgrad H on the symplectic manifold (M 2n , ω) is said to be completely Liouville integrable if there exist smooth functions f1 , . . . , fn on the manifold M 2n such that the following conditions hold: ˙ i.e., {H, fi } = 0, i = 1, . . . , n; (1) the functions f1 , . . . , fn are first integrals of the flow x, (2) the functions f1 , . . . , fn are pairwise in involution: {fi , fj } = 0; (3) the functions f1 , . . . , fn are functionally independent almost everywhere on M 2n , i.e., their differentials are linearly independent on an open everywhere dense set of full measure in M 2n ; (4) the vector fields sgrad fi are complete, i.e., they are defined at every instant of time. The set of functions f1 , . . . , fn satisfying the properties of the definition is called the complete involutive tuple of first integrals. Definition 3.3. The Liouville foliation corresponding to a completely integrable system is the partition of the manifold M 2n into connected components of common level surfaces of the integrals f1 , . . . , fn . Each leaf of the Liouville foliation is an invariant surface. The main role in the qualitative study of integrable Hamiltonian systems is played by the Liouville theorem.

438

Theorem 3.4. Let M 2n = {p, q} be the phase space of a Hamiltonian system with the standard Poisson bracket and a Hamiltonian H(p, q, t). Let the canonical equations ∂H dqi = , dt ∂pi dpj ∂H =− dt ∂qj

(21)

admit n integrals of motion f1 , . . . , fn in involution, i.e., ∂fi {fi , fj } = 0. + {fi , H} = 0, ∂t If on the set Mξ = {(p, q, t) ∈ R2n × R : fi (p, q, t) = ξi , i = 1, . . . , n}, the functions f1 , . . . , fn are independent, then Eqs. (21) are integrable in quadratures. The theorem formulated in [27] is a generalization of this theorem. Theorem 3.5. Let R2n be the phase space of a Hamiltonian system with the standard symplectic structure and the Hamiltonian H(p, q, t). Let this system admit n integrals of motion f1 , f2 , . . . , fn such that
{fi , fj } = ckij fk , ckij = const. k

R2n

× R : fi (p, q, t) = ξi , i = 1, . . . , n}, the functions f1 , f2 , . . . , , fn are If on the set Mξ = {(p, q, t) ∈ independent, the Lie algebra with structural constants ckij is solvable, and, moreover, ckij ξk = 0 for all i, j = 1, . . . , n, then solutions of the system lying on Mξ can be found in quadratures. We now formulate the generalized Liouville theorem, which describes the structure of the topology of the Liouville foliation in a neighborhood of a regular leaf. Theorem 3.6 (see [3, 8]). Let a completely regular integrable Hamiltonian system v = sgrad H be given on a symplectic 2n-dimensional manifold, and let Tξ be a regular level surface of the integrals f1 , . . . , fn . Then we have the following. (1) Tξ is a smooth manifold invariant with respect to the phase flow of v = sgrad H. (2) If the manifold Tξ is connected and compact, then Tξ is diffeomorphic to the n-dimensional torus T n . This torus is called the Liouville torus. (3) The Liouville foliation is trivial in a certain neighborhood U of the Liouville torus Tξ , i.e., it is diffeomorphic to the product of the torus T n by the disk Dn . (4) In a neighborhood U = T n × Dn , there exists a coordinate system s1 , . . . , sn , ϕ1 , . . . , ϕn called action-angle variables that have the following properties: ◦ s1 , . . . , sn are coordinates on the disk Dn and ϕ1 , . . . , ϕn are standard angular coordinates on n the torus
T , ϕ ∈ R/2πZ; ◦ ω= dϕi ∧ dsi ; ◦ the action-angle variables si are functions of the integrals f1 , . . . , fn ; ◦ in action-angle variables, the Hamiltonian flow of v is strengthened on each Liouville torus in the neighborhood U , i.e., the Hamiltonian equations become s˙ = 0,

ϕ˙ i = qi (s1 , . . . , sn ),

i = 1, 2, . . . , n.

On each torus, the flow of v defines a conditional periodic motion, and the trajectories are rectilinear windings (rational or irrational) of the torus. A detailed proof of this theorem is presented in [8]. 439

Remark 1. If the manifold Tξ is not compact, then it is diffeomorphic to the product of the k-dimensional torus by the (n − k)-dimensional Euclidean space Tξ  T k × Rn−k . Remark 2. In the Liouville theorem, the main role is played by the commutativity of the tuple of functions f1 , . . . , fn , i.e., the linear function space G spanned by f1 , . . . , fn is a commutative Lie algebra of dimension n. In some cases, Hamiltonian systems have a tuple of integrals f1 , . . . , fn that do not compose a commutative Lie algebra, i.e., they are not in involution. The Mishchenko and Fomenko theorem generalizes the Liouville theorem. Let M be a symplectic manifold of dimension 2n, and let f1 , . . . , fk : M → R be smooth independent functions. The linear span G over the field R of the functions f1 , . . . , fk is of dimension k. Let G be closed with respect to the Poisson bracket, {fi , fj } = ckij fk . Theorem 3.7. Assume that on the level set Mξ = {x ∈ M : fi (x) = ξi , 1 ≤ i ≤ k}, the differentials dfi are linearly independent and the algebra G satisfies the condition dim G + rank G = dim M.

(22) T r,

If Mξ is connected and compact, then it is diffeomorphic to the r-dimensional torus where r = rank G. If the functions f1 , . . . , fk are first integrals of a Hamiltonian system, then on Mξ , we can choose angular coordinates ϕ1 , . . . , ϕr mod 2π so that the Hamiltonian equations x˙ = sgrad H on T r become ϕ˙ s = ωs = const. The integral trajectories define a conditional periodic motion of the system, i.e., define a rectilinear winding of the torus T r . Remark 3. If the Lie algebra G of first integrals is commutative, then the condition dim G + rank G = dim M passes to the condition k + k = 2n. Therefore, if k = n, we obtain the classical commutative Liouville theorem. 4.

Momentum Mapping. Bifurcation Diagram

Let v = sgrad H be a Liouville integrable Hamiltonian system on a symplectic manifold M 2n . Let system v have a complete tuple of involutive integrals f1 = H, f2 , . . . , fn , functionally independent on M (almost everywhere). Then we can define a smooth mapping F : M 2n → Rn . Definition 4.1. The mapping F is called the momentum mapping. Definition 4.2. A point x in M is said to be critical (for F) if the rank of dF(x) is less than n. Its image F(x) in Rn is called a critical value. Definition 4.3. Let K be the set of all critical points of the momentum mapping in M . Then the set Σ = F(K) ⊂ R is called the bifurcation diagram [8]. Therefore, the bifurcation diagram is the set of all critical values of the momentum mapping. Consider a dynamical system with two degrees of freedom; then the set K of critical points of the momentum mapping is described as follows: K = {x ∈ M 4 : rank dF(x) < 2}. The bifurcation set Σ partitions the image of the momentum mapping into open two-dimensional sets, regions. The boundary of a region consists of a certain number smooth arcs from Σ and singular points of Σ. Singular points of the bifurcation diagram Σ are intersection points, tangent points, and break points of smooth arcs or isolated points not belonging to these arcs. If a point from the image of the momentum mapping lies exactly in the interior of a region, then its full inverse image consists of a certain number of Liouville 2-tori in M 4 . When the point considered varies inside the region, these tori move isotopically (deform) inside M 4 . If the point crosses Σ transversally passing to a neighboring region, then there arises 440

a bifurcation of the Liouville tori corresponding to the point from the image of the momentum mapping. Therefore, the bifurcation diagram allows us to trace surgeries of Liouville tori under the change of values of first integrals. The definition of bifurcation diagram was given for the case of a compact phase space of a Hamiltonian system. In problems of celestial mechanics, we also meet noncompact phase spaces. Therefore, we need a more general definition of the so-called extended bifurcation diagram. The momentum mapping for an integrable Hamiltonian system with two degrees of freedom is defined as the mapping F : M → R2 that, to a point x ∈ M of the phase space, puts in correspondence the pair of numbers (h, l), where h is the value of the Hamiltonian H and l is the value of an additional integral L at the point x. A point (h, l) is called a regular value of the momentum mapping F if its inverse image contains no critical points of the mapping F. A point (h, l) ∈ R2 is said to be regular if, for a certain neighborhood U of this point, in the plane of values of the momentum mapping F, its inverse image F −1 (U ) is homeomorphic to the direct product U × F −1 (h, l), and the mapping F on this inverse image is the projection on the first factor. In the general case, a point (h, l) can be regular but not a regular value, and also it can be a regular value but not a regular point (however, in the case of a compact phase space M , the fact that a point is a regular value implies that this point is regular). The bifurcation diagram of the momentum mapping is the set of points in the image in which the regular value property or regular point property is violated. (Sometimes, this set is also called the extended bifurcation diagram Σ , and the image of the set of critical points is called the bifurcation diagram; these concepts coincide in the case of a compact phase space.) In what follows, we will use the same notation Σ for the bifurcation diagram for the case of a compact and noncompact phase space, i.e., for the bifurcation diagram and the extended bifurcation diagram. Note that using the bifurcation diagram, we can construct a classification of regions of possible motion of the system under consideration and determine the type of motion for each of the regions [24]. Therefore, we can in fact perform a complete analysis of the dynamical system. 5.

Fomenko–Zieshang Topological Invariants

The theory of topological classification of integrable Hamiltonian systems was constructed by A. T. Fomenko and Kh. Zieshang in a number of works [16–19]. This theme was further developed in works [7, 9, 12, 32] of S. V. Matveev, A. B. Bolsinov, A. V. Brailov, and A. A. Oshemkov. In this section, we present the main notation, definitions, and results of this theory, which are necessary for the topological analysis of integrable problems of celestial mechanics. The Liouville theorem results in the assertion that a nonsingular compact level surface of first integrals of a completely integrable Hamiltonian system is the union of tori filled with conditionally periodic trajectories. There arises the question on the way in which the Liouville tori do surgery in a neighborhood of critical level surfaces of first integrals. Consider a Hamiltonian system v = sgrad H on a four-dimensional symplectic manifold M 4 . As is known, in the case of two degrees of freedom, for the integrability of the system v, it suffices to have two first integrals, i.e., the energy integral and only one additional integral F functionally independent of the energy integral (almost everywhere on the manifold M 4 ). Trajectories of the integrable Hamiltonian system with two degrees of freedom lie on common level surfaces of the Hamiltonian H and the additional integral F . Each nonsingular compact connected component of a common level surface of the functions H and F is a 2-dimensional torus. Therefore, the whole phase space is the union of these tori (Liouville tori) and certain singular leaves (containing points at which the functions H and F are dependent). Recall that such a partition of the phase space of the system is called the Liouville foliation. Its leaves are connected components of the sets {H(x) = h, F (x) = f }. The topology of the Liouville foliation contains information on the qualitative behavior of the system. We can give a convenient and compact description of this topology considering the Liouville foliation 441

not on the whole (4-dimensional) phase space but on its 3-dimensional submanifolds. Naturally, these submanifolds must contain each leaf of the Liouville foliation entirely. Therefore, we can consider them as inverse images of certain curves (on the plane) under the momentum mapping x → (H(x), F (x)). In particular, as these curves, we can consider straight lines {h = const}. In this case, we speak of the Liouville foliation on a (three-dimensional) isoenergetic surface Q3h = {H(x) = h}. Obviously, its leaves consist of connected components of the sets {F˜ (x) = f }, where F˜ is the restriction of the integral F to Q3h . In this paper, we will consider only nonresonant systems. Definition 5.1. A Hamiltonian H is said to be “nonresonant” on an isoenergetic surface Q3h if the Liouville tori on which the trajectories of the system compose dense irrational windings are everywhere dense in Q3h . A Liouville torus is nonresonant iff the closure of any integral trajectory of the field lying in it coincides with the whole torus. In the resonant case, the closure of a trajectory is a torus of strictly lesser dimension. Definition 5.2. An integrable system is said to be nonresonant on a manifold M 2n if almost all tori are nonresonant. Correspondingly, a system is said to be resonant if all its Liouville tori are resonant. Definition 5.3. An additional integral F is said to be “Bott” on an isoenergetic surface Q3h if critical points of the function f compose nonsingular critical submanifolds. A critical submanifold is said to be “nonsingular” if the restriction of the function f to any submanifold of complementary dimension transversal to it has a Morse singularity at the intersection point. The following assertion holds. Assertion 5.1. Any critical submanifold is either a circle S 1 (we call it a “nondegenerate critical circle”), or a two-dimensional torus T 2 , or a Klein bottle K 2 . In this paper, we consider those integrable systems which have no critical tori and Klein bottles on isoenergetic surfaces Q3h . Note that any nondgenerate critical circle is a periodic trajectory, and, therefore, there is a natural orientation on it given by the system. In physics and mechanics, there arises the following problem. Let two integrable Hamiltonian systems be given. It is required to show whether they are equivalent in the topological sense. In most cases, a practically unique method for solving this problem consists in calculating the corresponding Fomenko– Zieshang invariants. Definition 5.4. Let (v1 , Q1 ) and (v2 , Q2 ) be two nonresonant integrable Hamiltonian systems with Bott integrals considered on their own isoenergetic surfaces. Let the orientation of the isoenergetic surfaces Q1 and Q2 be fixed. We say that the systems (v1 , Q1 ) and (v2 , Q2 ) are Liouville equivalent if there exists a diffeomorphism τ : Q1 → Q2 preserving the structure of the Liouville foliation (i.e., it is fiberwise). Additionally, we require that τ preserves the orientation of 3-manifolds and the orientation of critical circles given by the Hamiltonian vector field. Definition 5.5. We say that an integrable Hamiltonian system is topologically stable on a given energy level Qh = {H = h} if, for a small movement of the energy level, the structure of the Liouville foliation on the isoenergetic surface does not change, in other words, for small ε, the systems (v, Qh ) and (v, Qh+ε ) are topologically equivalent. Remark 1. The topological stability is a generic property. Indeed, the topological type of a system can change only at some isolated points, i.e., there are finitely many bifurcation values of the energy for which the topology of the Liouville foliations changes jumpwise. The general method for verification of the topological stability consists in studying the bifurcation diagram Σ of the momentum mapping F : M 4 → R2 . In real situations, the bifurcation diagram is a set of smooth curves which can intersect and be tangent to each other at finitely many points. Namely, these points 442

are singular points of the bifurcation diagram Σ. Therefore, the topological stability of the energy level Qh = {H = h} means that the line {H = h} in the plane of integrals of motion intersects the bifurcation diagram transversally at nonsingular points. To describe the topological structure of the Liouville foliation on the isoenergetic surface Qh , it suffices to describe this foliation in (small) invariant neighborhoods of singular leaves and point out in what way the whole manifold Qh is glued from these neighborhoods. Consider all critical values c1 < c2 < · · · < cn of the function F and the singular levels F −1 (ci ) corresponding to them. The assumption that Q3h is compact and the integral F is Bott implies that there are finitely many critical submanifolds. In particular, there are finitely many critical values of the function F . Choose a positive ε so small that the segments Ik = [ck − ε, ck + ε],

k = 1, . . . , n,

are disjoint. The inverse images F −1 (Ik ) consist of finitely many connected pieces. Fix a certain critical value c and denote by Q3c a certain connected component of the inverse image F −1 ([c − ε, c + ε]). The boundary of the submanifold Q3c consists of finitely many Liouville tori. Denote by Lc the set Q3c ∩F −1 (c). We say that Lc is the “connected critical level corresponding to the critical value c” or we merely call Lc a “singular leaf.” The submanifold Q3c is called a “regular neighborhood of a singular leaf.” Definition 5.6. The neighborhood Q3c with the structure of the Liouville foliation is called a 3-atom. Or we say that a class of Liouville equivalence of a neighborhood of a singular leaf is a 3-atom. The topological structure of Q3c was described in [8] and [9] in detail. Note that if two regular neighborhoods of singular leaves have equivalent atoms, then there exists an orientation preserving homeomorphism of one to another that preserves the Liouville foliation and transforms nondegenerate critical circles into nondegenerate critical circles with preservation of the natural orientation on them given by the field of the system. In the problems under study, we meet only the following four atoms (see Fig. 5):

Fig. 5. Atoms A, B, A∗ , C2 . (1) the atom A is a solid torus (S 1 × D2 ) and the fibering is into concentric tori (a singular leaf is a circle, the axis of the solid torus); 443

(2) the atom B is a solid torus from which two thin solid tori are cut out, (N 2 × S 1 ) (a singular leaf is the direct product of a “figure eight curve” by the circle and N 2 is a neighborhood of the “figure eight curve”); (3) the atom A∗ is a solid torus from which one thin solid torus two times going around the axis is cut ˜ 1 ) (a singular leaf is the skew-product of a “figure eight curve” by a circle); out, (N 2 ×S (4) the atom C2 is a solid torus from which three thin solid tori are cut out, (M 2 × S 1 ) (a singular leaf is the direct product of “two intersecting circles” by a circle, M 2 is a neighborhood of “two intersecting circles”). Atoms describe bifurcations of the Liouville tori when passing a critical value of the function F˜ . The birth or disappearance corresponds to the atom A, the disintegration of one torus into two tori or the confluence of two tori into one torus corresponds to the atom B, the surgery of one torus into one torus corresponds to the atom A∗ , and the surgery of two tori into two tori corresponds to the atom C2 . The boundaries of atoms are glued by one-parameter families of Liouville tori that contain no singular leaves. The scheme of this gluing can be described by a directed graph, to each vertex of which we put in correspondence a certain atom, and to edges, we put in correspondence one-parameter families of tori. Such a graph is called the molecule of a given isoenergetic surface. The molecule describes the structure of the Liouville foliation with accuracy up to the so-called rough Liouville equivalence. This means that the gluings of boundary tori of atoms along edges are not unique. However, these gluings can be uniquely characterized by using certain numerical labels [8]. As a result, we obtain a labelled molecule (or a Fomenko–Zieshang invariant), which defines the type of the Liouville foliation uniquely with accuracy up to the Liouville equivalence.

Chapter 2 GENERALIZATION OF THE KEPLER PROBLEM TO SPACES OF CONSTANT CURVATURE 6.

Historical Essay

Before starting the presentation of this section on the construction of “non-Euclidean geometry” (the term was introduced by Gauss, a German mathematician), we can cite P. Beyl, an 18th century philosopher, who wrote that “the idea that the view passing from century to century and from generation to generation cannot be completely false is a pure illusion.” “It is well known that geometry preassumes that the concept of space, as well as the first main concepts needed for spatial constructions, are given in advance. It gives normal definitions of concepts, while the essential properties of the defined objects appear in the form of axioms. In this case, the interrelation between these prerequisites remains unknown: it is not clear whether and to what extent the relation between them is necessary; also it is not a priori clear whether this relation is possible.” (Riemann). “Among Euclid’s axioms, there is one axiom which mathematicians think is directly less obvious than others; for a long time they tend to reduce it to the other axioms, i.e., to prove it by using them.” (Einstein). This is the so-called axiom on parallels (fifth postulate). The exact formulation given by Euclid is as follows: If two straight lines meeting a third line compose inner angles to one of its side whose sum is less than two right angles, then these two lines continued unlimitedly meet one another to the side with which the sum of the angles is less than that of two right angles. In many editions, this assertion appears as an axiom. The following three postulates are learned in middle school. (1) Between two points, we can draw only one straight line. (2) A straight line is the shortest path between two points. 444

(3) Through a given point, we can draw only one straight line disjoint with a given straight line. The formulation of the latter postulate is equivalent to the Euclid fifth postulate in the formulation given by Plaifair (1748–1819), an English mathematician, in 1797 in “Elements of mathematics”: Through a point lying outside a straight line, we can draw only one parallel straight line, i.e., disjoint with the given straight line. V. Ya. Bunyakovskii, an outstanding Russian mathematician, formulated the fifth postulate as follows: An oblique line and a perpendicular to the same straight line always intersect when they are sufficiently continued. For thousands of years mathematicians tried to prove the axiom on parallels but have not succeeded. Henry Poincar´e (1854–1912) said that “For a long time, mathematicians sought the proof of the axiom known as Euclid’s postulate without success. The amount of effort made in this mystery hope cannot in fact be described. Finally, at the beginning of the last century, almost simultaneously, two scientists, Lobachevskii, a Russian scientist, and Bolyai, a Hungarian scientist, proved irrefutably that this proof is impossible; by this they got rid of discovers of geometry without Euclid’s fifth postulate; since that time, the Paris Academy of Sciences has received no more than one or two new proofs per year.” As for the proof of the fifth postulate, Lobachevskii wrote the following in his textbook on geometry written in 1823 in the chapter “On measuring of rectangles”: “The measuring of planes is based on the fact that two straight lines meet one another in the case where they are to one side of the third and when one of them is perpendicular and the other is oblique by an acute angle to the perpendicular. Up to now, there is no correct proof of this true fact; any given proofs can be named as explanations and cannot be considered as mathematical proofs as a whole.” (The manuscript of the textbook with the title “Geometry” was presented to Kazan’ University for publication at Government expense, but Lobachevskii’s colleagues did not give a positive review, and the manuscript was sent to M. L. Magnitskii, a guardian of the Kazan’ Educational Region. In turn, Magnitskii sent the manuscript to the Academician N. I. Fuss asking him to give a review of it. Fuss gave a very negative review, and Lobachevskii’s textbook on geometry was not published. The manuscript “Geometry” was published in 1909 for the first time through the efforts of Professor N. P. Zagosskin.) There is only one way to prove the impossibility of proving Euclid’s postulate; precisely, to assume that such a proof is impossible, i.e., this axiom does not reduce to the other axioms. Therefore, for this purpose, it is necessary to construct a new noncontradictory geometry with the only difference that the axiom on parallels is replaced by another axiom. For the first time, this conclusion was obtained by Lobachevskii, and independently of him, a new geometry was discovered by the Hungarian mathematician, military engineer J. Bolyai (1802–1860). Bolyai published the original results of his study in 1832 as a supplement to the first volume of a course of mathematics written and published by his father F. Bolyai (1775–1856). Therefore, the Bolyai study is known in the history of science by the name “Appendix.” For the first time, N. I. Lobachevskii reported on his new doctrine under the title “Exposition succinte des principes de la Geometrie, avec une demonstration rigoureuse du theoreme des paralleles” in February 12, 1825 at the session of the Department of Physics and Mathematics, Kasan’ University (the report was not published and was not preserved). Lobachevskii published a fragment of this report in 1829 in Kazan’ Vestnik published by the Imperial Kazan’ University under the title “On the Origins of Geometry.” The scientific community rejected the new doctrine. The death knell was dealt by Academician M. V. Ostrogradskii. In August 19, 1832, the Council of Kazan’ University sent (through Lobachevskii’s initiative) a copy to the Imperial St. Petersburg Academy of Science. In September 5, the Academy of Science gave this work to Academician Ostrogradskii for consideration. After a month, November 7, the Academician gave replied: the work does not deserve the attention of the Academy. The works of Lobachevskii devoted to the imaginary geometry were derided. It was said about the geometrical studies of Lobachevskii that a mountain gave birth to a mouse, and his contemporaries considered these studies trivial compared with Lobachevskii’ scientific authority in other fields of mathematics, since his view on the geometry of space was so surprising and nontraditional. Only many years after the death of Lobachevskii was his doctrine recognized by the scientific community. Lobachevskii took the first step towards changing the traditional view of space. 445

C

A B

Fig. 6 William Kingdon Clifford, an English mathematician (1845–1879), wrote the following about Lobachevskii: “As Vesalius was for Galenius, and Copernicus for Ptolemy, so was Lobachevskii for Euclid. There is an edifying parallel between Lobachevskii and Copernicus. Both are Slavs by birth. Both of them revolutionalized scientific ideas, and the greatness of these revolutions is of the same. The greatness of one or another revolution is in that they are revolutions in our understanding of the cosmos... Before Lobachevskii, it seemed that we have a real knowledge of something referring to the whole cosmos, something true in all nonmeasurability and the entire eternity. This conviction was refuted by Lobachevskii and his successors. A modern geometer knows nothing of the nature of the space existing in reality at an infinitely large distance, and he knows nothing of the properties of the existing space in the past or future of the eternity. Of course, he knows that the laws given by Euclid are true with accuracy up to what can be attained by the most exact experience, not only at places where we are, but at places at distances not imaginable for the astronomer. But he knows only the Here and Now . After him, there are the There and Then, for which he knows nothing but for which probably more will be known after us. Both revolutions, Copernicus’ and Lobachevskii’s, replace knowledge of the nonmeasurable and eternal by knowledge of the Here and Now , and, as a result, the idea of the universe, the macrocosmos as something available for human knowledge, was destroyed.” (“Philosophy of the Pure Science,” in the collections of Clifford papers “Lectures and Essays,” Second edition, London, Macmillan, 1886, pp. 212–214).1 Clifford’s views on the Lobachevskii geometry correspond to this spatial theory of the matter in which a revision of relativity theory is truthfully seen. To avoid being proofless, we present to the reader some of Clifford’s arguments. Assume that we have a thin tube bent in the form of a circle, and an infinitely thin worm of length AB creeps inside it; then we obtain a one-dimensional space. Also we put a label C on the circle. “What does the worm feel travelling along the tube?” (1) The worm notes when it crosses the label and when it returns to it (to the point C). (2) The worm deals with the same curvature everywhere. The next question is: “What conclusions can it make ?” (1) The space is bounded and the location is relative (it can be found by the arclength between C and the worm). (2) The space is the same everywhere, i.e., it has the same properties everywhere. The conclusions of the worm are more true than our conclusions on the space (which is assumed to be Euclidean). We make conclusions on the basis of experience in studying the surrounding space only, although we extend them to the whole universe; in contrast, the worm was at all points of the space. Moreover, if we remove the label C, then the worm will consider its space to be infinite. Since the worm deals with constant curvature only, it relates it with certain physical properties but not with the space itself. Then there is no difference (for the worm) between the circle and the line; it only feels a strangeness in its physical state (there are no advantages between states on the circle and the line). Therefore, the finiteness or infiniteness of the space follows from the postulate on the possibility of determining the position of a point in it. 1

Translated into English from the Russian translation.

446

Now let the worm creep along a tube of variable curvature; then it can determine its location by using the extent of curvature, by the extent of bend of the tube, and, therefore, we can relate the concept of relativity of location with the concept of the physical state of the worm. Let us go further in the conclusions of the worm. The worm can consider the space to be identical at all points and that the curvature is related to the changes in its organism. The greater the curvature, the more it feels fine (or vice versa). We can argue analogously in the case of a two-dimensional space. Therefore, a “flat” human can make a conclusion that his constitution is very variable and the space is identical. It should be noted that inhabitants living, for example, on a sphere (and their arguments) are not better than inhabitants living on the plane. What can we say about the space of three dimensions in which we live? We make the conclusion that it is identical based on the experience and observations of a bounded part of the space, which in fact is not true. Following Euclid, we say that parallel planes or two parallel straight lines extended arbitrarily far are disjoint. Probably, as Clifford’s worm, we relate the change of curvature to certain physical actions or to the changes in our constitution. Also, it should be noted that the curvature can change depending not only on the location but also on time. An important role in developing new views on the space was played by Bernhard Riemann, a German mathematician (1826–1866). Riemann showed that there exist lines and surfaces of various kinds, as well as spaces of three dimensions of various kinds, so that we can reveal only experimentally which kind is the space where we live. In particular, in the framework of experiments on the surface of a sheet of paper, the axioms of geometry on the plane are true, but we know that in reality, the sheet is covered by many small scars and furrows on which these axioms are not true. On the basis of the theory developed by Riemann, Clifford makes the following conclusions. (1) Small parts of the space are in fact analogous to mounds on a surface which is planar in the mean; precisely, the usual geometric laws are not true there. (2) The distortion or deformation property passes from one part of the space to another continuously in the form of a wave. (3) The change of curvature of the space is the thing realized in the phenomenon, which we call the motion of the matter in reality. For the first time, the problem of studying the dynamics in a space of constant curvature was posed by N. I. Lobachevskii, who studied the generalizations of the attraction law for a space of negative curvature. Lobachevskii wrote that “... in our mind, there can be no contradiction when we suppose that certain forces in nature follow one geometry, and others follow their own special geometry. To explain this idea, we suppose, as many people do, that attraction forces weaken owing to the propagation of their action along the sphere. In the usual geometry, the value of the sphere is taken to be equal to 4πr2 for the semiwidth r, which implies that the force must decrease with respect to the square of the distance. In the imaginary geometry, I have found the surface of the ball to be equal to π(expr − exp−r )2 , and, probably, molecular forces act in such a geometry...” Generally speaking, Lobachevskii speaks not of the potential but of the force. “Only forces produce everything: the motion, the velocity, time, the mass, and even the distance and angles. All are in close relation with forces...” “Certain cases justify such an opinion: for example, the value of the attraction force is expressed through the mass divided by the square of the distance.” “We now ask: in what way does the distance produce this force? How does this relation between so different subjects exist in nature? ...” “...we recognize one dependence from experiments, and owing to a deficiency of observation, another should be assumed in mind outside the visible universe or in a tight region of molecular attractions.” The primitive of the function 1 , r (e − e−r )2 447

written by Lobachevskii, is (with accuracy up to a coefficient) coth r. Precisely this potential is considered 1 at present as a true generalization of the Newtonian potential for the Lobachevskii space. r A further development of this topic is in the work [49] of N. E. Zhukovskii on the motion of a material pseudospherical plate on the Lobachevskii plane. An analog of the Newtonian potential for S3 (S3 is a three-dimensional space embedded into the Euclidean space in the standard way) is probably obtained by Schr¨ odinger in [37]. E. Schr¨ odinger considered the motion of an electron in the field of the hydrogen nucleus. In his work, he wrote that the true form of the Coulomb potential on the sphere corresponding to 1/r corresponds to V = −γ cot θ.

(23)

Here, γ is the constant defining the attraction and θ is the arc length of the large circle connecting gravitating material points on S3 , 0 ≤ θ ≤ π. 7.

Flat Kepler Problem

7.1. Description of the system. The Bertrand theorem. Celestial mechanics and the mechanics of space flight is based on the Newtonian gravitational law. In the Newton gravitation theory, the 1 gravitation is described by the potential V = −γ . The gradient of the potential taken with the opposite r sign is equal to the acceleration of a material point placed in the gravitational field. The law describing the field V in the case of absence of gravitating matter is determined by the Laplace equation ∆V = 0. In the case where there is a gravitating matter with density ρ, we have the Poisson equation ∆V = 4πγρ. Clearly, the latter equation is more general and contains the previous equation as a particular case. Consider the Kepler problem, a classical problem of celestial mechanics, whose results are well known but are necessary for the further presentation. We first study the motion of a material point in the central field. Definition 7.1. A vector field is said to be central with center at point O if it is invariant with respect to the group of motion preserving the point O fixed. The following theorem is well known. Theorem 7.1. Under the motion in the central field, the kinetic moment M = [r, r˙ ] with respect to the ˙ = 0, where r is the radius-vector with the origin at center of the field O does not change in time, i.e., M the center O of the field and r is its length. Indeed, ˙ = [˙r, r˙ ] + [r, ¨r]. M ∂V ˙ = 0. are collinear, which implies M In the central field, the vectors r and ¨r = − ∂r It follows from the theorem on conservation of kinetic moment of a material point with respect to the point O that [r, r˙ ] = c (24) This relation is called the area integral. Here, c is the vector constant of the area integral. The projections of the vector c on the axes of the system Oxyz are   ˙ c1 = y z˙ − z y, ˙ c2 = z x˙ − xz,   ˙ c3 = xy˙ − y x. 448

r(t + ∆t) r(t) ∆S

Fig. 7 Multiplying these relations by x, y, and z, respectively, we obtain c1 x + c2 y + c3 z = 0 for any instant of time. The following theorem holds. Theorem 7.2. Under the action of the central force, a point always moves along a flat trajectory. Indeed, in the vector form, we can write [r, r˙ ]r = Mr = 0. The trajectory plane passes through the center of the force and is perpendicular to the constant kinetic moment of the point. Clearly, its location is determined by the initial conditions. The equations of motion of the material point of unit mass in the Kepler problem in R3 are γ γ ∂V ¨r = − 3 r. (25) , V (r) = − , ∂r r r Write the components of the acceleration with respect to the axes x, y, and z of the fixed coordinate system: γx γy γz x ¨=− 3, y¨ = − 3 , z¨ = − 3 , r r r where r2 = x2 + y 2 + z 2 . We now consider the motion of the point in the central field on the plane (by Theorem 7.2). Let the motion be executed in the plane xOy. In the Kepler problem, it is natural to pass to the polar coordinates r, ϕ. Then the law of conservation of the kinetic moment is written as ¨r = −

xy˙ − y x˙ = r2 ϕ˙ = l, and the value of the kinetic moment l is a first integral. For the first time, the law of conservation of the kinetic moment was found by Kepler in observing the motion of Mars. Kepler formulated it in the following form. The sectorial velocity of each of the planets with respect to the Sun is constant. At equal instants of time, the radius-vector sweeps equal area (the second Kepler law). Proof. By the sectorial velocity C, Kepler means the speed of varying the area S(t) swept by the radiusdS . We have the expression vector: C = dt 1 ˙ + o(∆t). ∆S = S(t + ∆t) − S(t) = r2 ϕ∆t 2 Therefore, the sectorial velocity is 1 1 dS = r2 ϕ˙ = l. C= dt 2 2 Since l is a first integral, this implies that the sectorial velocity is constant and equals half of the value of the kinetic moment. Write the energy conservation law in polar coordinates: 1 1 h = r˙ 2 + r2 ϕ˙ 2 + V (r). 2 2 449

Therefore, the system considered admits the following two integrals of motion:  1 1  h = r˙ 2 + Veff , Veff (r) = r2 ϕ˙ 2 + V (r), 2 2 l = r2 ϕ. ˙ We can obtain from this the expression for the velocity:

r˙ = 2(h − Veff (r)). Rewrite the latter equation in the form



dt =

Since ϕ˙ =

(26)

(27)

dr

. 2(h − Veff (r))

(28)

dϕ l2 r˙ = 2 , dr r

(29)

we have l dϕ r2 = . dr 2(h − Veff (r)) As a result, we obtain the following equation of an orbit of the material point in the central field:

l/r2

dr. ϕ= 2(h − Veff (r))

(30)

In the general case, the orbit is not closed: the point oscillates between rmin and rmax , where r = rmin is the pericenter and r = rmax is the apocenter, r max dr

. (31) ∆ϕ = l 2 r 2(h − Veff ) rmin

Clearly, such an orbit is closed if

∆ϕ is a rational number, i.e., π m ∆ϕ = π, n

where m and n are integers. We now formulate the Bertrand theorem. Theorem 7.3. All bounded orbits in the central field are closed only in the following two cases: γ V (r) = − , γ ≥ 0 (Newtonian potential); r V (r) = kr2 , k ≥ 0 (Hooke potential). Proof. Following Arnol’d, we divide the proof into three parts [2], 1. Make the change x = l/r and rewrite Eq. (31) in the form x max

∆ϕ = xmin

dx

, 2(h − W )

.

(32)

  x2 l + , i.e., the angle between the pericenter and apocenter is equal to the half-period where W (x) = V x 2 of oscillations in a one-dimensional system with the potential energy W . 450

2. Let us find the angle ϕ for an orbit close to a circular orbit of radius a. Since the potential V (r) is attractive, the effective potential has a local minimum for r = a: l2 = 0, r = a. (33) r3 Choose the initial conditions in such a way that the particle moves along the circular orbit r = a. Clearly, in this case, l2 l2 = a3 V  (a). (34) V  (a) = 3 , a The angular frequency of rotation along the circular orbit is found fronm expression (31):  (r) = V  (r) − Veff

V  (a) l2 2 2  2 . (35) = ω , ω a = V (a), ω = 0 0 0 a4 a We now consider a trajectory close to circular. We expand the potential into a power series in r in a small neighborhood of the point r = a, restricting ourselves to first terms of the expansion. The equilibrium conditions imply that only the second derivative of the potential is different from zero. Then the frequency of radial oscillations is determined by the relation  l2 3V  (a)  , (36) (r)r=a = V  (a) + 3 = V  (a) + ω12 = Veff a4 a or   l  ϕ ≈ ϕcircle = π √ . (37) 2  r W r=a 3. The trajectory of the particle is closed if these frequencies are commensurable, i.e., m (38) ω1 = ω0 = νω0 , n where ν is a rational number independent of the form of the orbit and is a characteristic of the potential. We obtain from Eqs. (35), (36), and (38) that 3V  (r) V  (r) = ν2 , r r ν2 − 3  V  (r) = V (r). r Therefore, the potential V (r) is a power function: V  (r) +

V (r) = cβ −1 rβ ,

(39)

β = ν 2 − 2.

Indeed, V  (r) = crβ−1 , V  (r) = c(β − 1)rβ−2 . Now let us consider an arbitrary orbit. We obtain from relation (31) that r max dr π

. (40) ∆ϕ = = l 2 ν r 2(h − Veff ) rmin

Make the change of variables r = rmin /ρ; then dr = −

1 

∆ϕ = ρ0

Let β > 0, V = krν

2 −2

rmin dρ, and Eq. (40) is written in the form ρ2

dρ 2 2r2 2hrmin − min V − ρ2 2 l l2

=

π . ν

, k > 0. For h → ∞, we have ρ0 → 0; then we obtain

1 π dρ

= arcsin ρ|10 = . ∆ϕ = 2 1 − ρ2

(41)

(42)

0

451

2

Therefore, ν = 2, V = kr2 . Now let ν 2 − 2 < 0 and V (r) = −γρ2−ν , γ > 0. Letting h tend to zero, we obtain ν2

1

1 π π dρ ρ 2 −1

= dρ = 2 = . ∆ϕ = 2 2 ν ν ρ ρ−ν − 1 1 − ρν 0

0

Therefore, ν = 1. Therefore, only for the following two potentials are the finite orbits closed: γ V (r) = kr2 and V (r) = − , k, γ > 0. r This is what was required to prove. The proof of the theorem is given in [34]. 7.2. Kepler laws. We now study the motion of a material point of unit mass in the central field of the form γ V (r) = − r with a fixed center of force. In this case, the effective potential l2 l2 γ + = − 2r2 r 2r2 in depicted in Fig. 8. We obtain from formula (41) that Veff (r) = V (r) +

γ l − l . ϕ = arccos r γ2 2h + 2 l Introduce the notation l2 p= , γ Then

 1+

2hl2 = ε. γ2

(43)

p −1 p , r= . (44) ϕ = arccos r ε 1 + ε cos ϕ Equation (44) is the equation of a second-order curve in whose focus the origin is located; the constant p is called the orbit parameter , and the constant ε is called its eccentricity. The value of the constant of integration depends on the choice of the polar axis in the orbit plane. If we direct the polar axis to the point nearest to the center of the force, then the constant of integration vanishes. It is known from analytic geometry that the trajectory of form (44) is a hyperbola for ε > 1, a parabola for ε = 1, an ellipse for ε < 1, and a circle for ε = 0. Therefore, taking into account relation (44) and the form of the potential V = −γ/r, we can conclude that the trajectory of the point is a hyperbola for h > 0, a parabola for h = 0, an ellipse for (Veff )min < h < 0, and a circle for h = (Veff )min . In the case of repulsion (γ < 0), the point can move only along the hyperbola, since h > 0. Let us consider the motion along the ellipse. It follows from Eq. (44) that p p , rmax = . (45) rmin = 1+ε 1−ε The semi-axes of the ellipse are defined by the expression p p a= , b= √ , b2 = a2 (1 − ε2 ) (46) 2 1 − ε2 1−ε or γ l a= , b= √ . (47) 2h 2h 452

Fig. 8

Fig. 9

Therefore, the larger semi-axis a of the ellipse depends only on the total energy. In the case of a circular orbit where h = (Veff )min , r˙ = 0,

r = r0 = a = b.

In the first Kepler law discovered by him in observing the motion of Mars, it is asserted that each planet moves along an ellipse in one of the foci of which the Sun is located (first Kepler law). The eccentricities of planets are very small; therefore, Kepler formulated his first law as follows. Planets move around the Sun along a circle, but the Sun is not located in the center . By the way, if we assume that planets move in the central gravity field, then the first Kepler law implies the Newton gravitational law γ V =− . r 453

Introduce one more variable, the eccentric anomaly ξ, which is connected with ϕ by the relation cos ξ − ε sin ξ cos ϕ = , sin ϕ = 1 − ε2 . (48) 1 − ε cos ξ 1 − ε cos ξ Then the following relation holds: r = a(1 − ε sin ξ). (49) The eccentric anomaly is connected with time by the Kepler equation ξ − ε sin ξ = n(t − t0 ),



(50)

γ is the so-called mean motion and the constant t0 denotes the instant of passing through a3 the perihelion of the orbit (the nearest point of the orbit; the most distant point of the orbit is called the aphelion). It follows from (50) that the period of a complete turn of a planet along the ellipse is determined by the expression  where n =

a3 . γ From the latter expression, we obtain the third Kepler law T = 2π

(51)

T2 4π 2 a3 , . (52) = γ a3 γ The ratio between the square of periods of turn of the planet to the cubes of the large semi-axes of their orbits is constant and is the same for all planets (third Kepler law). T 2 = 4π 2

7.3. First integrals. Algebra of first integrals. Let R6 = (q, p) be the phase space of the Kepler problem. The Hamiltonian γ 1 H = p2 − 2 r is invariant with respect to the group of rotations SO(3) of the three-dimensional space. Therefore, the components of the momentum vector l = [q, p] (53) are preserved. In the Bertrand theorem, it is asserted that all finite trajectories, which exist for γ > 0 and correspond to the energy h < 0, are closed. This fact points to the existence of an implicit symmetry in the Kepler problem and, therefore, to the existence of additional integrals of motion. Indeed, in the Kepler problem, the components of the following Laplace–Runge–Lenz vector are also integrals of motion: γ (54) A = [l, p] − q. r The total number of functionally independent integrals of motion is equal to 5. The vector A lies in the orbit plane, and, moreover, its direction coincides with the principal or focal axis of the orbit. The eccentricity is proportional to the module of the Laplace vector: |A| 1 r ε= = [˙r, M] − . γ γ r The Poisson brackets for the integrals of motion A and l have the form {li , lj } = εijk lk ,

{li , Aj } = εijk Ak ,

{Ai , Aj } = −2hεijk lk ,

(55)

where εijk is a completely skew-symmetric tensor, ε123 = 1. Let m = −2h−1/2 A; then passing to the vector m, for h < 0, we obtain {li , lj } = εijk lk , 454

{li , mj } = εijk mk ,

{mi , mj } = εijk lk .

(56)

We see that this algebra is a Lie algebra of the Lie group SO(4). The following assertion holds. Assertion 7.1. In the Kepler problem, for a fixed energy h, the Lie algebra of the integrals Ai , lj is isomorphic to 1. so(4) for h < 0; 2. the Lie algebra of the group of motion in R3 for h = 0; 3. so(1, 3) for h > 0. The Bertrand theorem is a particular case of a more general assertion. In the case of motion of a nonrelativistic material point in the N -dimensional Euclidean space under the action of a conservative central force, the Hamiltonian is invariant with respect to the symmetry of a Lie group larger than SO(N ), the group of rotations around the center of the force, only in the following two cases: this is the case of the Kepler problem for which the higher symmetry group is SO(N + 1) for bounded states and SO(N, 1) for unbounded states, and the case of an isotropic oscillator for which the symmetry group is SU(N ). 7.4. Regularization. Certain solutions of equations of motion in the classical Kepler problem have a singularity for γ > 0, which corresponds to the fall of a material point onto the center. The velocity of motion is infinite at a singular point (i.e., the vector fields are not complete). However, after an appropriate regularization, this singularity can be removed. The regularization problem occupies a central place in celestial mechanics. One-dimensional case. To understand what happens at a singular point, we consider a system with one degree of freedom. Let a material point be attracted by a gravitational center located at the origin of the coordinate axis Ox according to the Newton law γ V =− , |x| where |x| is the distance from the material particle to the gravitating center. Then the equation of motion has the form γ ¨ = − 3 x. x |x| The total energy is defined by the expression γ x˙ 2 − . (57) 2 |x| Introduce a new independent variable in order to expand the time scale, and, moreover, so that the closer the particle to the center, the greater the deceleration of time: dt . (58) dt = |x|dτ ; dτ = |x| Further, define a new velocity as follows: dx x = dτ (where the prime denotes differentiation in the new independent variable τ ) and write the total energy with respect to the velocity x : γ x 2 = h. (59) − 2|x|2 |x| Also, make the change of the variable x by the formula h=

x = ±ω 2 .

(60)

Then the velocity and the energy become x = ±2ωω  ,

4ω 2 ω  2 γ − 2 = h. 4 2ω ω 455



ω

ω

γ/2 ω

ω

γ/h Fig. 10 And, finally, we obtain 2ω  − hω 2 = γ. 2

(61)

Consider two cases: bounded motion and unbounded motion. 1. Let h < 0; therefore, the potential energy is greater than the kinetic energy. Then we obtain from expression (61) that 2 2ω  + |h|ω 2 = γ. We see that this is the equation of an ellipse with center at the origin: ω2 ω 2 + γ γ = 1. 2 |h|

(62)

The greater h, the less stretched the ellipse. It is easy to obtain from the

latter expression that the singular state passes to a regular state with a finite velocity (ω = 0, ω  = γ/2), i.e., with fall onto the center, the material point has a finite velocity. We depict the case considered in the plane (ω, ω  ) (Fig. 10). 2. Let h > 0. Then we obtain from Eq. (61) the equation of a hyperbola: 2ω  − hω 2 = γ. 2

(63)

(The material point goes to infinity.) The situation seems to be more real if, with fall onto the center, the material point has zero velocity. Let the energy of the material point be defined by relation (59). We consider two cases again. 1. Let h < 0; then expression (59) can be written in the form   γ x 2 2 + |h| |x| − |x| = 0. 2 |h| After simple calculations, we obtain the equation of an ellipse   γ 2 γ2 x 2 + |h| |x| − = , 2 2|h| 4|h|2

  γ 2 |x| − x 2 2|h| + = 1. γ2 γ2 2|h| 4|h|2

(64)

The center of the ellipse has the coordinates (γ/2|h|, 0) (see Fig. 11). Obviously, the greater the energy, the lesser the value of the greater semiaxis of the ellipse. For x = 0, the velocity is x = 0. Therefore, we have removed the singularity. 2. Let h > 0. Then, from (59), we obtain the equation of a hyperbola  γ2 γ 2 x 2 =− . − h |x| + (65) 2 2h 4h Clearly, in this case, the motion is unbounded: the material point goes to infinity. 456

x

x x

x

Fig. 11 Two-dimensional case. We present a method for removing the singularity in the two-dimensional case [3]. Introduce a new independent variable z = q1 + iq2 . Then the equations of motion of the Kepler problem become γz ¨ = − 3. (66) z |z| Write the energy integral with respect to the new variable: h=

γ |z| ˙2 − . 2 |z|

(67)

Further, we make the change of the independent variable z and time t by the formulas z = ω2 ,

t =

dt = 4|ω 2 | = 4|z|, dτ

(68)

and write the energy integral in the form |ω  |2 = 4γ + 4h|ω 2 |. 2 We see that the point ω = 0 is regular. From (69), we obtain

(69)

ω  + 8|h|ω = 0.

(70)

This equation describes oscillations of a harmonic oscillator. Therefore, mapping (68) transforms the Kepler orbits of constant energy h < 0 into the orbits of the harmonic oscillator located on the energy level (69). Moser regularization. n-Dimensional case. In the work [31] of Moser, a natural regularization of equations of motion is described. Moser showed that after an appropriate compactification, the constant energy surface (h < 0) is topologically equivalent to the tangent bundle of unit vectors of the n-dimensional sphere Sn . He proved the following theorem. Theorem 7.4. For h < 0, the energy surface H = h can be mapped topologically in a one-to-one way onto the tangent bundle of unit vectors of Sn under the condition that one point of the sphere (the north pole corresponding to the center of forces) is punctured. Consider the sphere Sn embedded into the (n + 1)-dimensional space in the standard way:   n
n 2 2 S = ξ = (ξ0 , ξ1 , . . . , ξn ) : |ξ| = ξj = 1 .

(71)

0

In the ambient space, consider a dynamical system with the Hamiltonian 1 H = |η|2 |ξ|2 , 2 457

Fig. 12. Moser regularization in the two-dimensional case.

where ξ = (ξ0 , ξ1 , . . . , ξn ) and η = (η0 , η1 , . . . , ηn ) are (n + 1)-dimensional vectors of the coordinate and impulse. We can obtain the equations of motion using the Hamiltonian equations ξ  = |ξ|2 η,

η  = −|η|2 ξ.

(72)

Here, the differentiation is carried out in s, which plays the role of time. It follows from (72) that if the following conditions hold at the initial instant of time: |ξ|2 = 1,

(ξ, η) =

n


ξj ηj = 0,

(73)

j=0

then they hold at all subsequent instants of time. The manifold defined by Eq. (73) is the tangent bundle T Sn of the sphere Sn . Taking into account that |ξ|2 = 1, we rewrite Eq. (72) in the form ξ  + |η|2 ξ = 0. 1 2 |η| . 2 2 The unit tangent vectors {η : (η, ξ) = 0, |η| = 1} compose the constant energy surface H = h = 1/2. Consider the stereographic projection mapping the sphere Sn with the punctured point (1, 0, . . . , 0) on the n-dimensional Euclidean space (Fig. 12). The stereographic projection is described by the formulas This equation describes the geodesic flow on the sphere Sn = {ξ : |ξ|2 = 1} with the energy H =

xk =

ξk , 1 − ξ0

k = 1, . . . , n.

(74)

Under the stereographic projection, we also project the tangent bundle on the space R2n = {(x, y) : x, y ∈ Rn } so that the following condition holds: n
m=0

ηm dξm =

n


yk dxk .

(75)

k=1

We seek yk in the form yk = a(ξ, η)ηk + b(ξ, η)ξk , 458

k = 1, 2, . . . , n.

(76)

Here, a(ξ, η) and b(ξ, η) are functions of two variables. Write the necessary relations ξl dξl , ξ0 dξk (1 − ξ0 ) − ξk (−dξ0 ) dξk ξk (ξl dξl ) = − , dxk = 2 (1 − ξ0 ) 1 − ξ0 ξ0 (1 − ξ0 )2  n n n n 


η0 ξk dξk
ηk − ξk dξk . ηµ dξµ = η0 dξ0 + ηk dξk = −η0 + ηk dξk = ξ0 ξ0 dξ0 = −

0

1

1

(77)

1

Then we obtain from (76) the answer: yk = (1 − ξ0 )ηk + η0 ξk . The formulas of the inverse mapping from the space Rn onto the sphere Sn are ξk =

2xk , |ξ|2 + 1

ξ0 =

|x|2 − 1 , |x|2 + 1

(|x|2 + 1)2 k y − (x, y)xk , η0 = (x, y). 2 The Hamiltonian is rewritten in the form (|x|2 + 1)2 2 1 2 2 |y| = |ξ| |η| . F = 8 2 The Hamiltonian equations are ∂F ∂F , y = − . x = ∂y ∂x We further make the change √ (|x|2 + 1)|y| − 1. G = 2F − 1 = 2 Then Eqs. (81) pass to the equations ηk =

x =

∂G , ∂y

y = −

∂G , ∂x

(78) (79)

(80)

(81)

(82)

(83)

and the condition F = 1/2 passes to the condition G = 0. We now pass from the variable s to the variable t according to the formula

t=

|y| ds.

(84)

Then we obtain x˙ = |y|−1

∂G , ∂y

y˙ = −|y|−1

∂G . ∂x

(85)

For G = 0, |y|−1

∂H ∂G = , ∂y ∂y

|y|−1

∂H ∂G = , ∂x ∂x

where

1 √ 1 1 = ( 2F − 1) = |x|2 − |y|−1 . (86) 2 |y| 2 Finally, we make the change (x, y) → (−p, q); after that, we obtain the Hamiltonian of the Kepler problem: H = |y|−1 G −

1 1 1 =− . H = |p|2 − 2 |q| 2

(87)

Geodesic flows on the sphere are isomorphic to the Kepler problem. Therefore, transformations (74), (77), and (84) map the bundle of unit tangent vectors of the sphere into the 2n-dimensional phase space, and the large circles of the sphere Sn are mapped into Kepler ellipses on the energy surface H = h = −1/2. 459

To describe the flow near the singular point q = 0 corresponding to the north pole of the sphere, we use the transformation that maps the north pole into the south pole, i.e., ξ0 → −ξ0 ,

η0 → −η0 ,

ξk → ξk ,

ηk → ηk ,

k = 1, . . . , n.

The following transformation in the space (p, q) corresponds to this transformation: p q → |p|2 q − 2(pq)p, p → 2 . |p|

(88)

(89)

Under this transformation, the Kepler orbits pass to the Kepler orbits. The singular states (|p| = ∞, q = 0) of the system transform into the states (p = 0, |q| = 0). 8.

Dynamics in Spaces of Constant Curvature. Generalized Bertrand Theorem

We describe two approaches to the generalization of the Newtonian potential to the case of curved spaces, examining the example of spaces of constant curvature. 1. One of the approaches to the definition of the Newtonian potential V in a flat space consists in solving the Poisson equation, which transforms into the Laplace equation ∆V = 0 in the case of zero density of the matter. Along with constants, there are only the following spherical-symmetric solutions of the Laplace equation in the n-dimensional Euclidean space (with accuracy up to multiplication by a constant): r2−n for n ≥ 3 and ln r for n = 2. Therefore, in the three-dimensional space, the function 1 is harmonic. Therefore, if we consider plane motion under the action of a gravitational potential field r (for example, in the ordinary Kepler problem) as a reduction of the three-dimensional problem, then the 1 Newtonian potential can be defined as − in the two-dimensional case. r For the n-dimensional sphere (of constant curvature 1) and n-dimensional Lobachevskii space (of constant curvature −1), the solutions of the Laplace equation (invariant with respect to the attracting center) are the functions (tan r)2−n and (tanh r)2−n for n ≥ 3 and the functions ln tan r and ln tanh r for n = 2 (here, r is the attracting center; the constant entering the solution of the Laplace equation is obviously nonessential). As in the flat case, for the two-dimensional sphere, we can assume that the analog of the Newtonian potential (with accuracy up to a coefficient) is − cot r, i.e., a harmonic function on the threedimensional sphere. For the two-dimensional Lobachevskii space, the function − coth r (with accuracy up to a coefficient) is also an analog of the Newtonian potential. 2. Consider the motion of a material point p of unit mass in the field with a potential V depending on the distance between the particle and a fixed gravitational center P in a 3-dimensional space of constant curvature. We consider it as the sphere S3 or the upper flap of the hyperboloid H3 (depending on the sign of curvature) embedded into R4 or into the Minkowski space M 4 with coordinates q0 , q1 , q2 , q3 in the standard way. We pay primary attention to the case of negative curvature where the equation of the hyperboloid has the form q02 − q2 = R2 , making necessary remarks for the case of positive curvature. Here, R is the radius of curvature. For the sphere, the corresponding equation has the form 1 R2 = , q02 + q2 = R2 , λ where λ is the curvature. The problem considered is an analog of the classical problem of motion in the central field. Let θ be the arclength of the hyperbola connecting the points p and P . We place the center at the vertex of the hyperboloid. Then the potential V is a function depending on the angle θ only. The Laplace–Beltrami equation is   2 −2 ∂ 2 ∂V sinh θ = 0. (90) ∆ = R sinh θ ∂θ ∂θ 460

The function

γ coth θ + const (91) R is a solution of this equation. The constant is obviously nonessential, and the parameter γ plays the role of the gravitational constant. For a positive curvature, in relations (90) and (91), the functions involving the variable θ must be replaced by the corresponding trigonometric functions. The function V has a singular point of the Newton type for θ = 0. For the sphere, the potential is antisymmetric between two hemispheres. If γ is positive, then we have an attracting singularity θ = 0 (north pole) and a repelling singularity at the antipodal point θ = π (south pole). These two singular points can be treated as a source and a sink, since the phase flow through the boundary of any closed domain not containing gravitational centers is equal to zero. In the problem considered, the Lagrange function is 1 L = [−(q˙0 )2 + (q˙1 )2 + (q˙2 )2 + (q˙3 )2 ] − V. 2 The Lagrangian is defined in the ambient space where the metric is indefinite. It must be bounded on the tangent space of H3 . The metric induced by the pseudo-Euclidean metric on H3 is positive-definite, and, therefore, the kinetic energy is also a positive-definite quadratic form. The signature of the metric in the Minkowski space in this problem is g(−1, 1, 1, 1). If the signature of the metric in the Minkowski space is defined as g(1, −1, −1, −1), then the metric induced on H3 is negative-definite. To obtain a positive kinetic energy in this case, we must take the induced metric with the opposite sign. Let us introduce the pseudospherical coordinate system. The transition formulas are written as q1 = R sinh θ cos ϕ, q0 = R cosh θ, (92) q3 = R sinh θ sin ϕ sin ψ. q2 = R sinh θ sin ϕ cos ψ, V =−

Here, θ defines the length of the hyperbola (“meridian”) with respect to the pseudo-Euclidean metric passing from the upper flap of the hyperboloid to the variable point, i.e., the pseudospherical coordinates are analogous to the spherical coordinates (for the spherical coordinates, θ is the length of the meridian of the large disk passing from the north pole to the variable point). The metric induced in the space H3 (with respect to the coordinates R, θ, ϕ, ψ) is ds2 = R2 (dθ2 + sinh2 θdϕ2 + sinh2 θ sin2 ϕψ 2 ). In these coordinates, the Lagrangian is defined by 1 L = R2 (θ˙2 + sinh2 θ(ϕ˙ 2 + sin2 ϕψ˙ 2 ) − V. (93) 2 (For a positive curvature, in relation (93), the function sinh θ must be replaced by the corresponding trigonometric function.) It turns out that for the potential energy of form (91) (with a positive constant γ), all bounded orbits of a material point are closed. It is well known that all finite orbits are closed for the Newtonian potential (they are ellipses) (see Sec. 7). As J. Bertrand showed (in 1873), along with the Newtonian potential, there exists exactly one central potential field for which all finite trajectories are closed. This is the field generated by the Hooke potential V = kr2 /2, where k is a positive constant. For the sphere and the Lobachevskii space, the problem of description of potentials for which all finite trajectories are closed was solved in [14, 23, 26, 28]. Consider the following generalized Bertrand problem in spaces of constant curvature: find all potentials V (θ) for which all bounded orbits of a material point are closed . Theorem 8.1. The following potentials are solutions of the generalized Bertrand problem: V = −γλ1/2 coth θ, γ > 0, V = kλ−1 tanh2 θ/2, k > 0, in the Lobachevskii space H3 ; 461

Fig. 13. Gnomonic coordinates for the sphere and the pseudosphere. V = −γλ1/2 cot θ, γ > 0, V = kλ−1 tan2 θ/2,

k > 0, on the sphere S3 .

Proof. 1. The proof of the generalized Bertrand theorem for the Kepler problem on the sphere was given in [23] (1978). We present an analogous proof for the Lobachevskii space. Let the gravitational center be located at the vertex of the upper flap of the hyperboloid q02 − qi qi = 1/λ (the summation is carried out with respect to the repeated subscripts). Consider the projection of the hyperboloid on the tangent plane in the ambient space. Denote the Cartesian coordinates of this projection by xi ; they will be called the gnomonic coordinates. The relations between the gnomonic coordinates xi and the coordinates q0 , qi of a test particle on the hyperboloid are   3 
xi 1  2

qi = , q0 = R − qi2 = . (94) 2 1 − λ|x| λ(1 − λ|x|2 i=1 For the sphere, in relation (94), λ must have the opposite sign. Write the metric in the tangent space with respect to the gnomonic coordinates. We have the relations √ dxi λxi (x · dx) λ(x · dx) , dqi = + . dq0 = 3/2 1/2 2 2 (1 − λ|x| ) (1 − λ|x| ) (1 − λ|x|2 )3/2 Then the metric is written in the form dxdx (xdx)2 ds2 = −dq02 + dqi dqi = + λ. (95) 1 − λ|x|2 (1 − λ|x|2 )2 For the sphere, in the expression of the metric, the sign of the curvature λ changes to the opposite sign: dxdx (xdx)2 − λ. 2 1 + λ|x| (1 + λ|x|2 )2 It is easy to verify that in the limit as λ → 0, the metric becomes ds2 =

(96)

ds2 = dxdx. Obviously, the results can be generalized to any dimension of the Lobachevskii space Hn and that of the sphere Sn embedded into the corresponding spaces Rn1 and Sn+1 of dimension n + 1. The Lagrange function of a nonrelativistic particle of unit mass in the potential field V is defined by   1 ds 2 − V, L= 2 dt where s˙ 2 is found from (96) and (95). 462

Therefore, the impulse conjugate to x is defined by p=

∂L x(x · x) ˙ x˙ + λ, = ∂ x˙ 1 − λ|x|2 (1 − λ|x|2 )2

Lij = xi pj − xj pi = (1 − λr2 )−1 (xi x˙j − xj x˙i ). The Hamiltonian is written as 1 H = (1 − λr2 )(p2 − λ(x · p)2 ) + V (r). 2 Each projected orbit lies in the plane Lij xj = 0.

(97)

In the polar coordinates (r, θ) in this plane, the angular moment is r2 θ˙ 1 = L, where L2 = Lij Lij . 2 1 − λr 2 The energy conservation law is written in polar coordinates as   1 1 1 2 2 ˙2 r˙ + r θ + V (r) = h. 2 (1 − λr2 )2 1 − λr2 The differential equation of an orbit is     1 2 −4 dr 2 1 −2 +r L r + V (r) = h + λL2 , 2 dθ 2

(98)

where h is the total energy. An analogous expression was obtained in [23] for the sphere. This implies that the projected orbits are the same as for the potential considered in the Euclidean geometry, since the 1 curvature enters the right-hand side only in the form λL2 . Therefore, all orbits on the pseudosphere are 2 closed, and nonclosed projections of orbits correspond to the closed orbits on the sphere which intersect the equator. 2. Let us present one more solution of the Bertrand problem in the Lobachevskii space [26] (in contrast 1 to [26], we assume that the curvature is determined by a nonunuit radius λ = 2 ). R The following assertion holds. Lemma 8.1 (see [26]). If the potential energy depends only on the coordinate θ, then each of the orbits lies in a certain two-dimensional plane H2 ⊂ H3 passing through the point θ = 0. For a fixed plane H2 , the angular coordinates ϕ, ψ can be chosen so that this plane is defined by the equation ψ = const. Therefore, we have a system with two degrees of freedom. The Lagrangian is defined by 1 L = R2 (θ˙2 + sinh2 θϕ˙ 2 ) − V (θ). 2 Obviously, ϕ is a cyclic coordinate. By the Noether theorem, the first integral R2 sinh2 θϕ˙ = c corresponds to the cyclic coordinate. The Routh function Rc is 1 1 c2 − V (θ) = R2 θ˙2 − Veff . Rc = pϕ ϕ˙ − L = R2 θ˙2 − 2 2 2 2 2R sinh θ The function Veff = V (θ) +

c2 2R2 sinh2 θ 463

is a reduced potential. We obtain from the Routh equation that d ∂Rc ∂Rc − = 0, dt ∂ θ˙ ∂θ

R2 θ¨ = −

dVeff . dθ

Introduce new variables r and ρ by r = R tanh θ,

ρ=

1 1 = . r R tanh θ

If c = 0, then the motion is executed along a straight line in H2 passing through the point θ = 0. Let c = 0; then it follows from the expression of the first integral that ϕ is a monotone function of time. As a parameter on the orbit, we take the angular variable ϕ and seek the equation of the orbit in the form ρ = ρ(ϕ). Let the prime denote differentiation in the variable ϕ; then we obtain dρ θ˙ =− , dt R sinh2 θ

dt R2 sinh2 θ = , dϕ c

dρ dt dρ = , dϕ dt dϕ

˙ θR d dt θ¨ sinh2 θ 3 R . , ρ = (ρ ) =− c dt dϕ c2 Substituting the obtained expressions in the Routh equation, we obtain ρ = −

sinh2 θ dV . (99) R c2 dθ This equation can be rewritten in the form of the Clairaut equation in the problem of the motion in the central field on the Euclidean plane if we rewrite the right-hand side of the equation as   1   d 1 dV dU 1 ρ  V (θ) = U (r), =   =U R . 1 dθ dθ ρ cosh2 θ d ρ ρ + ρ =

Then we obtain 1 ρ + ρ = 2 2 U c ρ The energy integral is written in the form 

  1 . ρ

(100)

1 c2 γ h = θ˙2 R2 − + . 2 r 2R2 sinh2 θ Taking into account that γ = −γρ, r we rewrite the energy integral in the form −



ρc θ˙ = − , R

c2 2 c2 (ρ + ρ2 ) − γρ = h + λ. (101) 2 2 In fact, we obtain the equation of form (98). (For the sphere, an analogous equation can be written, and in this case, the curvature λ has the opposite sign.) Therefore, the equation of the orbit has the same form as in the problem of motion of a material point in the central force field with the potential U (r). The theorem is proved. Remark 1. It seems improbable that for an arbitrary metric, there exist (attracting) potentials for which all finite trajectories are closed. From the geometric viewpoint, the following inverse problem is of interest: describe all metrics for which there exist such potentials. 464

Remark 2. We mention one more possible interpretation of expressions for Newtonian potentials in the flat space (1/r), on the sphere (cot r), and in the Lobachevskii space (coth r). Consider the function r in each of these spaces; it is equal to the distance from the attracting center. Then it is easy to verify that 1 the function ∆r, where ∆ is the Laplace operator, coincides (with accuracy up to a coefficient) with in r the flat space, with cot r on the sphere, and with coth r in the Lobachevskii space; moreover, this is true for any dimension. 9.

Generalized Kepler Laws

The generalization of the Kepler laws to the sphere and Lobachevskii space was obtained by V. V. Kozlov in [26] and A. N. Chernikov in [14]. Let us present the Kepler laws for spaces of constant curvature. (1) Each of the planets moves along an ellipse (an ellipse is defined as the set of points the sum of distances from which to two given points, the foci, is constant), and the Sun is at one of the foci:  2hl2 l2 p , ε= 1+ 2 , p= , r= 1 + ε cos ϕ γ γ where h is the total energy of the planet, γ = const, l is the kinetic moment which is an integral of motion and depends on the initial conditions, and the angle ϕ is measured in the direction from the center of Earth to the point of the orbit nearest to the Sun, the perigee. For the sphere, we have    l2 2l2 ε= 1+ 2 h− λ . γ 2 Obviously, if ε < 1, then the orbits of a material point are quadrics on S2 (this motion corresponds to motion along an ellipse on the tangent space on which the sphere is projected), if ε = 1, then the trajectory of the material point is tangent to the equator of the sphere (this motion corresponds to motion along a parabola on the tangent space), and if ε > 1, then the trajectory intersects the equator (this motion corresponds to motion along a hyperbola on the tangent space). For the Lobachevskii space,    l2 2l2 ε= 1+ 2 h+ λ . γ 2 The curvature of the space does not violate the first Kepler law. (2) The second Kepler law is, in fact, formulated in the same way for the flat space and the Lobachevskii space. The sectorial velocity of each of the planets with respect to the Sun is constant (equals half of the kinetic moment). A small distinction exists for the sphere. An arc of the large circle connecting the gravitating center and an imaginary point distant from the center at twice greater distance than that of the real material point (if the angle θ > π/2, then the gravitating center and the antipodal center must be interchanged) sweeps equal areas on the sphere in equal intervals of time. (3) If we formulate the third Kepler laws as follows: the period of rotation of a planet along the orbit depends on only the total energy (on the large axis of the orbit), then it is formulated in the same way for all the cases considered. 10.

Bifurcation diagrams. Geometry of the Phase Space of the Generalized Kepler Problem

We consider bifurcation diagrams for the Kepler problem for the flat space [35], the sphere, and the Lobachevskii space. 465

The bifurcation diagram for the plane Kepler problem was given in the work [35] of Smale. The integrals of motion in this problem are defined by formulas (26). By definition, we find the set of critical values of the momentum mapping from the condition   p2ϕ pϕ γ − 3 0  pr r2 r r2 rang   < 2, 0 0 1 0 where pr = r˙ and pϕ = ϕr ˙ 2 are generalized impulses corresponding to the polar coordinates (r, ϕ). This implies that the set of critical values of the momentum mapping Σ1 is γ2 , l = 0. (102) 2l2 Also, the bifurcation set contains those points for which the “regularity” condition is violated. Therefore, the bifurcation set Σ is represented as Σ = Σ1 ∪ Σ2 , where Σ2 consists of the axis h = 0. For the Kepler problem on the sphere, the integrals of motion are written in the polar coordinates (r, ϕ) as follows:    1 1 1 2 2 2   h = 2 (1 + λr2 )2 r˙ + 1 + λr2 r ϕ˙ + V (r), (103) 2ϕ  r ˙  l = . 1 + λr2 The effective energy is l2 γ l2 (104) Veff = − + 2 + λ. r 2r 2 The set of critical values of the momentum mapping can be found from the condition   p2ϕ pϕ γ 2 2 2 2 2 pr (1 + λr ) 2λr(1 + λr )pr + 2 − 3 (1 + λr ) 2 0 r r r rang   < 2, 0 0 1 0 h=−

˙ 2 are the generalized impulses corresponding to the polar coordinates (r, ϕ). where pr = r˙ and pϕ = ϕr Therefore, the set of critical values is defined by the expression l2 γ2 λ, l = 0. (105) + 2l2 2 Note that in this problem, all orbits are closed, since the sphere (configuration space) is a compact manifold. Those projections of trajectories on the tangent plane which are nonclosed (hyperbolas) correspond to the closed orbits on the sphere intersecting the equator. In the limit, as λ → 0, the orbits on the sphere pass to the following Kepler plane orbits: bounded orbits (from the upper hemisphere) and unbounded orbits (from the lower hemisphere where a repelling singularity is located). In this case, the set of critical values of the momentum mappings defines the bifurcation diagram. For the Kepler problem in the Lobachevskii space, the integrals of motion are written in the form  l2 2 1 1 l2  2  , ρ= = h + 2 λ = 2 (ρ + ρ ) − γρ, r R tanh θ (106)  ϕ ˙  l = . ρ2 Analogously, we find the set of critical values of the momentum mapping. We obtain from the condition   pρ 2 ρ − γ pϕ 0 l 2 r2 <2 rang  l 0 0 1 0 h=−

466

(here pρ and pϕ are the generalized impulses corresponding to the polar coordinates (ρ, ϕ)) that h=−

γ2 l2 − λ, 2l2 2

l = 0.

γ Also, the bifurcation set contains the straight line h = − separating the bounded and unbounded R motions. Definition 10.1. A motion is said to be bounded (the elliptic-type motion by analogy with the plane Kepler problem) if a material point always remains in a certain finite domain, i.e., sup |r| < ∞ (where t≥t0

|r| is the distance between the material point and the attracting center) and unbounded if the latter condition does not hold. Definition 10.2. A motion of a material point in the Kepler problem in the Lobachevskii space is called γ γ a parabolic-type motion if h = − , |r| → ∞; it is a hyperbolic-type motion if h > − , |r| → ∞. R R Correspondingly, the initial velocity is called the parabolic or hyperbolic velocity. Assertion 10.1. We have the following types of motion in the Kepler problem in the Lobachevskii space: γ the elliptic type for h < − ; R γ the parabolic type for h = − ; R γ the hyperbolic type for h > − . R Proof. Since the kinetic energy is always nonnegative, the following inequality must hold naturally during the whole time of motion: −V + h ≥ 0. (107) Therefore, h≥−

γ coth r. R

As |r| → ∞, we obtain

γ (108) h≥− . R For values of the energy satisfying inequality (108), the material point goes to infinity independently of γ what initial position it occupied, i.e., the motion is unbounded; it is parabolic for h = − and hyperbolic R γ γ for h > − . Obviously, for h < − , the condition sup |r| < ∞ holds, i.e., the motion is elliptic. In R R t≥t0 the limit, as R → ∞, we obtain the plane problem, where the value h = 0 of the energy separates the bounded and unbounded motions. γ l2 γ2 It is easy to calculate that the straight line h = − is tangent to the curve h = − 2 − λ at the R 2l 2 points with the coordinates      √  γ √ γ −γ λ, √ , −γ λ, − √ . λ λ Therefore, we can formulate the following assertion. Assertion 10.2. The bifurcation set for the Kepler problem is in the Euclidean space γ2 h = − 2 , l = 0, h = 0; 2l 467

l ∅ 1

2

1

2

h

∅

Fig. 14. Bifurcation diagram for the plane Kepler problem. l

S

H

∅

l

∅ 1

h

1 1

1 ∅

2

h

2

∅

Fig. 15. Bifurcation diagram for the Kepler problem on the sphere and in the Lobachevskii space. in the Kepler problem on the sphere l2 γ2 λ, + 2l2 2 in the Kepler problem in the Lobachevskii space h=−

l = 0;

γ l2 γ2 λ, h = − , l = 0. − 2 2l 2 R Obviously, the bifurcation diagrams in the Kepler problems for the sphere and the Lobachevskii space pass one to another under the change of the curvature λ. If by λ we mean a quantity taking positive, as well as negative, values, then the bifurcation sets coincide for the spaces of constant curvature (except for the straight line h = 0 separating the bounded and unbounded motions). If λ = 0, then we obtain the plane Kepler problem. We now introduce the concept of integral manifold. Let a dynamical system, i.e., a smooth vector field on a smooth manifold M , dx = ξ(x), (109) dt be given. The manifold M is the phase space of the dynamical system considered. Let system (109) admit first integrals I1 , . . . , In : M → R. Then their common level Ik = {x ∈ M : Ii (x) = ki , i = 1, . . . , n} (110) (where ki = const) is a subset of the phase space invariant with respect to the phase flow of system (109), i.e., each of the trajectories passing through a point belonging to set (110) is contained in it entirely. Set (110) is called an integral manifold of system (109). Now, our goal is to study the topology of the mapping of the phase space into a linear space for the Kepler problem in the Euclidean space, on the sphere, and in the Lobachevskii space, i.e., we seek the h=−

468

Fig. 16

(h, l) ∈ Σ1

(1)

(1) l = 0

Fig. 17. Projections of Liouville tori and cylinders on the configuration space (for h < 0). topological type of the integral manifold. For fixed values of the first integrals h, l, system (26) defines in R4 a surface Ih,l invariant with respect to the phase flow which is an integral manifold. Proposition 10.1. One of the following cases holds for the integral manifold of the plane Kepler problem: 1. the cylinder Ih,l = SO(2) × R if h ≥ 0; 2. the torus Ih,l = S 1 × S 1 if Veff (min) < h < 0; γ2 3. the circle if (h, l) belongs to the curve h = − 2 ; 2l 4. Ih,l = ∅ if h < Veff (min) . The proof is rather obvious; it is given in [35] and visually depicted in Fig. 16(a). The Liouville foliation is the direct product of the fibered 2-disk by the circle in the case where h < 0. When a point in the image of (h, l) crosses the straight line h = 0, the type of motion changes: the bounded motion passes into unbounded; moreover, the type of integral manifold changes, and the torus is transformed by surgery into the cylinder; moreover, the derivative of the effective potential tends to zero as r → ∞. Also, consider the case where l = 0, i.e., pϕ = 0. In this case, the effective potential energy is depicted in Fig. 16(b). The Liouville foliation is analogous for h < 0; it is the direct product of the fibered 2-disk depicted in Fig. 16(b) by the circle. For h ≥ 0, motion is possible for any values of the coordinates. The projections of Liouville tori and cylinders are shown in Figs. 17 and 18. Consider the Kepler problem on the sphere using the gnomonic coordinates, i.e., consider the projections on the tangent space from the center of the sphere on the ambient space. The advantage of this projection is that in this case, the free motion of a material point on the sphere (the uniform speed motion along a large circle) projects on the rectilinear motion (not uniform) on the tangent space. That is, the projected orbits are the same as in Euclidean geometry: the curvature influences only on the velocity of the projected motion. As was always said earlier, in the Kepler problem on the sphere, all orbits are closed (the bifurcation set does not contain the straight line separating the bounded and unbounded motions). If the 469

(2)

(2) l = 0

Fig. 18. Projections of Liouville tori and cylinders on the configuration space (for h ≥ 0). orbit on a sphere intersects the equator, this corresponds to the going to infinity in the Euclidean space. In the inverse image of the momentum mapping, we have the torus (region 1). In the case where (h, l) γ 2 l2 belongs to the curve h = − 2 + λ, a quadric or circle is the inverse image in the gnomonic coordinates, 2l 2 since pr = 0. Proposition 10.2. One of the following cases holds for the integral manifold in the Kepler problem on the pseudosphere: γ 1. the cylinder if h ≥ − (region 2); R γ 2. the torus Ih,l = S 1 × S 1 if Veff (min) < h < − (region 1); R γ2 l2 3. the quadric if (h, l) belongs to the curve h = − 2 − λ; 2l 2 4. Ih,l = ∅ if h < Veff (min) . 11.

Regularization of the Kepler Problem on the Sphere

We perform the regularization of the Kepler problem on the two-dimensional sphere [42]. Consider the 1 gravitational center located at the pole of the sphere q02 + q12 + q22 = with the coordinates (q0 , q1 , q2 ) = λ   1 √ , 0, 0 and locally pass to the gnomonic coordinates x1 , x2 . λ Introduce the complex variable z = x1 + ix2 . Then, in the gnomonic variables, the energy of the system can be represented as (111) h = (1 + λr2 )(z˙ 2 + λ(z, z˙ )2 ) + V (r). Introduce the change of the variable z and time t according to the relations dt = 4|ω 2 | = 4|z|; dτ 1 (112) ωω  ; z˙ = 2ω ω˙ = 2|ω 2 | γ r = |z|; V (r) = . r Then the expression for h can be rewritten in the form ' (2 ) %& 1$ + 4γ = 4h|ω 2 |. 1 + λ|ω 2 |2 |ω  |2 + λ|ω 2 | Re(ωω  ) (113) 2 Obviously, after the change of the coordinates and time, the equations of motion have no singularity. z = ω2 ,

12.

t =

Lie Algebra of First Integrals

As was noted above, in the Kepler problem in the Euclidean space, there exist five independent first integrals. The Runge–Lenz–Laplace vector, which lies in the orbit plane parallel to the principal axis at 470

each of its points, has the Cartesian components γxi . (114) r The generalization of expression (114) for the sphere was found in [23]. The first quantity in the expression for Ai is preserved under the free motion of a particle, and it consists of the generators Lij and the impulses pj . Under the free motion of the particle on the sphere, the conservation law of the linear moment is replaced by the conservation law of the vector Ai = −Lij pj +

π = p + λx(x · p).

(115)

The components of this vector are proportional to the corresponding generators of the geometric symmetry group SO(N + 1). These generators are components of the angular moment in the ambient space: πi = λ1/2 L0i . Making the same change in expression (114), we can obtain the desired generalization of the Runge–Lenz– Laplace vector in the Kepler problem on the sphere: γxi . (116) Ai = −Lij πj + r The length of the Runge–Lenz–Laplace vector is defined by the expression A2 = γ 2 + 2HL2 − λ(L2 )2 .

(117)

The Hamiltonian is written in the form γ 1 H = (π 2 + λL2 ) − . 2 r The commutation relations between L and A are as follows [20, 21, 23]: {Ai , Aj } = −2εijk Lk (H − λL2 ), {Li , Lj } = εijk Lk ,

{Li , Aj } = εijk Ak

(118)

(119)

(here, we have used the notation Lij = εijk Lk ). In the spaces of constant curvature, the sign inside the brackets is positive. 13.

Neumann Problem

We consider the Neumann problem on the motion of a point on the n-dimensional unit sphere: Sn = {(x, x) = 1},

x = (x0 , x1 , . . . , xn ) ∈ Rn+1

(120)

in the force field with potential, which is an analog of the Hooke potential on the sphere. Recall that the Bertrand problem on S3 has two solutions k V = −γ cot θ, V = tan2 θ, γ, k > 0. 2 The second solution is an analog of the Hooke potential, or, in other words, it is an analog of the potential of an elastic spring. This function has a singularity on the equator θ = π/2. We place the elastic attracting (or repelling) centers at the points with the coordinates (±1, 0, . . . , 0), . . . , (0, . . . , 0, ±1).

(121)

Let rp = (x0 , . . . , xn ) be the radius-vector of the particle and let ri be the radius-vector of the elastic attracting center. Then we can write the relations  sin θi = 1 − x2i , cos θi = ri , rp  = xi , tan2 θi =

1 − x2i 1 = 2 − 1. 2 xi xi 471

This implies (see the generalization of the Bertrand theorem to spaces of constant curvature) that an analog of the Hooke potential is (with accuracy up to a nonessential additive constant) the function αi (122) V (x) = − 2 , i = 0, 1, . . . , n. xi We now consider the motion on Sn in the field with the potential equal to the sum of potentials generated by each of the n + 1 elastic attracting (or repelling) centers: n

1
αi . W (x) = 2 x2i i=0

We find from the constraint equation (x, x) = 1 that (x, ˙ x) ˙ + (x, x ¨) = 0. Write the Lagrange equations [48] with a multiplier λ: αi x ¨i = 2 + λxi , i = 0, 1, . . . , n. xi

(123)

(124)

Then we can find the multiplier λ from Eqs. (123) and (124): λ = −(x, ˙ x) ˙ − 2W (x). We see from the latter equation that the multiplier λ is a constant equal to twice the energy of the system taken with the minus sign. Therefore, the system on Sn falls into n + 1 disconnected oscillators and can be integrated in qudratures. Indeed, each of the equations in (124) has the first integral αi (125) x˙ 2i − λx2i + 2 = hi , xi where hi = const, h0 + h1 + · · · + hn = −2λ. It is easy to obtain the equations of motion from Eqs. (125):  d 2 (xi ) = 2 λx4i + hi x2i − αi . (126) dt In the general case, these equations on the 2n-dimensional phase space T Sn have n(n + 1) quadratic integrals αj αi Mij = (x˙ i xj − x˙ j xi j)2 + xi 2 + xj 2 , i, j = 0, 1 . . . n. xj xi Among these integrals, 2n − 1 are independent, which is a consequence of the existence of symmetry. In turn, the existence of symmetry leads to the fact that all trajectories of the system considered are closed curves. Note that the system with n + 1 gravitational centers on the sphere and the potential be an analog of the Kepler potential on the sphere is not integrable.

Chapter 3 TWO-CENTER PROBLEM ON THE SPHERE The problem of two fixed centers consists in finding the motion of a material point attracted by two fixed point masses according to the Newton law. For the first time, this problem was studied and reduced to quadratures by Euler in 1760. The qualitative analysis of this problem was in the plane case considered by K. Charlier [13] (1907), H. Tallquist [38] (1927), and N. K. Badalyan [5] (1934, 1939; this analysis is also 472

presented in the book [15] of A. G. Duboshin (1964)). In the work [1] of V. M. Alekseev, the generalized spatial two-fixed-centers problem is studied, and the qualitative analysis and the classification of types of motion are presented. On the three-dimensional sphere (of constant curvature 1), an analog of the Newtonian potential generated by a gravitational center is cot θ, where the angle θ is equal to the distance to the center. The odinger in integrability of the Kepler problem on the three-dimensional sphere S3 was proved by E. Schr¨ [37]. Schr¨ odinger considered the motion of an electron in the potential field of the nucleus of a hydrogentype atom from the viewpoint of quantum mechanics. V. V. Kozlov and O. A. Kharin proved the integrability of the problem on the motion of a material point on the sphere S2 in the field of two fixed Newtonian centers in [28]. The works [40, 42, 44, 47] study the two-center problem in spaces of constant curvature and the Lobachevskii space. In the work [46] of T. G. Vozmishcheva and A. A. Oshemkov, a topological analysis of the two-center problem on the two-dimensional sphere was carried out. The Fomenko–Zieshang invariants completely describing the topology of Liouville foliations of isoenergetic surfaces Q3 were constructed. All types of motion (regular motions and the limit motions corresponding to bifurcations of Liouville tori) were described on the configuration space. The performed topological analysis of two-center problem on the sphere shows that this problem and the two-center problem on the plane are similar from the topological viewpoint. In some sense, the plane problem is the limit of the two-center problem on the sphere as the radius of the sphere tends to infinity. An analogous situation also holds for the Kepler problem. At the same time, we note that the study (in particular, a computer analysis) of the two-body problem in spaces of constant curvature carried out in [36] allows us to suppose that this problem is not integrable. 14.

Plane Two-Fixed-Centers Problem

In this section, we describe the statement of the two-center problem on the plane, present various forms of writing this system in the Liouville form, and also give the description of the bifurcation diagram for the momentum mapping generated by the Hamiltonian and an additional integral. 14.1. Statement of the problem. The Hamiltonian of a natural mechanical system is equal to the sum of the kinetic energy and the potential energy: H = T + V. We consider the problem of the motion of a material point in the flat space in the field generated by two fixed attracting centers. In this case, the kinetic energy T is given by the Euclidean matrix, and the potential energy V is equal to the sum of the potentials generated by each of the centers. In the flat case (for a flat metric), as the potential generated by one of the attracting centers, we consider the Newtonian γ potential V = − , where γ is a constant positive coefficient characterizing the attraction force. r Consider two fixed centers P1 and P2 , respectively. Draw the abscissa through these centers in the direction P1 P2 and place the origin of coordinates O at the middle of the segment connecting these two centers. Denote by c the distance from one of the centers to the origin. The force function in this problem obviously has the form γ1 γ2 (127) V =− − . r1 r2 Here, the constants γ1 and γ2 characterize the attraction force between the material point and the gravitational centers P1 and P2 , respectively, r1 and r2 are the distances of the moving point to the centers, and, as is seen from Fig. 19, they are defined by the expressions r12 = (x + c)2 + y 2 , r22 = (x − c)2 + y 2 .

(128)

473

Fig. 19. Two-center problem in a flat space. Write the Lagrangian of this system. We consider the Newtonian potential system, which is a particular case of a Lagrangian system. Every Lagrangian mechanical system is given by the configuration space (in this case, the configuration space is Euclidean) and the Lagrange function given on its tangent bundle. The Lagrange function is equal to the difference of the kinetic energy and the potential energy: 1 L = (x˙ 2 + y˙ 2 + z˙ 2 ) − V. 2 Introduce the following new coordinates (see Fig. 19): x = x ,

y = y  cos ϕ,

z = y  sin ϕ.

(129)

Then we obtain the following expression for the Lagrangian: 1 γ1 γ2 L = (x˙ 2 + y˙ 2 + y 2 ϕ) ˙ + + . (130) 2 r1 r2 By the axial symmetry of the problem, the angular coordinate ϕ does not enter the Lagrange function, and, therefore, it is cyclic (recall that a coordinate is cyclic iff it does not enter the Lagrange function, ∂L = 0). By the Noether theorem, the impulse corresponding to the cyclic coordinate is the first i.e., ∂ϕ integral ∂L pϕ = const = = y 2 ϕ. ˙ ∂ ϕ˙ In this case, the variation of other coordinates in time is the same as in a system with two independent coordinates in which pϕ plays the role of a parameter. Using the Routh method, we can exclude the cyclic coordinate:   2 p γ1 γ2 1 ϕ x˙ 2 + y˙ 2 − 2 + + . (131) R= 2 y r1 r2 The main difficulty in integrating differential equations consists in introducing convenient variables. The change of variables makes the problem visual and simplifies the analysis. So, for example, the wellknown Kepler problem is considerably simplified with passage to the polar coordinates (r, ϕ). Many problems of Hamiltonian mechanics are solved by using the elliptic coordinates introduced and studied by Jacobi. In the case of two fixed centers, it is also natural to pass from the Cartesian coordinates to the Jacobi elliptic coordinates, the roots (with respect to ω) of the equation y2 x2 + = 1, a21 − ω a22 − ω

(132)

where the constants a1 and a2 will be defined later, so that for any ω relation (132) defines a conical section whose foci coincide with the fixed attraction centers. Setting ω = 0, we find that the semiaxes a1 474

and a2 of ellipse (132) and the semifocus distance c are connected by the relation a21 − a22 = c2 . Further, we obtain from Eq. (132) the following transition formulas for the roots ω1 and ω2 : x2 =

(a21 − ω1 )(a21 − ω2 ) , a21 − a22

y2 =

(a22 − ω1 )(a22 − ω2 ) . a22 − a21

(133)

Then expressions (128) are rewritten in the form r12

2   2 2 = (x + c) + y = a1 − ω1 + a1 − ω2 ,

r22

2   = (x − c) + y = a21 − ω1 − a21 − ω2 .

2

2

2

(134)

2

We now set a1 = 0 (then a2 = ic) and −ω1 = λ2 , −ω2 = µ2 . As a result, Eqs. (134) become r1 = λ + µ,

r2 = λ − µ.

(135)

The inverse formulas are written as 1 1 µ = (r1 − r2 ). (136) λ = (r1 + r2 ), 2 2 The geometric treatment of these formulas is as follows. The equation λ = const represents an ellipsoid of revolution around the axis Ox whose focuses are located at the gravitational centers. The equation µ = const represents a hyperboloid of revolution around the axis Ox with the same focuses. The equation ϕ = const is the equation of a plane passing through the axis Ox. The coordinates of a point are defined by the intersection of these surfaces. The minimum value r1 + r2 corresponds to the case where a test particle is on the line connecting two centers r1 + r2 = 2c. The maximum value r1 − r2 corresponds to the case where the test particle is on one of the centers, i.e., r1 − r2 = c. This implies that the obtained quantities vary in the limits −c ≤ µ ≤ c ≤ λ. (137) In this case, the elliptic coordinates and the Cartesian coordinates are connected by the relations λ2 µ2 (λ2 − c2 )(µ2 − c2 ) 2 ≥ 0, y = ≥ 0. c2 −c2 Taking (135) into account, we write the Routh function (131) in the new coordinates: R = T − Veff ,   c2 k1 λ − k2 µ p2ϕ λ˙ 2 µ˙ 2 1 2 2 , + + − R = (λ − µ ) 2 λ2 − c2 c2 − µ2 λ2 − µ2 2 (λ2 − c2 )(c2 − µ2 ) x2 =

where 1 T = (λ2 − µ2 ) 2



µ˙ 2 λ˙ 2 + 2 2 2 λ −c c − µ2

 ,

c2 k1 λ − k2 µ p2ϕ Veff = − 2 . + λ − µ2 2 (λ2 − c2 )(c2 − µ2 ) Here, k1 = γ1 + γ2 and k2 = γ1 − γ2 . The corresponding energy integral is h = T + Veff .

(138)

(139)

(140)

The system with the Routh function (138) can be reduced to the Liouville form by using one more change of variables [48]:



dµ dλ √

, ξ= , (141) η= 2 2 2 λ −c c − µ2 475

so that λ = c cosh η, µ = c cos ξ. As a result, we obtain the following Routh function: p2ϕ c2 k1 c cosh η − k2 c cos ξ 1 (142) 2 − 2 . R = (c2 cosh2 η − c2 cos2 ξ)(η˙ 2 + ξ˙2 ) + 2 2 2 2 2 2 2 2 2 2 c cosh η − c cos ξ (c cosh η − c )(c − c cos ξ) Therefore, the variables are separated. Proposition 14.1. Along with integral (140), the system with the Routh function (142) has the following additional integral :   1 2 (λ − µ2 ) (c2 − µ2 )η˙ 2 + (λ2 − c2 )ξ˙2 2   2 1 1 k1 λ(c − µ2 ) + k2 µ(λ2 − c2 ) p2ϕ c2 − = L. (143) + + λ2 − µ2 2 f1 (η) f2 (ξ) Proof. Denote f1 (η) = λ2 − c2 and f2 (ξ) = c2 − µ2 and write the Routh equation   ∂R d ∂R − = 0. dt ∂ η˙ ∂η Precisely, p2ϕ c2   k1 λ − k2 µ ∂f1 d k1 ∂λ ∂f1 1 ∂f1 2 ˙2 [(f1 + f2 )η] η˙ + ξ + − − 22 = 0. ˙ − 2 dt 2 ∂η (f1 + f2 ) ∂η f1 + f2 ∂η f1 f2 ∂η ˙ as a result, we obtain the expression Multiply the latter equation by (f1 + f2 )η; d ˙ (f1 + f2 )η˙ ((f1 + f2 )η) dt   p2ϕ c2 p2ϕ c2         k1 λ − k2 µ ∂f1 ∂λ ∂f1 1 − η˙ (f1 + f2 ) η˙ 2 + ξ˙2 − η˙ = 0. + 2 − k1 η˙ − 22 ∂η  2 f1 + f2 f1 f2  ∂η f1 ∂η     The quantity in braces in the obtained expression is the total energy h of the system by the equation for η, and, therefore, it is constant. Therefore, we can represent expression (143) as   p2ϕ c2  d   1 (f1 + f2 )2 η˙ 2 − f1 h − k1 λ + 2  = 0.  dt 2 f1  The expression in parentheses is an integral independent of h. Then substituting the expression for h in this formula and using Eq. (141), we finally obtain relation (143). The latter can be expressed through the variables λ and µ once again:   2 k1 λ(c2 − µ2 ) + k2 µ(λ2 − c2 ) c − µ2 ˙ 2 λ2 − c2 2 1 2 2 − (λ − µ ) − µ ˙ λ 2 λ2 − c2 c2 − µ2 λ2 − µ2   p2ϕ c2 1 1 = L. (144) + − 2 2 2 2 λ −c c − µ2 To reduce the problem to quadratures, we multiply the energy integral by (λ2 −c2 ) and add the obtained expression to expression (144). As a result, we obtain   p2ϕ c2 c2 − µ2 ˙ 2 1 2 2 λ (λ − µ ) 1 + 2 = L + h(λ2 − c2 ). − k λ + 1 2 λ − c2 2(λ2 − c2 ) 476

R∗

S∗

λ

c

−c

λ2

µ1

c µ

O

Fig. 20 Further, after multiplying by (λ2 − c2 ), we obtain p2ϕ c2 1 2 (λ − µ2 )2 λ˙ 2 = (λ2 − c2 )(n + h(λ2 − c2 ) + γk1 λ) − . (145) 2 2 The potential in this problem has a singularity at the point with the coordinates λ = 0, µ = 0. Introduce a new independent variable τ by dt = (λ2 − µ2 ) dτ. The introduced variable monotonically increases when t grows. We carry out analogous calculations for the coordinate µ and write the equations of motion dµ dλ

= dτ, = dτ, R(λ) S(µ)

(146)

where R(λ) and S(µ) are polynomials of the fourth degree: R(λ) = 2(λ2 − c2 )(λ2 h + k1 λ + (L − hc2 )) − pϕ c2 , S(µ) = 2(µ2 − c2 )(µ2 h − k2 µ + (L − hc2 )) − pϕ c2 . Integrating Eqs. (146), we can obtain expressions for λ(τ ) and µ(τ ) in terms of elliptic functions. 14.2. Qualitative analysis. Obviously, the motion is possible only in those regions of the space in which the conditions R ≥ 0 and S ≥ 0 hold and also conditions (137) hold. In other words, the elliptic coordinates µ, ν must satisfy the inequalities max{λ1 , c} ≤ λ ≤ λ2 ,

max{µ1 , −c} ≤ µ ≤ min{µ2 , c}.

(147)

where λ1 and λ2 are nearest-to-each-other real roots of the equation R(λ) = 0 between which R(λ1 ≤ λ ≤ λ2 ) ≥ 0, and, moreover, λ ∈ [λ1 , λ2 ]. The case of µ1 and µ2 is analogous. Therefore, the boundaries of the range of variables λ and µ can depend on constants of first integrals in a complicated way, which stipulates various types of motion. Let us construct the bifurcation diagram on the plane h, l = L − hc2 of first integrals for the particular case pϕ = 0, where the point executes the motion in one fixed plane. Taking into account condition (137), we obtain that λ1 , λ2 , µ1 , and µ2 are roots of the quadratic trinomials R∗ (λ) = hλ2 + k1 λ + l and S ∗ (µ) = −hµ2 + k2 µ + l ; moreover, in the real motion, it must be R∗ (λ) ≥ 0 and S ∗ (µ) ≥ 0. In Fig. 20, we present one of the possible situations (for h < 0) of the behavior of the functions R∗ and S ∗ . The shaded regions correspond to the zones of possible motions. The motion on the configurational space depicted in Fig. 21 corresponds to the indicated zones. The presented type of motion corresponds to Region 1 on the bifurcation diagram (Fig. 22). Consider the following situations. It follows from the expressions R∗ (λ) = 0 and S ∗ (µ) = 0 that    1 2 −k1 ± k1 − 4lh , λ1,2 = 2h 477

Fig. 21 l 5 6

4

7

3

8 h

2 1

Fig. 22. Bifurcation diagram for the two-center problem in the flat case.

µ1,2

1 = 2h

   2 k1 ± k2 − 4lh ;

l l , λ2 = ∞ and µ1 = , µ2 = ∞. Our aim is to depict on the plane (h, l) the sets k1 k2 of those pairs of constants which correspond to singular types of motion of the point. The latter arise when we have multiple roots, i.e., in the following cases: 1. λ1 or λ2 = c; 2. µ1 or µ2 = −c, or µ1 , or µ2 = ; 3. λ1 = λ2 or µ1 = µ2 . The lines 1. l = −c2 h − ck1 ; 2. l = −c2 h − ck2 and l = −c2 h + ck2 ; 3. 4lh = k12 and 4lh = k22 correspond to the indicated cases. if h = 0, then λ1 = −

Proposition 14.2. The bifurcation set in the two-center problem on the plane consists of two hyperbolas 4lh = k12 ,

4lh = k22

(148)

and the straight lines l = −c2 h − ck1 , 478

l = −c2 h − ck2 ,

l = −c2 h + ck2 ,

h = 0.

The straight line h = 0 separates the bounded and unbounded motions. The bifurcation diagram is presented in Fig. 22. We have eight distinct regions of possible motion (RPM are marked by digits). We do not dwell on the analysis of types of motion on the configuration space, since an analysis is presented in the book [15] in detail. 15.

Description of the System on the Sphere. Reduction

Let r1 and r2 be the radius-vectors (with origins at the center of the sphere) of fixed attracting centers, and let r be the radius vector of a test particle. Then the potential of the two-center problem on the sphere is γ2 γ1 cot θ2 , (149) V = − cot θ1 − R R where R is the radius of the sphere, γ1 and γ2 are positive constants characterizing the attraction force, and θi is the angle made by the vectors ri and r. Denote by P1 and P2 the attracting centers, and by Q1 and Q2 the points diametrally opposite to them. 1 It is seen from the relation for the potential V that this potential has singularities of type − at the r 1 points P1 and P2 and singularities of type at the points Q1 and Q2 , i.e., for the Newtonian potential, r the existence of an attracting center leads to an additional repelling center (at the antipodal point). Consider the three-dimensional sphere of radius R embedded into the space R4 with coordinates q0 , q1 , q2 , q3 in the standard way. Its equation is (q0 )2 + (q1 )2 + (q2 )2 + (q3 )2 = R2 . Let the attracting centers be located at the points with coordinates r1 = (α, β, 0, 0) and r2 = (−α, β, 0, 0), α > 0, β > 0, α2 + β 2 = R2 ; the material point moves under the action of them. Introduce the spherical coordinate system. The transition formulas are written as q0 = R cos θ, q2 = R sin θ sin ϕ cos ψ,

q1 = R sin θ cos ϕ, q3 = R sin θ sin ϕ sin ψ.

Then the Jacobi matrix is   cos θ −R sin θ 0 0  sin θ cos ϕ  R cos θ cos ϕ −R sin θ sin ϕ 0   sin θ sin ϕ cos ψ R cos θ sin ϕ cos ψ R sin θ cos ϕ cos ψ −R sin θ sin ϕ sin ψ  . sin θ sin ϕ sin ψ R cosh θ sin ϕ sin ψ R sin θ cos ϕ sin ψ R sin θ sin ϕ cos ψ After calculations, we obtain the Jacobian equal to J = R3 sin2 θ sin ϕ. The metric induced on the three-dimensional sphere is written as ds2 = R2 (dθ2 + sin2 θdϕ2 + sin2 θ sin2 ϕdψ 2 ). The Lagrangian L is defined by the expression 1 L = R2 (θ˙2 + sin2 θ(ϕ˙ 2 + sin2 ϕψ˙ 2 )) − V 2 with respect to the spherical coordinates R, θ, ϕ, ψ.

(150)

(151)

Theorem 15.1. A material point in the two-center problem in the space S3 moves in the same way as in the two-dimensional system (on the unit two-dimensional sphere S2 : y 2 + x2 + z 2 = 1) with the energy 1 h = (y˙ 2 + x˙ 2 + z˙ 2 ) + Veff , 2 where the effective potential energy is defined by the expression Veff = −γ1 cot θ1 − γ2 cot θ2 +

p2ϕ . 2z 2 479

Proof. Without loss of generality, we assume here that the sphere is of unit radius. Let the material point move on the unit three-dimensional sphere in the field of two fixed centers. Let us pass to the new variables q0 = x, q1 = y, q2 = z cos ϕ, and q3 = z sin ϕ. The Lagrange function with respect to the new variables is 1 (152) L = (x˙ 2 + y˙ 2 + z˙ 2 + z 2 ϕ˙ 2 ) − V. 2 Since the Lagrangian is independent of the variable ϕ, the system admits the symmetry group g α : ϕ → ϕ + α, x → x, y → y, z → z. The vector field ∂/∂ϕ corresponds to this group. According to the Noether ∂L is conserved. The coordinate ϕ is said to be cyclic in mechanics. We can theorem, the quantity pϕ = ∂ ϕ˙ exclude it using the Routh method: ∂L = pϕ = const, (153) ∂ ϕ˙ 2 where pϕ is the generalized impulse corresponding to the coordinate ϕ and depending on the initial conditions. As a result, we obtain the following Routh function:

where Veff

1 R = (x˙ 2 + y˙ 2 + z˙ 2 ) − Veff , 2 is the reduced potential of the form

(154)

p2ϕ . (155) 2z 2 Therefore, we have reduced the problem on the motion of a material point on the sphere in the field of two fixed centers to the two-dimensional case, i.e., to the motion on the two-dimensional sphere x2 + y 2 + z 2 = 1 in the field with the reduced potential (155). Veff = −γ1 cot θ1 − γ2 cot θ2 +

16.

Integrals of the System

The coordinates on the sphere in which the Hamiltonian of the two-center problem has the Liouville form were indicated in [28]. These are the spheric-conic coordinates ξ, η. They are defined as follows. Consider the following equation (with respect to λ): y2 z2 x2 = 0. + + λ − α2 λ + β 2 λ It is easy to verify that its roots have distinct signs. Denote them by ξ 2 and −η 2 and obtain the coordinates (ξ, η), where 0 ≤ ξ ≤ α and 0 ≤ η ≤ β. The lines (ξ = const, η = const) of intersection of the sphere with two families of confocal cones with vertices at the centers of the sphere and the quadric are coordinate lines of this system. In many respects, the geometric properties of these lines on the sphere are analogous to those of the ordinary ellipses and hyperbolas on the plane. Moreover, they are “Kepler” orbits in the problem of motion of a point on the sphere in the field generated by a single center (see [26]). The formulas expressing the Cartesian coordinates in terms of the spheric-conic coordinates are 1 2 (α − ξ 2 )(α2 + η 2 ), α2 1 y 2 = 2 (β 2 + ξ 2 )(β 2 − η 2 ), β R2 z2 = 2 2 ξ2 η2 . α β

x2 =

Obviously, extracting roots, we obtain distinct signs for x and y depending on which hemisphere we consider, the left or the right, the upper or the lower. Therefore, we obtain the following expressions for 480

the transition formulas:

1 sign(x) (α2 − ξ 2 )(α2 + η 2 ), α

1 y = sign(y) (β 2 + ξ 2 )(β 2 − η 2 ), β R ξη. z= αβ The introduced curvilinear variables are orthogonal and satisfy the conditions x=

ξ 2 ≤ α2 ,

η2 ≤ β 2.

(156)

(157)

Consider the potential (149): V =

−γ1 cos θ1 sin θ2 − γ2 cos θ2 sin θ1 . R sin θ1 sin θ2

Clearly, cos θi = ±αx + βy,

sin2 θi = 1 − cos2 θi .

Then we obtain the following relations: sin θ1 sin θ2 = ξ 2 + η 2 ,

cos θ1 = sign(x) (α2 − ξ 2 )(α2 + η 2 ) + sign(y) β 2 + ξ 2 )(β 2 − η 2 ),

sin θ1 = sign(x) (α2 + η 2 )(β 2 + ξ 2 ) − sign(y) (α2 − ξ 2 )(β 2 − η 2 ),

cos θ2 = −sign(x) (α2 − ξ 2 )(α2 + η 2 ) + sign(y) β 2 + ξ 2 )(β 2 − η 2 ),

sin θ2 = sign(x) (α2 + η 2 )(β 2 + ξ 2 ) + sign(y) (α2 − ξ 2 )(β 2 − η 2 ). The Hamiltonian function with respect to the new coordinates is defined by the expression H = T + Veff , (α2 − ξ 2 )(β 2 + ξ 2 ) 2 (α2 + η 2 )(β 2 − η 2 ) 2 pξ + pη , 2(ξ 2 + η 2 ) 2(ξ 2 + η 2 )

−sign(y)(γ1 + γ2 ) (α2 + η 2 )(β 2 − η 2 ) = R(ξ 2 + η 2 )

−sign(x)(γ1 − γ2 ) (α2 − ξ 2 )(β 2 + ξ 2 ) p2ϕ α2 β 2 (ξ −2 + η −2 ) + , + R(ξ 2 + η 2 ) 2R(ξ 2 + η 2 )

T = Veff

(158)

where T and Veff are the kinetic and reduced potential energies and pξ and pη are the impulses corresponding to the coordinates ξ and η. (The coefficient γ1 corresponds to the attracting center with the coordinates (α, β, 0, 0), and the coefficient γ2 corresponds to the attracting center with the coordinates (−α, β, 0, 0)). The functions sign(x) and sign(y) describe the potential in a quarter of the plane: x > 0, y > 0; x < 0, y < 0; x > 0, y < 0; x < 0, y > 0. In the coordinates (ξ, η), the Hamiltonian has the Liouville form, which allows us to write the integrals of the system considered: the energy integral h = T + Veff ,

(159)

and two Liouville integrals (which are linearly independent) p2ϕ α2 β 2 ξ −2 1 sign(x)(γ1 − γ2 ) 2 (α − ξ 2 )(β 2 + ξ 2 ) + − hξ 2 , I1 = (α2 − ξ 2 )(β 2 + ξ 2 )p˙2ξ − 2 R 2R p2ϕ α2 β 2 η −2 1 sign(y)(γ1 + γ2 ) 2 I2 = (α2 + η 2 )(β 2 − η 2 )p˙2η − − hη 2 . (α + η 2 )(β 2 − η 2 ) + 2 R 2R

(160)

481

The additional integral can be written in the following symmetric form: (α2 − ξ 2 )(β 2 + ξ 2 )η 2 2 (α2 + η 2 )(β 2 − η 2 )ξ 2 2 pξ − pη 2(ξ 2 + η 2 ) 2(ξ 2 + η 2 )  2 α2 β 2 η −2

p ϕ ξ2 sign(y)(γ1 + γ2 ) (α2 + η 2 )(β 2 − η 2 ) − 2

L=  + 

R(ξ 2 + η 2 )

p2ϕ α2 β 2 ξ −2 2 2 2 2 −sign(x)(γ1 − γ2 ) (α − ξ )(β + ξ ) + 2

 η2

. R(ξ 2 + η 2 ) In this problem, the potential has a singularity at the point with the coordinates ξ = 0, η = 0. Instead of t, we introduce the new variable τ (the new time monotonically increases as t grows) by +

ξ2 + η2 dτ. dt = 2(γ1 + γ2 ) l h ,l→ , and reduce the problem to the quadratures: γ1 + γ2 γ1 + γ2

dξ dη = R(ξ), = S(η), dτ dτ where R(ξ) and R(η) are irrational functions defined by Make the change h →

(161)

R(ξ) = (α2 − ξ 2 )(β 2 + ξ 2 )R∗ (ξ), S(η) = (α2 + η 2 )(β 2 − η 2 )S ∗ (η), where

(α2 − ξ 2 )(β 2 + ξ 2 ) ,

R sign(y) (α2 + η 2 )(β 2 − η 2 ) S ∗ (η) = −l + hη 2 + . R ∗

2

R (ξ) = l + hξ +

sign(x)K

(162)

γ1 − γ 2 is a parameter and h and l are integrals depending on the initial data. As was said γ1 + γ2 above, the functions R∗ and S ∗ describe the motion in a quarter of the sphere: x > 0, y > 0; x < 0, y < 0; x > 0, y < 0; x < 0, y > 0. When the coordinates ξ and η assume the values ±α and ±β, respectively, the particle passes from one quarter to another (156). When these values are attained, the functions R∗ and ∗ , R∗ , S ∗ , and S ∗ , S ∗ describing the change of ξ and η must be changed, and we have four functions R+ − + − i.e., four combinations of R∗ S ∗ for each of the quarters of the sphere depending on sign(x) and sign(y). As was already noted, the properties of coordinate lines of the coordinate system (ξ, η) are analogous to those of ellipses and hyperbolas on the plane. For example, for each point of the coordinate line {η = const}, the sum of distances (on the sphere) from this point to the attracting centers P1 and P2 and also the difference between the distances from this point to the points P1 and Q2 are constant. An analogous property also holds for the coordinate lines {ξ = const}. Using this fact, we can write the Hamiltonian and the integral of the problem considered in a more visual coordinates q1 and q2 , where Here, K =

q1 = θ2 − θ 1 ,

q2 = θ 2 + θ 1 ,

and θ1 and θ2 are angular values of the arcs connecting the point considered with the centers P1 and P2 . Let pϕ = 0. Introducing the impulses p1 and p2 corresponding to the coordinates q1 and q2 , we can write the Hamiltonian and the integral as 2(cos q1 − cos δ) 2 2(cos δ − cos q2 ) 2 (γ1 − γ2 ) sin q1 + (γ1 + γ2 ) sin q2 p + p − , H= cos q1 − cos q2 1 cos q1 − cos q2 2 R(cos q1 − cos q2 ) 482

L=

2 cos q2 (cos δ − cos q1 ) 2 2 cos q1 (cos δ − cos q2 ) 2 γ1 sin(q1 + q2 ) + γ2 sin(q2 − q1 ) , p1 − p2 + cos q1 − cos q2 cos q1 − cos q2 R(cos q1 − cos q2 )

where δ denotes the angular value of the arc between the centers P1 and P2 . The coordinates q1 , q2 (as well as the coordinates ξ, η) are naturally not global coordinates on the sphere (there are no such coordinates on the sphere). The coordinate system (q1 , q2 ) has singularities at the points of intersection of the sphere with the plane connecting the centers. In particular, the same coordinates q1 , q2 correspond to points of the sphere symmetric with respect to this plane. However, if the variables separate, then their coordinate lines are uniquely defined (if the system is nonresonant) by the system itself, since they bound the projections of the Liouville tori on the configuration space. Therefore, for the system considered, any other “good” (i.e., separated) variables must have the form of functions q˜1 (q1 ) and q˜2 (q2 ). Let us describe coordinates u, v on the sphere (with the same coordinate lines), which are more convenient for calculating topological invariants of the system. It is well known that the two-dimensional sphere admits a two-sheeted covering by a two-dimensional torus with four branch points. The covering can be chosen so that the attracting centers P1 and P2 and the points Q1 and Q2 diametrally opposite to them are branch points and the inverse images of the “ellipses” {q1 = const} and {q2 = const} are the coordinate lines of the (global) angular coordinates u, v on the torus. This covering can be described by using Jacobi elliptic functions (see, e.g., [4]). Consider the mapping of the torus T 2 with angular coordinates u, v into the space R3 with the Cartesian coordinates x, y, z given by the relations x = R sn(u, k1 ) dn(v, k2 ), y = R sn(v, k2 ) dn(u, k1 ),

(163)

z = R cn(u, k1 ) cn(v, k2 ). (where, as above, δ is the angular value of the arc between the centers P1 and P2 ). In what follows, for brevity, we do not indicate the module, meaning that for the Jacobi functions of the variable u, the module equals k1 , and for the Jacobi functions of the variable v, it is equal to k2 . Using the properties of the Jacobi functions, we easily verify that under the mapping given by relations (163), the image of any point of the torus (u, v) is a point on the sphere {x2 + y 2 + z 2 = R2 }. Moreover, two points of the torus are mapped into each point of the sphere (except for the points P1 , P2 , Q1 , Q2 ). Therefore, mapping (163) is a two-sheeted covering of T2 → S2 branched in four points. It is convenient to imagine this covering as follows. The first two of the relations (163) define a continuous one-to-one mapping of the rectangle {|u| ≤ K1 , |v| ≤ K2 } onto the disk {x2 + y 2 ≤ R2 }, where K1 and K2 are complete elliptic integrals of the first kind corresponding to the modules k1 and k2 . Moreover, the vertices of the rectangle transform into the points of the boundary circle with the coordinates (±α, ±β). This mapping extends to the whole plane R2 (u, v) by using symmetries with respect to sides of the rectangle. Taking into account the third relation in (163), we obtain the mapping of the plane R2 (u, v) onto the sphere {x2 + y 2 + z 2 = R2 }. In Fig. 23, we depict the partition of the plane R2 (u, v) into rectangles with sides 2K1 an 2K2 each of which is mapped onto a hemisphere (dotted rectangles are mapped onto the “upper” hemisphere {z ≥ 0}, and nondotted are mapped onto the “lower” hemisphere {z ≤ 0}). In this case, “black” vertices of rectangles are mapped into the attracting centers P1 and P2 , and “white” vertices into the repelling centers Q1 and Q2 . Since the functions snu and cnu are of period 4K1 and the functions snv and cnv are of period 4K2 , the described mapping of the plane R2 (u, v) onto the sphere S2 defines the mapping T2 → S2 , where the torus T2 can be considered as a rectangle in the plane R2 (u, v) with sides 4K1 and 4K2 (consisting of two dotted rectangles and two nondotted rectangles with a common vertex) in which the pair of opposite sides are identified by shifts. The central symmetry of the plane R2 (u, v) with respect to each of the vertices of rectangles (see Fig. 23) defines the involution σ : T2 → T2 with four fixed points. The quotient space T2 /σ is the configuration 483

v K2 K1

u

Fig. 23 space S2 of the problem considered. Therefore, instead of the motion of the point on the sphere S2 , we can consider the motion of the point on the torus T2 , taking into account the action of the involution σ. More precisely, this procedure will be described in Sec. 17. The Hamiltonian and the integral in the variables u, v (i.e., those describing the motion of the point on the torus T2 ) are (γ − γ2 ) sin δsnudnu + (γ1 + γ2 ) cos δsnvdnv p2u + p2v %− 1 $ % , H= $ 2 2 2 2 2 sin δcn u + cos δcn v R sin2 δcn2 u + cos2 δcn2 v L=

(γ − γ2 ) cos δsnudnucn2 v − (γ1 + γ2 ) sin δsnvdnvcn2 u cot δcn2 vp2u − tan δcn2 up2v $ % − 1 $ % , 2 tan δcn2 u + cot δcn2 v R tan δcn2 u + cot δcn2 v

where pu and pv are the impulses corresponding to the coordinates u and v. 17.

Regularization

The potential V of the problem considered has singularities at four points on the sphere (the attracting centers P1 and P2 and the repelling centers Q1 and Q2 ). Moreover, at the points P1 and P2 , the function V tends to −∞, and at the points Q1 and Q2 , it tends to +∞. Since the kinetic energy T is always positive and the total energy H = T + V is constant along the trajectories of the system, this implies that the particle moving on the sphere in the field generated by the potential V never attains the points Q1 and Q2 . For the points P1 and P2 , the situation is opposite: for any location of the particle on the sphere, we can define an initial velocity such that the particle attains one of the attracting centers for a finite time. In this case, at the “instant of attaining the attracting center,” the velocity of the particle becomes infinitely large (since T + V = const). Therefore, the two-center problem on a sphere is described by the Hamiltonian system on the cotangent bundle T ∗ S2 of the two-dimensional sphere with the Hamiltonian H = T + V , where T is a quadratic function with respect to impulses (the standard metric on the sphere) and V is the function on the sphere given by formula (149). However, in this approach, the phase space of the system is not the whole manifold T ∗ S2 , since the function V is not defined at the four points P1 , P2 , Q1 , and Q2 of the sphere (and, therefore, the function H is not defined on the four planes that are the leaves of the cotangent bundle T ∗ S2 over these four points). Denote by S0 the sphere S2 with four points P1 , P2 , Q1 , and Q2 removed. Then the phase space of the system is T ∗ S0 . As was already mentioned above, the Hamiltonian vector field w = sgrad H on T ∗ S0 defining the system is not complete. Therefore, although the system has an additional integral L, it is not Liouville integrable. Nevertheless, as will be shown below, after a certain regularization, the qualitative behavior of the system becomes the same as for Liouville integrable Hamiltonian systems (almost all trajectories are conditionally periodic windings of tori). Note that the method for regularization of the system described below is analogous to the regularization suggested by T. Levi-Civita for the classical Kepler problem (see [29]). 484

Consider the Hamiltonian H as a function of the variables (u, v, pu , pv ). The system with such a Hamiltonian can be considered as a Hamiltonian system on the cotangent bundle of the torus T2 . Introduce the notation (164) λ(u, v) = sin2 δcn2 u + cos2 δcn2 v. Then the Hamiltonian becomes (γ1 − γ2 ) sin δsnudnu + (γ1 + γ2 ) cos δsnvdnv p2 + p2v − , H= u 2λ(u, v) R · λ(u, v)   ∂H ∂H ∂H ∂H ∗ 2 and the coordinates of the vector field W = sgrad H on T T are equal to ,− . , ,− ∂pu ∂pv ∂u ∂v In the phase space T ∗ T2 , the vector field W has singularities at those points at which λ(u, v) = 0, ˜ = λ(u, v) · sgrad H. In the i.e., at points of the form (±K1 , ±K2 , pu , pv ). Consider the vector field W coordinates (u, v, pu , pv ) it is as follows:    γ1 − γ 2 2 pu , pv , sin δcnu (2dn u − 1) − 2 sin δsnudnu · h , R   (165) γ1 + γ 2 2 (2dn v − 1) − 2 cos δsnvdnv · h , cos δcnv R ˜ where h = H(u, v, pu , pv ) is the value of the Hamiltonian at the point (u, v, pu , pv ). The vector field W also has singularities at the points (±K1 , ±K2 , pu , pv ), since the Hamiltonian H is not defined at those ˜ to the isoenergetic points at which λ(u, v) = 0. Denote by Wh the restriction of the vector field W ∗ 2 surface Qh = {H = h} ⊂ T T . Now, the vector field Wh has no singularities (it is defined on only the three-dimensional surface Qh ). It is given by the relation (165) and, in particular, is defined at points of the form (±K1 , ±K2 , pu , pv ) lying on the surface Qh : Wh (±K1 , ±K2 , pu , pv ) = (pu , pv , 0, 0). Clearly, the integral trajectories of the field Wh coincide (with accuracy up to change of the parameter) with the integral trajectories of the initial vector field W = sgrad H on T ∗ T2 , since the multiplication of dt = λ(u(t), v(t)), where the field W by the function λ(u, v) can be interpreted as the change of time dτ (u(t), v(t), pu (t), pv (t)) is a trajectory of the field W . On the other hand, the vector field Wh on the surface Qh coincides with the restriction to this surface of a certain Hamiltonian vector field defined on the whole phase space T ∗ T2 . Clearly, such an extension is not unique. For example, as such a field, we can take the field sgrad Fh , where $ % p2 + p2v − h sin2 δcn2 u + cos2 δcn2 v Fh = λ(H − h) = u 2 (166) γ1 + γ 2 γ1 − γ 2 sin δsnudnu − cos δsnvdnv. − R R Then sgrad Fh = λ sgrad H + (H − h)sgrad λ. Since {Fh = 0} = {H = h}, the vector field sgrad Fh is tangent to the surface Qh and coincides with the field Wh on it. The integral L of the initial system is obviously an integral of the Hamiltonian system with the Hamiltonian Fh on the surface {Fh = 0}. Therefore, after the described regularization, on each isoenergetic surface Qh , the topological properties of systems with the Hamiltonian H on T ∗ T2 are the same as for the usual integrable Hamiltonian systems. In particular, nonsingular integral manifolds of the system are Liouville tori, and surgeries of these tori can be described by Fomenko–Zieshang invariants. Up to now, we, in fact, speak of the regularization of the system on T ∗ T2 , which arose from the consideration of the (branched) covering of the sphere S2 by the torus T2 . This covering is defined by the involution σ : T2 → T2 described in Sec. 16. The involution σ naturally extends to the involution σ ∗ : T ∗ T2 → T ∗ T2 . Now, to return to the system on the sphere (precisely this system is the main object of our study), we must take into account the action of the involution σ ∗ on T ∗ T2 . 485

Since the involution σ : T2 → T2 is generated by the central symmetry of the plane R2 (u, v) with respect to the point (K1 , K2 ) (or any other vertex of the rectangles in Fig. 23), the involution σ ∗ has the following form in the coordinates (u, v, pu , pv ): σ ∗ : (u, v, pu , pv ) → (2K1 − u, 2K2 − v, −pu , −pv ).

(167)

Therefore, the involution σ ∗ has exactly four fixed points (±K1 , ±K2 , 0, 0). Note that the quotient space T ∗ T2 /σ ∗ is not a manifold. We now fix a certain value h and consider the function Fh given by relation (166). It is easy to see that the surface {Fh = 0} is invariant with respect to the involution σ ∗ and does not contain the points (±K1 , ±K2 , 0, 0) (by a direct calculation, we verify that at these four points, the values of the function γi Fh are equal to ± sin δ, where i = 1, 2). Therefore, the quotient spaces {Fh = 0}/σ ∗ can be considered R as isoenergetic surfaces of the initial system on the sphere after the regularization. Moreover, we also easily verify that the vector field on the surface {Fh = 0} is invariant with respect to the involution σ ∗ , which allows to consider the vector field wh = Wh /σ ∗ as a result of the regularization of the initial vector field w = sgrad H on the isoenergetic surface Qh = {H = h} ⊂ T ∗ S2 . Therefore, the presented arguments lead to the following assertion describing the regularization of the two-center problem on the sphere, i.e., the Hamiltonian system w = sgrad H on the cotangent bundle T ∗ S2 of the sphere with the Hamiltonian H = T + V , where the function T quadratic with respect to impulses is defined by the standard metric on the sphere of radius R in R3 and the function V is given by relation (149). Theorem 17.1 (on the regularization). Let h be a regular value of the Hamiltonian H and let Qh = {H = h} ⊂ T ∗ S2 be the corresponding isoenergetic surface. On the surface Qh , let us consider the vector field wh = λ sgrad H, where λ is function (164) on the sphere S2 . Let f : T2 → S2 be a (branched ) two-sheeted covering (163), σ ∗ : T ∗ T2 → T ∗ T2 be the corresponding involution (167), and Fh be the function defined by relation (166) on the cotangent bundle T ∗ T2 to the torus. On the surface {Fh = 0} ⊂ T ∗ T2 , let us consider the vector field Wh = sgrad Fh . Then (1) the surface {Fh = 0} ⊂ T ∗ T2 is a closed three-dimensional manifold on which the involution σ ∗ acts without fixed points; (2) the vector field Wh on the surface {Fh = 0} has no singularities and is invariant with respect to the involution σ ∗ ; (3) mapping (163) induces a diffeomorphism of the quotient space (with respect to the involution σ ∗ ) of the surface {Fh = 0} without points lying in four leaves over the branch points of the mapping f onto the surface {H = h}, and, moreover, this diffeomorphism transforms the vector field Wh into the vector field wh . Note that the irregular values h of the Hamiltonian H are explicitly written in constructing the bifurcation diagram. Also, we note that the indicated procedure for regularization of the system can be described without passing to the two-sheeted covering T2 → S2 . However, it turns out that the consideration of such a “covering” system considerably simplifies the calculation of the Fomenko–Zieshang invariants. 18.

Bifurcation Diagrams

The phase space is not compact for the problem considered. Moreover, even the isoenergetic surfaces are not compact because the Hamiltonian has singularities (as a function on T ∗ S2 ). However, we will study the topology of the “regularized” system for which the isoenergetic surfaces are compact. Precisely, instead of the surface {H = h}, we will consider the surface {Fh = 0} and study its foliation into level surfaces of the additional integral. After that, taking into account the action of the involution, we obtain the description of an analogous foliation for the regularized isoenergetic surface of the initial system. 486

In this case, as an additional integral on the surface {Fh = 0}, instead of the function L having a singularity, we can take another function that has no singularity. For example, as such a function, we can take the function sin2 δcn2 u − cos2 δcn2 v Fh . Kh = 2L + sin2 δcn2 u + cos2 δcn2 v Calculating, we obtain   γ −γ γ1 + γ2 p2u − p2v 1 2 2 2 2 2 + h cos δcn v − sin δcn u − sin δsnudnu + cos δsnvdnv. (168) Kh = 2 R R The following assertion is verified by a direct calculation. Lemma 18.1. Let Fh and Kh be functions on T ∗ T2 given by formulas (166) and (168), respectively. Then (1) {Fh , Kh } ≡ 0; (2) the set {Fh = 0, Kh = k} is given by the equations 2(γ1 − γ2 ) sin δsnudnu + k, R (169) 2(γ1 + γ2 ) 2 2 2 cos δsnvdnv − k; pv = 2h cos δcn v + R (3) under the diffeomorphism described in item (3) of Theorem 17.1, the points of the set {Fh = 0, Kh = 2l} transform into the points of the set {H = h, L = l}. p2u = 2h sin2 δcn2 u +

Therefore, the construction of the bifurcation diagram of the momentum mapping for the (regularized) two-center problem on the sphere can be carried out as follows. For each value h, find critical points of the function Kh restricted to the surface {Fh = 0} and then collect the obtained critical values in the curves with the parameter h on the plane R2 (h, k). The second assertion of the lemma means, in fact, that these critical values correspond to “multiple roots” of the functions on the right-hand sides of Eqs. (169). Finding those pairs (h, l) (where k = 2l) for which such roots exist, we obtain an answer.

Fig. 24. Bifurcation diagram for the case 0 < δ < π/2. The bifurcation diagrams are depicted in Fig. 24. The upper diagram is depicted for the case 0 < δ < π/2. For δ > π/2, the qualitative form of the bifurcation diagram is the same, but singular points (i.e., tangent points and intersection points of bifurcation curves) can be located in a distinct way depending 487

Fig. 25. Bifurcation diagrams for the case δ > π/2.

on the relations between cos δ and γ1 /γ2 . Two examples of bifurcation diagrams for the case δ > π/2 are presented in Fig. 25. As will be clear from what follows, for studying the topology of the system considered on a certain isoenergetic surface Q3h = {H = h}, only the mutual location of the vertical line {h = const} and singular points of the bifurcation diagram on the plane R2 (h, l) is essential. Using the explicit formulas presented below, we easily verify that there are 12 variants of such a location. They correspond to isoenergetic surfaces with distinct Fomenko–Zieshang invariants (see Chapter 1). The vertical lines hi , 1 ≤ i ≤ 12, are indicated in Fig. 24. Moreover, here (and everywhere in what follows), it is assumed that γ1 > γ2 . Theorem 18.1. The equations of bifurcation curves in the two-center problem on the sphere are (γ1 −γ2 )2 , R2 (γ1 +γ2 )2 Γ2 : (2l−h cos δ)2 −h2 = , R2 γ1 +γ2 Γ3 : l= sin δ, 2R Γ1 : (2l−h cos δ)2 −h2 =

γ1 −γ2 sin δ, 2R γ1 −γ2 sin δ, Γ5 : l= − 2R γ1 +γ2 Γ6 : l= − sin δ. 2R Γ4 : l=

Also, write the bifurcation set as 1 1 4h2 α2 β 2 + 2 4lh(β 2 − α2 ) − R2 R 1 1 4h2 α2 β 2 + 2 4lh(β 2 − α2 ) − Γ2 : 2 R R αβ Kαβ , Γ4 : l = , Γ3 : l = R R αβ Kαβ Γ6 : l = − . Γ5 : l = − , R R

Γ1 :

1 2 4l + K 2 = 0, R2 1 2 4l + 1 = 0, R2

The tangent points of the hyperbola Γ1 with the straight lines Γ4 and Γ5 have the coordinates     γ1 − γ 2 γ1 − γ2 γ1 − γ 2 γ1 − γ 2 cot δ, sin δ , d= cot δ, − sin δ . c= − R 2R R 2R 488

(170)

The tangent point of the hyperbola Γ2 with the straight line Γ3 has the coordinates   γ1 + γ2 γ 1 + γ2 cot δ, sin δ . b= − R 2R The intersection points of the hyperbola Γ1 with the straight lines Γ3 and Γ6 have the coordinates   √ (γ1 +γ2 ) cos δ+2 γ1 γ2 γ1 +γ2 , sin δ , a= − R sin δ 2R   √ (γ1 +γ2 ) cos δ+2 γ1 γ2 γ1 +γ2 e= ,− sin δ . R sin δ 2R Two irregular values h of the Hamiltonian correspond to the latter two points. Other values of the Hamiltonian are regular. The bifurcation set partitions the image of the momentum mapping into regions so that under the motion of the point inside such a region, the Liouville tori in the image of this point have no bifurcation. Therefore, a certain number of Liouville tori corresponds to each of the regions. These numbers can be found by studying functions (169). In Fig. 24, the numbers of the regions are marked (by the digits 1–6) and the numbers of the Liouville tori corresponding to them in the inverse image are indicated (in regions 1, 3, and 5 there is one torus, and in regions 2, 4, and 6 there are two tori). 19.

Topological Analysis of the Two-center Problem on the 2-Sphere

In this section, we calculate the Fomenko–Zieshang invariants for the integrable Hamiltonian system considered. 19.1. Bott property of the integral. As was noted above, the Fomenko–Zieshang invariants are defined for isoenergetic surfaces on which the additional integral has the Bott property. For arbitrary integrable system, the verification of the Bott property for the additional integral is as usual nontrivial. However, for system in which the variables are separated (as in the case considered), the calculations are simplified considerably. In the problem considered, the verification of the Bott property in fact reduces to the verification of the nondegeneracy condition for critical points of the functions (of one variable) defined by the right-hand sides of relations (169). Calculating them, we obtain the following assertion. Proposition 19.1. The additional integral has the Bott property on all isoenergetic surfaces Qh = {H = h}, except for those for which h is the abscissa of one of the tangent points of the bifurcation curves. 19.2. Regions of possibility of motion. It is convenient to carry out the study of topology of natural mechanical systems by using the projection on the configuration space. In this case, each Liouville torus projects on a certain domain called a region of possibility of motion (for the values h and k of the Hamiltonian H and the integral K defining this torus). We first consider the projections of the Liouville tori on the configuration space T2 , i.e., for the “covering system.” Each such torus is given in T ∗ T2 by Eq. (169) with certain constants h and k, i.e., by the equations p2u = f (u),

p2v = g(v).

Since f and g are functions of distinct variables, the projection of the Liouville torus is a certain domain on T2 bounded  by lines of the form {u = const} and {v = const}. In this case, four  the torus

points u, v, ± f (u), ± g(v) ∈ T ∗ T2 project into each interior point (u, v) of this domain, and on the boundary of the domain, these points are glued into two points or one point. Therefore, the form of the projection of the Liouville torus given by the constants h and k is completely determined by the signs of the expressions f (u) and g(v) from formulas (169) and is the same for all 489

points (h, k) lying in the same region from the partition of the plane R2 (h, k) produced by the bifurcation diagram. The bifurcation diagram partitions the region R2 (h, k) into six regions marked by digits in Fig. 24 (here, k = 2l; see the Lemma 18.1). Finding the signs of the expressions f (u) and g(v) for each of these regions, we obtain the following projections of the Liouville tori (see Fig. 26).

Fig. 26. Projections of the Liouville tori on T2 . In Fig. 26, we depict the torus T2 in the form of a rectangle composed of four rectangles of the plane with coordinates (u, v) (see Fig. 23). In this case, the black and white vertices have the same meaning as in Fig. 23, and the left lower and right upper rectangles in Fig. 26 correspond to the dotted rectangles in Fig. 23. The dotted regions in Fig. 26 depict the projections of the Liouville tori. Several Liouville tori immediately project on some of dotted regions in Fig. 26. In  each of the six

cases, their number is uniquely defined by the form of the projection. Indeed, four points u, v, ± f (u) , 

± g(v) projected into an interior point (u, v) of the dotted region are glued as follows when this  

point approaches the boundary of the region: on the horizontal segments, the point u, v, f (u), g(v)    



is glued with the point u, v, f (u), − g(v) and the point u, v, − f (u), g(v) is glued with the    



point u, v, − f (u), − g(v) ; on the vertical segments, the point u, v, f (u), g(v) is glued with      



u, v, − f (u), g(v) and the point u, v, f (u), − g(v) is glued with u, v, − f (u), − g(v) . We obtain from this that one torus is projected on the region in Fig. 26 (1), on each of the two domains in Figs. 26 (2) and 26 (5) also one torus is projected, on the domain in Fig. 26 (3), two tori are projected, and on each of two domains in Fig. 26 (4) and (6), two tori are projected. The surgeries of the Liouville tori with passage of the point (h, k) through bifurcation curves can also be defined by considering the surgeries of the corresponding regions of possibility of motion depicted in Fig. 26. After that, it is easy to calculate the Fomenko–Zieshang invariants describing the Liouville foliation on the surfaces {Fh = 0} for various h. 490

Recall that our primary goal is the study of the topology of the system on T ∗ S2 but not on T ∗ T2 , i.e., on the surfaces {Fh = 0}/σ ∗ . (The calculation for this case is more complicated and is described below in detail.) Therefore, for the “system on the torus T2 ,” we give only the answer. Theorem 19.1. For the system with the Hamiltonian Fh on T ∗ T2 , the complete list of Fomenko–Zieshang invariants describing the Liouville foliation on the surfaces {Fh = 0} (for various values of the parameters γ1 , γ2 , δ, and h) consists of 12 molecules enumerated in Table 1 (the number i in this table corresponds to the straight line hi in Fig. 24). The regions of possibility of motion for the (regularized) two-center problem on the sphere are obtained from the regions of possibility of motion depicted in Fig. 26 as a result of their factorization by the action of the involution σ : T2 → T2 . They are presented in Sec. 20 (see Fig. 37). The number of Liouville tori for each of the six cases corresponding to six regions on the plane R2 (h, k) can be found by taking into account the action of the involution σ ∗ : T ∗ T2 → T ∗ T2 (see (167)). As a result, we obtain that in cases 1, 2, 3, and 5 in Fig. 37, one Liouville torus is projected on each of the regions (disks), and in cases 4 and 6, two tori are projected on each of the regions (annuli). Note that in cases 4 and 6, the projections of trajectories lying on the same torus pass to the projections of trajectories lying on another torus projected on the same annulus with variation of the direction on them. 19.3. Construction of admissible coordinate systems. To calculate the Fomenko–Zieshang invariants (labelled molecules) describing the topology of the Liouville foliation, on the isoenergetic surfaces Qh = {H = h}, we need to carry out the following: (1) find the type of surgeries of the Liouville tori in the inverse image of the point (h, k) moving along the straight line {h = const} on the plane R2 (h, k) when this points crosses the bifurcation diagram; (2) describe the gluings of the boundary Liouville tori for the inverse images of the segments {|k − k1 | ≤ ε}, {|k − k2 | ≤ ε} lying on the straight line {h = const}, where k1 and k2 correspond to two sequential bifurcations (i.e., for a pair of atoms connected by an edge). To describe the gluings of the boundary Liouville tori, we need to specially choose basic cycles on them (to construct admissible coordinate systems), and after that, to write the gluing matrices in these bases. The general rule for choosing such circles is described in [8]. In the calculations carried out in each concrete case, we will formulate the condition necessary for these cycles. Clearly, for constructing the molecules corresponding to arbitrary h, it suffices to describe the bifurcations and the admissible coordinate systems for each of the bifurcation curves, i.e., for all possible passages i → j (from the region with number i to the region with number j) through a certain bifurcation curve. In this case, it is assumed that we always pass from the “lower” region to the “upper” region (this is correct, since in the case considered, the bifurcation curves have no vertical tangents; see Figs. 24 and 25). We see from Figs. 24 and 25 that there arise the following 12 variants: 1 → 2, ∅ → 1,

2 → 3, 2 → ∅,

3 → 4, 3 → ∅,

1 → 3, 4 → ∅,

5 → 1, ∅ → 5,

6 → 5, ∅ → 6,

where ∅ denotes the region consisting of points with empty inverse images. The projections of the Liouville tori depicted in Fig. 26 have a simple form (roughly speaking, under these projections, tori “fold” along two or four cycles transforming into “rectangular” regions). Moreover, their surgeries are also well described in terms of these projections. However, to describe the admissible coordinate systems on some torus, it is convenient to “unfold” the torus transforming it into a plane rectangle (two or four times greater than the projection, respectively). After that, we need to take into account the action of the involution σ ∗ on this “unfolding” in order to obtain the image of the Liouville torus of the system considered on the sphere. In Fig. 27, we depict the described procedure for case (1) from Fig. 26 We first “unfold” the torus, obtaining from four small rectangles overlapping each other a single rectangle four times greater than each of them. In this case, we assume that the location of one of the small rectangles does not change (it is dotted in the figure), i.e., as before, the axis u is its horizontal axis, and the axis v is its vertical axis. Let 491

Fig. 27. “Unfolding” and factorization of Liouville tori. this rectangle consist of points for which pu , pv ≥ 0. Therefore, after the first step, the torus is depicted in the form of a large rectangle whose opposite sides are identified by using shifts parallel to the axes. At the second step, we factor this torus by the involution σ ∗ . Here, it is necessary to take into account that we consider the involution σ ∗ but not σ. Therefore, we first need to consider not the central symmetry with respect to the center of the dotted rectangle but its composition with the central symmetry with respect to the center of the large rectangle (which corresponds to the replacement of (pu , pv ) by (−pu , −pv )). We obtain that the involution σ ∗ on this torus is a shift by the vector coinciding with the diagonal of the dotted rectangle. Therefore, the Liouville torus for the system on the sphere is depicted in the form of a rectangle (consisting of two small rectangles) whose sides are glued as is shown in Fig. 27 (we should identify the sides marked by the same letters). Analogously (in the form of unfoldings), we also depict the Liouville tori for other cases (2)–(6) in Fig. 26. It should be noted here that we depict the unfoldings of the Liouville tori after the factorization by the involution σ ∗ . In this case, the dotted part on the unfolding contains those points for which pu , pv > 0. A unique case where in such an approach there arises an ambiguity is case (5) in Fig. 26. Here, it is assumed that under factorization, the dotted part of the unfolding contains those points satisfying the condition pu , pv > 0 which are projected on the left (dotted) rectangle in Fig. 26 (5). Moreover, for all unfoldings, except for cases (1) and (2), the sides of rectangles are identified by horizontal and vertical shifts, and for cases (1) and (2), in the way depicted in Fig. 27. We always will use these agreements in what follows in depicting the admissible coordinate systems on the unfoldings of Liouville tori. Let us make one more remark. For each of the boundary atoms, the basis cycles λ and µ of an admissible coordinate system must be oriented in an appropriate way. If we assume that the orientation in the phase space of a Hamiltonian system is given by the form ω ∧ ω (where ω is the symplectic structure) and the orientation on the isoenergetic surfaces is defined by the “normal” grad H (with respect to a certain metric), then on the Liouville tori there arises a natural orientation defined by the “normal” grad K. It turns out that under the method for depicting the Liouville tori described above, this orientation can be defined in the following way on the unfoldings: the orientation defined on the dotted rectangles of the unfolding of the torus with coordinates u, v is assumed to be positive. Clearly, in this way, the orientation is defined on all tori. Further, on the boundary tori of an atom, we choose basis cycles λ and µ in such a way that on “upper” tori, they define the positive orientation, and on “lower” tori they define the negative orientation (recall that in this problem, we always consider bifurcations under the passage from the “lower” region of the bifurcation diagram to the “upper” region of it). A pair of cycles on each of the upper tori of the atom is denoted by (λ− , µ− ), and a pair of cycles on each of the lower tori is denoted by (λ+ , µ+ ). We now pass to the description of admissible coordinate systems for the listed 12 cases The cases i → ∅ or ∅ → i. These bifurcation correspond to the atom A. This atom has a single boundary torus, which either contracts onto the axis circle or, otherwise, is born from it in the process of bifurcation. Under the projection on the configurational space, this circle projects on a vertical or horizontal segment. These two possibilities correspond to the case where, in approaching a point to the 492

λ+

λ+ λ+

λ− µ− λ−

µ+

∅→1

µ+

2→∅

λ+

µ−

µ+ 3→∅ µ−

µ− µ+

λ+

λ−

µ+ 4→∅

λ− ∅→5

λ−

∅→6

Fig. 28. Admissible coordinate systems on atoms A. boundary bifurcation curve, either the function f (u) or the function g(v) tends to zero. Finding which of these two possibilities is realized, we obtain that in the cases ∅ → 1, ∅ → 5, and ∅ → 6, the critical circles project on vertical segments, and in the cases 2 → ∅, 3 → ∅, and 4 → ∅, they project on the horizontal segments. The rule for choosing the cycle λ for the atom A is as follows: this cycle must contract into a point when the torus contracts into the axis of the complete torus (atom A). The additional cycle µ is chosen so that in this contraction of the torus, its direction tends to the direction of the flow sgrad H on the critical circle (and that the pair λ, µ is regularly oriented: see above). Note that the cycle µ is not defined uniquely. From this, taking into account the rule for choosing the orientation and that pu , pv ≥ 0 for the dotted rectangular, we obtain admissible coordinate systems on the boundary torus of the atom A for all six cases. The answer is presented in Fig. 28. Recall that for first two surgeries depicted in 28, the gluing of rectangles is carried out as is shown in Fig. 27. Therefore, in particular, the cycle λ for the surgery ∅ → 1 in Fig. 28 consists of two pieces. Case 1 → 2. In this case, we have the following: one torus (which was in region 1) does not change, but there arises the second torus. This happens exactly in the same way as in the surgery ∅ → 1 described above (see Fig. 28). Case 2 → 3. In this case, two tori deform into one. This bifurcation is described by the atom B. The rule for choosing cycles for the atom B is as follows: if we consider the atom B as a (trivial) bundle over the circle on the figure-eight curve N 2 , then the cycles λ must coincide with a fiber of this bundle and be codirected to the flow on the critical cycle, and the cycles µ must bound a certain two-dimensional surface which is a transversal section of this bundle. Moreover, as above, the condition of choosing a right orientation of the basis λ, µ remains. Analogously to the above, we prove that the critical circle is projected on a vertical segment. Therefore, as fibers, we can take vertical segments. Then the cycles µ cut out on three boundary tori of the atom B by the global section of the bundle can be represented as follows. If on the Liouville tori, we consider the cycles a = {u = const} and b = {v = const}, then the cycles µ have the form a + b (without orientation at present). The above family of cycles µ is glued into the global section and, moreover, transversally to the fibers {u = const}. Taking into account the agreement of orientation, we obtain admissible coordinate systems in the case considered (Fig. 29). Case 3 → 4. As in the previous case, this bifurcation is described by the atom B, but here, on the contrary, one torus falls into two. 493

Fig. 29. Admissible coordinate system on the atom B (case 2 → 3). The rule for choosing the cycles λ and µ in this case is the same as for the atom B in case 2 → 3. Obviously, the critical circle is projected on a horizontal segment. Therefore, the cycles λ have the form {v = const}. Each of the dotted rectangles in Fig. 26(3,4) depicts the projections of two Liouville tori (pu > 0 for one torus, and pu < 0 for another), but these tori are identified under the factorization. Therefore, we can assume that for unfoldings of the boundary Liouville torus (of the atom B considered) the dotted part corresponds to points at which pu , pv > 0 as usual.

Fig. 30. Admissible coordinate system on the atom B (case 3 → 4). As the cycles µ cut out by the global section, we can take the cycles {u = const}. Taking into account the rule for choosing the orientations on the cycles λ and µ, we obtain admissible coordinate systems on three boundary tori of the atom B (see Fig. 30). 494

Case 6 → 5. Here, as in two the previous cases, the surgery of two tori into one is described by the atom B. The critical circle is projected into a horizontal segment; therefore, the cycles λ are given by the condition {v = const}. We can choose the global section in such a way that the cycles on all tori (the cycles µ) cut out by it are depicted by vertical segments.

Fig. 31. Admissible coordinate system on the atom B (case 6 → 5). As above, taking into account the rule for choosing the orientation of the cycles λ and µ, we obtain the answer (see Fig. 31). Case 1 → 3. In this case, we have the surgery of one torus into one torus described by the atom A∗ . On this torus, the structure of the oriented Seifert foliation with one singular fiber (critical circle) is defined. In this case, the cycles λ are uniquely defined as leaves of this foliation. Obviously, if the tori are depicted as unfoldings, then we can consider vertical segments as leaves. In this case, the cycles λ are oriented from the bottom to the top (as on the dotted part of the unfolding pu , pv > 0). One of the cycles µ+ and µ− can be chosen arbitrarily (but, certainly, in such a way that the corresponding pair of cycles λ and µ composes a basis on the torus and has the true orientation). Choose the cycle µ− on the upper torus of the atom A∗ as a horizontal segment (see Fig. 32). After fixing the cycle µ− , the rule for choosing the cycle µ+ is as follows. On the boundary of a neighborhood of the critical circle (singular fiber of the Seifert foliation), we have two uniquely defined cycles: λ is a leaf; κ is a cycle contracted to a point in this neighborhood. Let us orient the cycle κ in such a way that the pair of cycles λ, κ is positively oriented (recall that the cycle λ is oriented in such a way that, on the singular leaf, the direction is given by the flow sgrad H). Then a certain cycle µ is uniquely defined on the boundary of a neighborhood of the singular leaf by the condition λ + 2µ = κ. Now, removing the neighborhood of the singular leaf from the 3-atom A∗ , we must construct the section of the Seifert foliation (without singular leaves now) coinciding with the cycle µ− on the upper torus of the atom A∗ and with the cycle µ on the boundary of the neighborhood of the singular leaf. This section is uniquely defined and thus cut out a certain cycle on the second (lower) torus of the atom A∗ . Orienting this cycle in accordance with the rule for orientation, we obtain the cycle µ+ . The process of constructing the cycle µ+ is depicted in Fig. 32(a) in the form of unfoldings, and in Fig. 32(b) it is depicted in a more visual form by using a “doubled” section of the Seifert foliation (see [8]). The dotted arrows in Fig. 32(a) depict a shift in a direction transversal to the family of Liouville tori 495

Fig. 32. Constructing an admissible coordinate system for the atom A∗ . (along the flow ±grad K); they correspond to dotted parts of the cycle κ in Fig. 32(b). Also, we note that the lateral sides of unfoldings in Fig. 32(a) correspond to cycles cut out on the boundary Liouville tori of the atom A∗ by separatrices. In this case, we must take into account that for unfolding (3), the lateral sides are glued by horizontal shifts as usual, and for (1), they are glued as shown in Fig. 27.

Fig. 33. Admissible coordinate system on the atom A∗ ( 1 → 3). The final answer, the admissible coordinate system on the atom A∗ in case 1 → 3, is depicted in Fig. 33. Case 5 → 1. Here, as in the previous case, we have the surgery of one torus into one torus described by the atom A∗ . The rule for choosing the cycles λ and µ and arguments are analogous to the previous case. Therefore, we present the final answer only (see Fig. 34). 19.4. Constructing molecules and calculating labels. Therefore, for all possible bifurcations of Liouville tori, we have described admissible coordinate systems on boundary tori of the corresponding atoms. Now, we can imagine the isoenergetic surface as a result of gluings of boundary tori of atoms. Then on each such torus, there arise two pairs of basis cycles, λ− , µ− and λ+ , µ+ . Expressing the cycles λ+ , µ+ through the cycles λ− , µ− , we obtain gluing matrices for each edge of the molecula. 496

Fig. 34. Admissible coordinate system on the atom A∗ (case 5 → 1). An example of constructing the molecule is depicted in Fig. 35. This molecule corresponds to the line h4 in Fig. 24. The line h4 intersects the bifurcation diagram at four points at which we have the following surgeries: ∅ → 1 → 3 → 4 → ∅. For each of the three one-parameter families of the Liouville tori, we must take the cycles λ− , µ− and λ+ , µ+ constructed in Subsec. 19.3 (see Figs. 28, 30, and 33 for this example). Analogously, we construct the molecules for all the lines h1 –h12 marked in Figs. 24 and 25. Molecules, together with gluing matrices, completely describe the topology of the Liouville foliation on isoenergetic surfaces. However, gluing matrices are not uniquely defined (by the nonuniqueness of choosing admissible coordinate systems). Therefore, it is convenient to give a final answer in the form of labelled molecules (Fomenko–Zieshang invariants). The labels coding the information about gluing matrices are now uniquely defined. For the definition of labels and formulas for their calculation, see [8]. Theorem 19.2. For the two-center problem on the two-dimensional sphere, the complete list of Fomenko– Zieshang invariants describing the Liouville foliation on isoenergetic surfaces {H = h} (for various values of the parameters γ1 , γ2 , δ, and h) consists of 12 molecules enumerated in Table 2 (the number i in the table corresponds to the line hi in Fig. 24). For each of the 12 cases, in Table 2, except for molecules with labels presented in the second column, we depict in the third column the same molecules with gluing matrices for admissible coordinate systems constructed in Subsec. 19.3. Note that in the case of the Lobachevskii space, for a bounded motion in the two-center problem (h < −1; see [44]), we also have molecules 1, 2, and 3 from Table 2 (see Chapter 2). The bifurcation diagram for the problem considered has singular points (points of transversal intersections and tangent points of bifurcation curves). It is convenient to describe the topology of the Liouville foliation in a neighborhood of these points by using circular molecules that correspond to three-dimensional surfaces in the phase space of the system which are the inverse images of small circles centered at these singular points (see [8, 10]). Recall that we are studying the topology of the system obtained as a result of regularization of the two-center problem (see Sec. 17). In particular, the topology of the Liouville foliation was separately studied on each isoenergetic surface but not in the whole phase space. Therefore, strictly speaking, for such an approach, circular molecules for the system considered are not usual circular molecules. However, we can formally construct them analogously to what was done for isoenergetic molecules. 497



λ+ µ+



λ+ µ+



λ+ µ+



 =



 =



 =

0 1 1 0



λ− µ−

0 −1 −1 0

−1 2 0 1





λ− µ−



λ− µ−





Fig. 35. Example of constructing the molecula (for the line h4 ). ∞

∞

∞

∞ ∞

Fig. 36. Circular molecules. Circular molecules (with r-labels) for all five cases marked by the letters a, b, c, d, and e in Fig. 24 are presented in Fig. 36. They have the same form as circular molecules for typical singular points occurring in the classical cases of integrability (see [8, 10]). For the points a and e, the circular molecules coincide with circular molecules of center-saddle type, for the point b, they coincide with circular molecules of singularity type called a “pitch-fork,” and for the points c and d, they coincide with circular molecules that are hyperbolic and elliptic “period-doubling,” respectively. 20.

Motions on the Configuration Space

We consider motions and limit motions on the configuration space arising under bifurcations of Liouville tori. 498

A definite type of motion corresponds to each region on the bifurcation diagram. The projections of Liouville tori for each of these six regions are presented in Fig. 37 (the numbers in Fig. 37 correspond to the numbers of regions in Fig. 24).

Fig. 37 Note that the projections of Liouville tori can go around to the equator of the sphere, which in the cases of a flat space and the Lobachevskii space corresponds to going to infinity. The cases where a material point goes around to the equator and remains in the upper hemisphere all the time are separated by the line l = hβ 2 , which is one of the asymptotes of the hyperbolas. Indeed, for η 2 = β 2 , the material point crosses the equator (which corresponds to going to infinity). In particular, in cases 1, 3, and 4, the projections of Liouville tori can have the form depicted in Fig. 38.

Fig. 38 Let us describe the type of motion for each of the regions. In region 1 (Fig. 24), we have the satellite motion (an analogous motion exists for the flat space and the Lobachevskii space [15, 44]): a planet moves near one of the centers, and the orbit everywhere densely sweeps the whole region (Fig. 37(1)). If a point x ∈ R2 is located to the left from the line l = −hα2 , then on the configuration space, this case corresponds to the motion on the right hemisphere only. When intersecting the line, the material point goes around to the left hemisphere. In region 2, we observe an analogous satellite motion that is possible near two centers now. In region 3, the motion is lemniscate-like: the trajectory can everywhere densely sweep the region. If a point x ∈ R is located to the right from the line l = hβ 2 , then we obtain the region containing the whole upper hemisphere. In the case of Euclidean space and the Lobachevskii space, the corresponding region coincides with the whole configuration plane. 499

In region 4, the projection of the torus on the configuration plane is an annulus containing two attracting centers. To the right from the line l = hβ 2 , under an analogous motion in the flat space and the Lobachevskii space, the trajectory goes to infinity. In region 5, the point moves in a curvilinear rectangle whose sides are quadrics. In the case of the Euclidean space and the Lobachevskii space, the unbounded region on the configurational space (between two hyperbolas) always corresponds to this rectangle. In region 6, as in case 4, we have an annulus, but in this case, it is located between the attracting centers. This region also corresponds to going to infinity. Both cases 5 and 6 correspond to the same case in the Euclidean space and the Lobachevskii space. Also, it is of interest to consider limit motions, since they give a visual imagination of surgeries of Liouville tori under bifurcations coded by using Fomenko–Zieshang invariants.

y

y

c

b

y

x

x

x

a

y

x

d

Fig. 39 Consider the motions related to the fall on the center. The zero root of the function R corresponds to γ1 − γ 2 β 2 − α 2 (i.e., to the left of a tangent point with the hyperbola). points on the line Γ5 for h < R(γ1 + γ2 ) 2αβ Here, the material point moves along a part of the arc connecting the attracting center and the antipodal center. The motion along the arc is bounded by the quadric η = const, the root of the function S (see Fig. 39(a)). The 3-atom A (the minimal critical circle of the integral, the birth of a Liouville torus) corresponds to this motion. When crossing the line Γ4 , γ1 − γ 2 β 2 − α 2 , γ1 + γ2 2Rαβ one more Liouville torus is born (the 3-atom A). The motion is as follows: the material point either moves near one of the centers (satellite motion) or along a part of the arc connecting the attracting and antipodal centers in another hemisphere (see Fig. 39(b)). When crossing the line Γ3 ,    γ1 − γ 2 2 −(β 2 − α2 ) − R2 1 − γ1 + γ2 , h< 2Rαβ we have an analogous motion but along parts of the arc connecting two attracting centers (see Fig. 39(c)). If    γ1 − γ 2 2 2 2 2 −(β − α ) − R 1 − γ1 + γ2 β 2 − α2
500

Fig. 40 surrounding two attracting centers; the motion is also periodic (a part of the curve Γ2 between regions ∅ and 4; see Fig. 40). We now consider the motions for bifurcations described by saddle atoms (unstable periodic trajectories and the asymptotic trajectories corresponding to them). In the case where the root of the function S equals zero, and a part of the line Γ3 , h>−

β 2 − α2 , 2Rαβ

moves along a spiral, the material point asymptotically tends to the arc connecting two centers winding on it and not attaining it at a finite instant of time (to the other side, the motion is bounded by the quadric η = const, see Fig. 41(a)). Multiple roots of the function R (ξ1 = ξ2 ) and one root of the function S correspond to points on a part of the curve Γ1 (between regions 2 and 3) (l > 0). The following asymptotic motion is possible: the point executes pendulum-like oscillations inside the quadric defined by the root of S and asymptotically approaches the quadric (ξ = ξ1 = ξ2 ) to the side of one of the centers (Fig. 41(b)).

Fig. 41 The zero root of the function R (ξ1 = ξ2 ) and one root of the function S correspond to points on a part of the curve Γ4 between regions 1 and 3, h>−

γ1 − γ2 β 2 − α2 , γ1 + γ2 2αβR

l > 0.

The following asymptotic motion is possible: the point executes pendulum-like oscillations inside the quadric defined by the root of S and asymptotically approaches a part of the arc connecting the gravitating and antipodal centers (Fig. 41(c)). The zero root of the function S and two roots of the function R correspond to points on a part of the curve Γ6 (between regions 5 and 6) (l < 0). The following asymptotic motion is possible: the point 501

y

y x

x

Fig. 42 executes pendulum-like oscillations inside the quadrics defined by the roots of R and asymptotically approaches the arc connecting two antipodal centers (Fig. 41(d)). Finally, consider equilibrium points. Point 1 (intersection point of the hyperbola Γ1 and the line Γ3 ). This is a point of stable equilibrium (Fig. 42) and is located on the arc connecting two attracting centers at such a distance that the gravitational forces are balanced. Point 2 (intersection point of the hyperbola Γ1 and line Γ6 ) is located to the opposite side.

Chapter 4 TWO-CENTER PROBLEM IN THE LOBACHEVSKII SPACE For the first time, the problem of studying the dynamics in a space of constant negative curvature was posed by N. I. Lobachevskii. As is well known [14], the function coth r is an analog of the Newtonian 1 in the Lobachevskii space. A further development of this theme is found in the work of potential r N. E. Zhukovskii [49] devoted to the motion of a pseudospherical plate on the Lobachevskii plane. In this chapter, we consider a generalization of the two-center problem to the Lobachevskii space [40, 41, 44], i.e., to H3 . Here, H3 is the Lobachevskii space defined as the upper flap of the three-dimensional hyperboloid embedded into the Minkowski space M 4 . 21.

Description of the System. Reduction

Consider the upper flap of the hyperboloid H3 embedded into the four-dimensional Minkowski space with coordinates x1 , x2 , x3 , x4 in the standard way. The equation of the hyperboloid is

M4

(x1 )2 − (x2 )2 − (x3 )2 − (x4 )2 = R2 .

(171)

Let a material point move under the action of the Newtonian attraction of two fixed centers located at points with coordinates r1 = (β, α, 0, 0) and r2 = (β, −α, 0, 0), where β 2 − α2 = R2 . The potential energy of the material point in the field of these centers is equal to γ2 γ1 coth θ2 , (172) V = − coth θ1 − R R where θi is the angle between the radius-vectors of the gravitating center and the moving material point; the capacity of these two centers is denoted by γi . Here, coth θi =

1 r, ri  = 2 (x1 x1i − x2 x2i − x3 x3i − x4 x4i ), |r||ri | R

where ·, · is the inner product in the Minkowski space. The Lagrange function is 1 L = [−(x˙ 1 )2 + (x˙ 2 )2 + (x˙ 3 )2 + (x˙ 4 )2 ] − V. 2 502

Table 1 Introduce the pseudospherical coordinate system. The formulas of change are such that x1 = R cosh θ,

x2 = R sinh θ cos ϕ,

x3 = R sinh θ sin ϕ cos ψ,

x4 = R sinh θ sin ϕ sin ψ

The Jacobi matrix has the form analogous to the case of the sphere, but instead of sin θ and cos θ, we must write sinh θ and cosh θ. Recall that θ defines the length of the hyperbola (“meridian”) in the pseudo-Euclidean metric from the pole of the upper flap of the hyperboloid to the variable point, i.e., the pseudospherical coordinates are 503

Table 1. (Continuation) analogous to the spherical coordinates (for the spherical coordinates, θ is the length of the meridian, the great circle from the north pole of the sphere to the variable points). The metric induced in the space H3 (with respect to the coordinates R, θ, ϕ, ψ) is written in the form ds2 = R2 (dθ2 + sinh2 θdϕ2 + sinh2 θ sin2 ϕdψ 2 ). The Lagrange function L is rewritten in the following way in the pseudospherical coordinates: 1 L = R2 (θ˙2 + sinh2 θ(ϕ˙ 2 + sin2 ϕψ˙ 2 ) − V. 2 504

(173)

(174)

Table 2.

Without loss of generality, we consider the unit pseudosphere. Theorem 21.1. A material point in the two-center problem in the space H3 moves in the same way as in the two-dimensional system (on the unit two-dimensional pseudosphere H2 : {y 2 − x2 − z 2 = 1}) with 505

Table 2. (Continuation) 506

Table 2. (Continuation)

507

the energy 1 h = (−y˙ 2 + x˙ 2 + z˙ 2 ) + Veff , 2 where the effective potential energy is defined by p2ϕ . 2z 2 Proof. By symmetry of the problem, introduce new variables according to the formulas Veff = −γ1 coth θ1 − γ2 coth θ2 +

x1 = y,

x2 = x,

x3 = z cos ϕ,

x4 = z sin ϕ.

(175)

Then taking into account that L = T − V , we see that the Lagrange function in these variables becomes 1 L = (x˙ 2 − y˙ 2 + z˙ 2 + z 2 ϕ˙ 2 ) + γ1 coth θ1 + γ2 coth θ2 . (176) 2 Obviously, the coordinate ϕ is cyclic, and hence, according to the Noether theorem, there exists the integral ∂L = z 2 ϕ˙ = pϕ . ∂ ϕ˙ Write the Routh function as 1 R = (−y˙ 2 + x˙ 2 + z˙ 2 ) − Veff , 2 where p2ϕ Veff = V + 2 . 2z This implies that the energy of the particle is p2ϕ 1 (177) h = (−y˙ 2 + x˙ 2 + z˙ 2 ) + V + 2 . 2 2z Therefore, we have reduced our problem to the problem of motion of a material point on the unit two-dimensional pseudosphere H2 : {y 2 − x2 − z 2 = 1} in the field with the reduced potential Veff . The theorem is proved. 22.

Integrals of the System

The Hamiltonian of this problem has the Liouville form in the pseudosphero-conical coordinates ξ, η [44]. We define ξ and η as roots of the equation f (λ) =

−(λ − ξ 2 )(λ + η 2 ) y2 z2 x2 = . − + λ − α 2 λ− β 2 λ λ(λ − α2 )(λ − β 2 )

(178)

The coordinate surfaces are described by the equations x2 z2 y2 x2 z2 y2 = + ; = + ; ξ2 − β2 ξ 2 − α2 ξ2 η2 + β 2 η 2 + α2 η 2 y 2 − x2 − z 2 = R2 .

(179)

The transition formulas from the Cartesian coordinates to the pseudosphero-conical coordinates can be obtained from relations (178): %$ % 1 $ x2 = 2 α2 − ξ 2 α2 + η 2 , α %$ % 1 $ 2 y = 2 β 2 − ξ2 β 2 + η2 , (180) β R2 z2 = 2 2 ξ2 η2 . α β 508

In this case, the following conditions hold: ξ 2 ≤ α2 , Extracting roots, we obtain the expressions

0 ≤ η < ∞.

(181)

(α2 − ξ 2 )(α2 + η 2 ) , x = sign(x) α

(β 2 − ξ 2 )(β 2 + η 2 ) Rξη y= , z= . β αβ

(182)

In this problem, the coordinate lines are lines of intersection of the pseudosphere with two families of confocal cones. The coordinates ξ and η are orthogonal in the sense of the metric of the pseudo-Euclidean space. Clearly,

cosh θ1,2 = ∓αx + βy ∓ (α2 − ξ 2 )(α2 + η 2 ) + (β 2 − ξ 2 )(β 2 + η 2 ), cosh2 θ1,2 − sinh2 θ1,2 = 1. The Hamiltonian of the problem with respect to pseudosphero-conical coordinates is   1 (α2 − ξ 2 )(β 2 − ξ 2 ) 2 (α2 + η 2 )(β 2 + η 2 ) 2 pξ + pη + Veff , H= 2 ξ2 + η2 ξ2 + η2 where Veff

−(γ1 + γ2 ) (α2 + η 2 )(β 2 + η 2 ) − sign(x)(γ1 − γ2 ) (α2 − ξ 2 )(β 2 − ξ 2 ) = R(ξ 2 + η 2 ) pϕ2 α2 β 2 (ξ −2 + η −2 ) ; + 2R(ξ 2 + η 2 )

(183)

(184)

here, pξ and pη are impulses corresponding to the coordinates ξ, η. The function sign(x) describes the potential on half of the upper flap of the hyperboloid (x > 0 or x < 0). This nonuniqueness is connected with different methods for extracting roots in (180). Obviously, the Hamiltonian in the coordinates ξ, η has the Liouville form, and hence we can write the additional integral (α2 − ξ 2 )(β 2 − ξ 2 )η 2 2 (α2 + η 2 )(β 2 + η 2 )ξ 2 2 pξ − pη 2(ξ 2 + η 2 ) 2(ξ 2 + η 2 ) 

p2ϕ α2 β 2 η −2 ξ2 (γ1 + γ2 ) (α2 + η 2 )(β 2 + η 2 ) − 2

L=  +  −

R(ξ 2 + η 2 )

p2ϕ α2 β 2 ξ −2 sign(x)(γ1 − γ2 ) (α2 − ξ 2 )(β 2 − ξ 2 ) + 2 R(ξ 2 + η 2 )

 η2 .

Also, write the two dependent Liouville integrals as

sign(x)(γ1 − γ2 ) (α2 −ξ 2 )(β 2 − ξ 2 ) p2ϕ α2 β 2 (ξ −2 ) ξ˙2 (ξ 2 + η 2 )2 1 + + − hξ 2 , I1 = 2 (α2 − ξ 2 )(β 2 − ξ 2 ) R 2R

(γ1 + γ2 ) (α2 + η 2 )(β 2 + η 2 ) p2ϕ α2 β 2 (η −2 ) η˙ 2 (ξ 2 + η 2 )2 1 I2 = + + − hη 2 . 2 (α2 + η 2 )(β 2 + η 2 ) R 2R

(185)

Since I1 + I2 = 0, we introduce the notation l = I1 = −I2 . 509

Consider the case where pϕ = 0. Introducing the new independent variable τ instead of t using the h l ξ2 + η2 dτ and making the change h → , l → , we reduce the substitution dt = γ + γ γ + γ2 2(γ1 + γ2 ) 1 2 1 equations of motion to

dη dξ (186) = R(ξ), = S(η), dτ dτ where R(ξ) and S(η) are the irrational functions R(ξ) = (α2 − ξ 2 )(β 2 − ξ 2 )R∗ (ξ),

S(η) = (α2 + η 2 )(β 2 + η 2 )S ∗ (η),

and define

(187)

(α2 − ξ 2 )(β 2 − ξ 2 ) , R

(α2 + η 2 )(β 2 + η 2 ) ∗ 2 S (η) = −l + hη + . R ∗

2

R (ξ) = l + hξ +

sign(x)K

γ1 − γ 2 is a parameter and h and l are the values of the integrals depending on the initial γ1 + γ2 conditions. The function R describes the motion on half of the hyperboloid, x > 0 and x < 0, respectively: when the coordinate ξ assumes the value ±α, the particle passes from one region to another (see Eq. (182)). For these values, the function R describing the change of ξ must be changed. Clearly, the function S is the same in both regions. Here, K =

Definition 22.1. The motion of a material point in the two-center problem on the pseudosphere is called γ 1 + γ2 , |ri | → ∞, and it is called a motion of hyperbolic type if a motion of parabolic type if h = − R γ 1 + γ2 h > − , |ri | → ∞. Correspondingly, the initial velocity is called the parabolic and hyperbolic R velocity. Assertion 22.1. In the two-center problem on the pseudosphere, we have the following types of motion: γ 1 + γ2 ; the elliptic type for h < − R γ1 + γ2 ; the parabolic type for h = − R γ1 + γ2 the hyperbolic type for h > − . R Proof. Since the kinetic energy is always nonnegative, the following inequality must hold during the whole time of motion: −V + h ≥ 0. (188) Therefore, h≥−

γ2 γ1 coth r1 − coth r2 . R R

For |ri | → ∞, we obtain γ1 + γ2 . (189) R For values of the energy satisfying inequality (189), the material point goes to infinity independently of γ1 + γ2 and hyperbolic for its initial position, i.e., the motion is unbounded; it is parabolic for h = − R γ 1 + γ2 h− . In the limit as R → ∞, we obtain the flat case where the value of the energy h = 0 separates R the bounded and unbounded motions. h≥−

510

The two-center problem on the pseudosphere is described by the Hamiltonian system on the cotangent bundle T ∗ H2 of the pseudosphere with Hamiltonian (172). As in the case of the sphere [46], in this problem the potential has singularities at the points where the attracting centers are located. For any location of the particle, we can give an initial velocity such that the particle reaches the attracting center at a finite time, and the velocity of the particle at the instant of fall on the center becomes infinite, i.e., the vector field is not complete, and, formally speaking, integrable. Let  the system is not Liouville  ∂H ∂H ∂H ∂H ,− , ,− . In the phase space, ω = sgrad H be a vector field on T ∗ H2 with coordinates ∂pξ ∂pη ∂ξ ∂η the vector field ω = sgrad H has singularities at points at which λ(ξ, η) = ξ 2 + η 2 = 0. Consider the vector field ω ˜ = λ(ξ, η)sgrad H and its restriction ωh to the isoenergetic surface {Qh = H = h} ⊂ T ∗ H2 . The regularization is carried out similarly to [46]. In this case, the change of time has the form dt = λ(ξ(t), η(t)). dτ As a result of the regularization, the vector field ωh = λsgrad H has no singularities. The integral trajectories of the field ωh coincide with the integral trajectories of the initial vector field ω on T ∗ H2 . The vector field ωh on the surface Qh coincides with the restriction to the same surface of the Hamiltonian vector field defined on the whole phase space. As such a field, we take the field sgrad Fh , where 1' Fh = λ(H − h) = (α2 − ξ 2 )(β 2 − ξ 2 )p2ξ 2 ( +(α2 + η 2 )(β 2 + η 2 )p2η − h(ξ 2 + η 2 ) sgn(x)(γ1 −γ2 ) 2 2 2 2 (γ1 +γ2 ) 2 2 2 2 (α +η )(β +η ) − (α −ξ )(β −ξ ) − R R (here, we set pϕ = 0). The additional integral of the initial system is the integral of the Hamiltonian system with the Hamiltonian Fh = λ(H − h) on the surface Fh = 0. The Bott property of the additional integral of the problem considered is verified similarly to that in the two-center problem on the sphere [46], i.e., we verify the nondegeneracy of critical points of the functions R and S. An analogous assertion holds. The additional integral has the Bott property on all isoenergetic surfaces Qh = {H = h}, except for those which correspond to the lines h = const passing through tangent points of bifurcation curves. 23.

Bifurcation Diagrams

M 2n

be a smooth symplectic manifold, and let v = sgrad H be a Hamiltonian dynamical system Let with the Hamiltonian H. Consider a regular common level surface Mξ of functions f1 , . . . , fn that are in involution pairwise and independent almost everywhere on M , i.e., Mξ = {x ∈ M |fi = ξi , i = 1, . . . , n}. It is known that if the manifold Mξ is not compact, then each connected component of Mξ is diffeomorphic to Rk × T n−k ; if the manifold Mξ is compact, then each connected component of Mξ is diffeomorphic to the n-dimensional torus T n . In this problem, we have both these cases. Theorem 23.1. The bifurcation set in the two-center problem on the pseudosphere has the form 1 4h2 α2 β 2 + R2 1 Γ2 : 2 4h2 α2 β 2 + R Kαβ , Γ3 : l = − R Γ1 :

1 1 4lh(α2 + β 2 ) + K 2 + 2 4l2 = 0, 2 R R 1 1 4lh(α2 + β 2 ) + 1 + 2 4l2 = 0, R2 R αβ Kαβ Γ4 : h = −1, , Γ6 : l = . Γ5 : l = R R

(190)

The line h = −1 (after the redesignation) separates the bounded and unbounded motions. 511

h1

h2

h3 l 2

2

1 3

2

5 4

Ã5

6 7

1

Ã6 h

1

Ã3

8

Ã4

Ã1

Ã2

Fig. 43. Bifurcation diagram in the two-center problem on the pseudosphere.

The tangent points of the hyperbola Γ1 with the lines Γ3 and Γ6 have the coordinates 

K(α2 + β 2 ) Kαβ ,− 2Rαβ R



 ,

−K(α2 + β 2 ) Kαβ , 2Rαβ R

 ,

(191)

respectively. The intersection point of the hyperbola Γ1 with the line Γ5 has the coordinates 

 √ −(α2 + β 2 ) − 1 − K 2 αβ , . 2Rαβ R

The bifurcation diagram is depicted in Fig. 43. The regions of possibility of motion are marked by numbers. Vertical lines in the bifurcation diagram correspond to isoenergetic surfaces Qh .

24.

Classification of Motions on the Configuration Space. Limit Motions

The classification of regions of possible motions is presented in Fig. 44 (bounded motions are presented under numbers 1–4, and unbounded motions are presented under numbers 5–8). The coordinates of tangent points (191) depend on the parameter K. If the fixed centers are comparable in mass, then the parameter K is small, and the tangent point of the hyperbola Γ1 with the line Γ3 is located to the right from the line h = −1 separating the bounded and unbounded motions (see Fig. 45). In this case, there arises one more type of motion arising from the satellite motion (see Fig. 44(N2)) as η → ∞; this motion is analogous to the motion in region 7. In the limit, when the centers are identical, i.e., K = 0, in the bifurcation set, the hyperbola Γ1 degenerates into asymptotes, and the lines Γ3 and Γ6 pass to the line l = 0. 512

Fig. 44. Regions of possible motion in two-centers problem on the Lobachevskii plane.

Fig. 45. Bifurcation diagram for the two-center problem on the pseudosphere (when the centers are identical). In the flat case, there are eight regions with distinct types of motion. The bifurcation set has the following form (we use the same notation as above): Γ1 : 4lh = K 2 ,

Γ2 : 4lh = 1,

Γ3 : l = −hc − c,

Γ4 : l = −hc2 − cK 2 ,

2

Γ5 : l = −hc + cK , 2

2

(192)

Γ6 : h = 0

Here, the attracting centers are located at the points with the coordinates (c, 0) and (−c, 0), c = const. The classification of regions of possible motion is analogous, i.e., we have the same types of motion for the corresponding regions in the flat case, as well as in the case of the Lobachevskii plane. The regions of possible motion are the same in the configuration plane if we take into account that “ellipses” and “hyperbolas” on the Lobachevskii plane are defined in the same way as on the Euclidean plane. However, 513

it is seen from Figs. 43 and 22 that their location is different from the flat case: in the curved space, region 5 neighbors the region of impossible motion, and in the flat case, they have no common points. Obviously, the trajectories of motion in these problems are different. In the case of repelling centers, we obtain analogous results by reflection with respect to the axis Oz. The equilibrium point lies on the bifurcation diagram at the intersection point of the hyperbola Γ1 and the line Γ5 . In the two-center problem on the sphere, the bifurcation set has the form presented in [46]. Consider the passage to the limit as the radius of the sphere tends to infinity: R → ∞. Using the gnomonic coordinates and formulas (195) (applied in the case of the sphere), we obtain the relations R 2 c2 R4 2 2 , β = R − α = . R 2 + c2 R 2 + c2 Substituting these relations in (170) and taking into account that R → ∞, we obtain the set α2 =

Γ1 : 4h2 c2 + 4lh + K 2 = 0, Γ2 : 4h2 c2 + 4lh + 1 = 0, Γ3 : l = c, Γ4 : l = Kc, Γ6 : l = −Kc. Γ5 : h = 0,

(193)

l The curves Γ1 and Γ2 are hyperbolas with the asymptotes h = 0 and h = − 2 , and, moreover, the line c h = 0 belongs to the bifurcation set, since it is separating between bounded and unbounded motions for the flat case. Moreover, it is seen from qudratures (186) that one of the coordinates η becomes unbounded; this is the cause of disappearance of the line Γ5 in the bifurcation set in passing to the limit. Let us describe the motions on the configuration space. We do not describe bounded motions, since they are analogous to motions in the case of the flat space and the sphere for regions 1, 2, 3, and 4 described in [46]. By definition, in regions 5, 6, 7, and 8 we observe motion of hyperbolic type (h > −1; see Fig. 43). In region 5, we have ξ 2 ≤ α2 and η1 ≤ η ≤ ∞, i.e., motion is possible in the entire hyperboloid lying outside the ellipse η = η1 . If at the initial instant of time, the material point moves in direction of the attracting centers, then the trajectory winds around the ellipse, is tangent to it at a certain instant of time (η˙ = 0), and then, winding from it, goes to infinity. In region 6, the coordinates satisfy the inequalities ξ 2 ≤ α2 and 0 ≤ η ≤ ∞. In this case, motion is possible on the whole configuration space. In region 7, we have −α ≤ ξ ≤ ξ1 and 0 ≤ η ≤ ∞, i.e., the region of possible motion is bounded by the hyperbola ξ = ξ1 . The motion is executed in the part of the Lobachevskii plane where the center P1 is located, and the region of motion is bounded by the branch of the hyperbola located either in the left part of the pseudosphere or by the branch of the hyperbola located in the right part of the pseudosphere. The motion is as follows: the particle bends the attracting center and goes to infinity oscillating between the branches of the “hyperbola” (motion of oscillation type). In region 8, we have 0 ≤ η ≤ ∞ and ξ1 ≤ ξ ≤ ξ2 , i.e., the region of possible motion lies between the hyperbolas, and the motion is of oscillation type between two bounding hyperbolas. 25.

Description of Noncompact Bifurcations

Let us describe noncompact bifurcations. Assume that we pass from the “lower” region to the “upper” region and from the “left” region to the “right” region intersecting the line h = −1 on the bifurcation diagram. Obviously, there arise the following variants: ∅ → 7, 8 → 7, 7 → 6, 6 → 5, ∅ → 8, 5 → 4, 6 → 3, 7 → 1, 5 → ∅. 514

Fig. 46. Case ∅ → 7.

Case ∅ → 7. When crossing the bifurcation line l = − −1 < h <

Kαβ , ξ = 0, and the condition R

K(α2 + β 2 ) 2Rαβ

dξ = 0 (this is easily calculated; see Fig. 46(b)). In this dτ case, the limit motion on the configuration space is depicted in Fig. 46(a) and is a fall on the center, i.e., if the velocity of the material point is directed to the side of increasing distance from the center, then the particle goes to infinity; if the velocity is directed to the opposite side, then the particle falls on the center, is reflected from it (at the center, the pseudosphero-conical coordinates vanish ξ = 0, η = 0, the components of the velocity satisfy ξ˙ = 0, and η˙ changes sign; see Fig. 46(b)), and goes to infinity. At the Kαβ , there arises a instant of bifurcation, the phase space is a straight line. In crossing the line l = − R cylinder. In region 7, the regular surface is a cylinder. Since the coordinate η satisfies the condition 0 ≤ η ≤ ∞, this surface is organized as follows: two semicylinders are obtained by multiplying the circle by two rays (where η˙ > 0, 0 ≤ η < ∞ and η˙ < 0, 0 ≤ η < ∞), then these two semicylinders are glued by their bases, and we obtain the surface S 1 × R1 (on two semicylinders, the velocity η˙ has opposite signs; on the configuration space, this corresponds to motion in which the material point crosses the coordinate η = 0). Consider a nonsingular level line close to a minimum or maximum point. This line is homeomorphic to the circle. As the regular value tends to a maximum or minimum, the circle contracts into a point. In this case, the two-dimensional disk is foliated into concentric circles with a common center. This case corresponds to a bifurcation of type A in the compact case. If the phase space is not compact, then the singular fiber is not a circle but a line, the axis of the cylinder which is foliated into concentric cylindrical surfaces (R1 × D2 ). Under the bifurcation, the cylinder either is contracted into a point or a cylinder is holds, the component of the velocity satisfies

515

born. This motion corresponds to periodic motion in the compact case if we remove a single point. This bifurcation is called An .

Fig. 47. Case ∅ → 8. Case ∅ → 8. Points on the hyperbola Γ1 from the bifurcation set correspond to multiple roots of the function R∗ . When crossing the point (h, l) of the hyperbola Γ1 , the material point crosses the line connecting two centers on the configuration space and goes to infinity; in this case, the component of the velocity satisfies ξ˙ = 0 (Fig. 47(b)). In the phase space, at the instant of bifurcation, two cylinders contract into their axes under the passage 8 → ∅ (or, respectively, there arise two cylinders under the passage ∅ → 8), i.e., we have two lines for which ξ˙ = 0, ξ = ±ξ1 (see Fig. 47(a)). The regular surface in region 8 is two cylinders (i.e., there are two connected components) each of which is organized exactly in the same way as the connected component in region 7; each of the cylinders is glued from two semicylinders by bases. Kαβ , ξ = 0, when the condition Case 8 → 7. In crossing the bifurcation line l = − R K(α2 + β 2 ) h> 2Rαβ holds, one of the “hyperbolas” on the configuration space (see 44(N 8)) degenerates into the positive part of the axis Ox with origin at the corresponding center (the projection of the curve x2 − y 2 = −1 on the plane zOx). The material point crosses the line connecting the centers and asymptotically approaches the axis Ox (the motion is bounded by the second “hyperbola”) being tangent to the bounding “hyperbola” (see Fig. 48(a)). At the instant of bifurcation, the two cylinders deform into one (or vice versa) as is shown in Fig. 48(c). The two cylinders are obtained in the following way: the figure eight curve is multiplied by the ray [0, ∞), which defines the positive component of the velocity η, ˙ and by the ray [0, ∞), which defines the negative component of the velocity −η, ˙ and then the obtained semicylinders with a common ruling are glued by the base. In this case, at the instant of bifurcation, the level curve is the figure eight curve (see Fig. 48(b)), and the critical line is a straight line. This bifurcation corresponds to bifurcation of type B in the compact case (recall that the atom B is a complete torus from which two thin complete tori (N 2 × S 1 ) are cut out, the singular leaf is the direct product of the figure eight curve by the circle, and N 2 is a neighborhood 516

Fig. 48. Case 7 → 8. of the figure eight curve). In the noncompact case, the Liouville equivalence class of a neighborhood of a singular leaf is a complete cylinder from which two thin cylinders (N 2 × R1 ) are cut out, and the singular leaf is the direct product of the figure eight curve by a straight line. Denote the bifurcation of such a form by Bn . Case 7 → 6. When the point (h, l) crosses the line Γ6 (h > −1), we observe the following motion on the configuration space. The “hyperbola” depicted in Fig. 44(N7) degenerates into a ray emanating from the gravitational center (x < 0), and the material point crosses the line connecting the centers and asymptotically approaches the ray going to infinity (see Fig. 49(a)). One more cylinder is born in phase space (Fig. 49(b)). (Compare the bounded motion where under the intersection of the lines Γ3 and Γ6 , first one torus is born, and then the second.)

Fig. 49. Case 7 → 6. Case 6 → 5. When the point (h, l) crosses the line l = αβ, on the configurational space there exist trajectories of two types: one trajectory is a segment connecting two centers (η = 0), and the motion is periodic along this segment; the second trajectory is a spiral whose coils become tighter when the trajectory approaches the above segment (Fig. 50(a)). In regions 5 and 6, we have two connected components: two cylinders, and at the instant of bifurcation, two cylinders deform to two cylinders (Fig. 50(c)). As was pointed out above, the line h = −1 separates the bounded and unbounded motions. In Fig. 51, we show the surgeries of the cylinders into tori (in the case of the compact phase space) when the point αβ crosses the image of the momentum mapping of the separating line h = −1. Correspondingly, for l > R (case 5 → 4), two cylinders deform into two tori (Fig. 51(a)). The lateral surfaces of each of the cylinders 517

Fig. 50. Case 6 → 5. αβ Kαβ < l < (case R R Kαβ Kαβ 0, η → ∞ and η˙ < 0, η → ∞ are tangent (as in the case 5 → 4; Fig. 51(a)). are tangent to each other at infinity, and, as a result, we obtain a torus. For

Fig. 51. Surgery of cylinders into tori.

26.

Passage to the Limit

We write the metric and the potential energy of the system in the gnomonic coordinates (we consider the projection of the pseudosphere on the tangent space from the center of the pseudosphere in the ambient 518

space); the Cartesian coordinates of this projection are denoted by x1 , x2 . In the gnomonic coordinates, αR let the attracting centers be at the points with coordinates (a, 0), (−a, 0) (where a = ). It is known β that (r(i) · r) , i = 1, 2, (194) cosh θi = (i) |r | |r| where the inner product is understood in the sense of the metric of the space R21 . Write the coordinates of vectors of a test particle r and one of the centers for the pseudosphere r1 (the treatment for another center is analogous; the difference is the sign of the parameter a):   x2 1 x1 , , r= , λ(1 − λ|x|2 ) 1 − λ|x|2 1 − λ|x|2   (195)   1 a (1) R2 + α2 , α, 0 = ,0 . r = ,√ λ(1 − λa2 ) 1 − λa2 As a result, using formula (194), we obtain 1 − λax1 √ , cosh θ1 = 1 − λ|x|2 1 − λa2

(1 − λax1 )2 − (1 − λ|x|2 )(1 − λa2 )

√ , sinh θ1 = 1 − λ|x|2 1 − λa2 1 − λax1 coth θ1 = , 2 (1 − λax1 ) − (1 − λ|x|2 )(1 − λa2 ) 1 + λax1 . coth θ2 = 2 (1 + λax1 ) − (1 − λ|x|2 )(1 − λa2 ) Using the obtained formulas, we can write the expression for the potential energy. If a = 0, we obtain the Kepler problem. Indeed, in this case, the gravitating centers move to the origin, and we obtain γ γ 1 + γ2 =− . V =− r |x|2 If λ → 0, i.e., the radius of the sphere tends to infinity, then we obtain γ1 γ2 1 1 V = −γ1 − γ2 =− − . 2 2 2 2 r1 r2 (a − x1 ) + x2 (a + x1 ) + x2 Therefore, we obtain the Kepler potential once again. We can obtain analogous expressions for the sphere if we replace hyperbolic functions by trigonometric functions, and the sign of λ by the opposite sign. Therefore, as a result, we can conclude that integrable problems, i.e., the Kepler problem and the two-center problem, pass from one to another in the space of constant curvature when the following two parameters change: the curvature of the space and the distance between the centers. In particular, we have shown that the Kepler problem and the two-center problem pass to classical problems on the plane when the curvature vanishes. Remark 1. It is easy to see that the results of passage to the limit extend to any dimension of the sphere embedded into the corresponding spaces of dimension n + 1, i.e., Sn and the pseudo-Euclidean space Rn−1 1 in Sn+1 and Rn1 . 519

27.

Comparative Analysis of the Topology of Liouville Foliations of the Two-Center Problem

Recall that a partition of a manifold Q3h into Liouville tori (in the compact case), cylindrical surfaces (in the noncompact case), and connected components of critical level surfaces of the additional integral l (i.e., connected components of inverse images of critical values of the integral l) is called a Liouville foliation on Q3h . Systems are said to be Liouville equivalent if they have the same Liouville foliation. Let v1 be an integrable Hamiltonian system on M 41 (resp. on Q31 ), where M 41 is the cotangent bundle of the sphere T ∗ S2 with Hamiltonian HS2 = T + V , where the function T quadratic in impulses is defined by the metric on the sphere S2 in R3 , and the function V is given by relation (149). Let v2 be an integrable Hamiltonian system on M 42 (resp. on Q32 ), where M 42 is the cotangent bundle of the pseudosphere T ∗ H2 with Hamiltonian HH2 = T + V , where the kinetic energy T is defined by the standard metric on the pseudosphere H2 in R21 , and the potential energy is given by relation (172). Let v3 be an integrable Hamiltonian system on M 43 (resp. on Q33 ), where M 43 is the phase space in the problem of two fixed centers on the plane R2 with Hamiltonian HR2 = T + V , where the kinetic energy T is defined by the standard Euclidean metric on the plane R2 , and the potential is Newtonian, i.e., is given by γ1 γ2 V =− − . r1 r2 Theorem 27.1. The system v1 under the condition that the material point lies in the upper hemisphere l all the time, i.e., for energies satisfying the condition h < 2 , the system v2 under the condition h < −1, β and the system v3 under the condition h < 0, i.e., in the case of bounded motion, are Liouville equivalent. Proof. As was noted in [46], the projections of Liouville tori in the problem of two fixed centers on the sphere can go around to the equator of the sphere, which in the cases of the flat space and the Lobachevskii space corresponds to going to infinity. The cases where the material point goes around to the equator and hβ 2 on the bifurcation diagram always remains in the upper hemisphere are separated by the line l = R (l is the value of the additional integral). For η 2 = β 2 , the material point crosses the equator, which corresponds to going to infinity. For h < −1, all trajectories in the two-center problem on the pseudosphere are bounded, i.e., we have motion of elliptic type. The phase space for these energies is compact (after regularization), and hence the system satisfies the Liouville theorem. Isoenergetic surfaces are foliated into Liouville tori. Analogously, for the flat two-center problem for h < 0, all trajectories are compact, and there is motion of elliptic type. The phase space for these energies is also compact (after regularization), and the system satisfies the Liouville theorem. Isoenergetic surfaces are also foliated into Liouville tori. Precisely owing to this, such conditions are imposed on the energy. It is known that if two regular neighborhoods of singular leaves have equivalent atoms, then there exists an orientation-preserving homeomorphism of one onto another that preserves the Liouville foliation and transforms nondegenerate critical circles into nondegenerate critical circles with preservation of the natural orientation on them given by the field of the dynamical system. To describe the topological structure of the Liouville foliation on the isoenergetic surface Qh , it suffices to describe it in small invariant neighborhoods of singular leaves and show in what way the whole manifold Qh is glued from these neighborhoods, i.e., to construct topological invariants, the Fomenko–Zieshang invariants. To prove the Liouville equivalence of two dynamical systems, we need to show that the topological invariants are the same for these systems. The study of the topology of natural mechanical systems is carried out by using the projection on the configuration space. Each Liouville torus is projected on the region of possible motion. The projections of 520

the tori for the systems v1 , v2 , and v3 (under the conditions imposed on the energy) on the configuration space (sphere, pseudosphere, and plane) are topologically the same, as well as the types of motion for these regions. A certain number of Liouville tori correspond to each of these regions; the numbers of Liouville tori are the same for the three systems considered for each region of possible motion in the inverse image (one torus in regions 1 and 3 and two tori in regions 2 and 4). The number of tori for the system v1 was found in [46], for the system v2 , it is found from the analysis of the functions R and S, and for the system v3 , the number of tori is found by using quadratures presented in [15]. Moreover, the number of critical points and critical circles and the number of their projections on the configuration space for limit motions coincide. The choice of the basic cycles of the admissible coordinate system in gluing invariant neighborhoods of singular leaves is also analogous. Therefore, the topological invariants are the same for the systems considered (under the conditions imposed on the energy); these are molecules 1, 2, and 3 (see Chapter 3). We have shown that the systems considered have the same Liouville foliations, and hence they are Liouville equivalent.

Chapter 5 MOTION IN THE NEWTONIAN AND HOMOGENEOUS FIELD IN THE LOBACHEVSKII SPACE 28.

Reduction. Analog of Homogeneous Field

R2 ,

there exists a limit case of the two-center problem where one of the centers goes to For the plane infinity. In this case, the mass of the center grows with constant traction. There arises the problem on the motion of a material point in the field of the Newtonian center and the homogeneous field. This problem was studied by Lagrange for the first time; he reduced it to quadratures. The classification of regions of possibility of motion was carried out by V. V. Beletskii [6]. There exists an analog of the Lagrange problem in quantum mechanics, the problem of splitting energetic levels of hydrogen in the homogeneous electric field (Stark effect) [11]. The flat Lagrange problem is described by the Hamiltonian [6, 34] γ 1 V = − + f q1 . (196) H = (p21 + p22 ) + V (q1 , q2 ), 2 r The axis q1 is directed along the constant vector f of the acceleration. Clearly, the equations of motion are written as  q1 q2 q¨2 = −γ 3 , r = q12 + q22 . q¨1 = −γ 3 − f, r r This problem admits the following additional integral of motion 1 I = A1 + f q22 , 2 where A1 is the component of the Laplace–Runge–Lenz vector. Therefore, the equations of motion are integrated by using the method of separation of variables. The parabolic coordinate system is most natural in the Lagrange problem. Indeed, for a sufficiently 1 1 large distance from the attraction center, f  2 (since f is constant and 2 can be arbitrarily small). r r Therefore, the motion is close to motion under the action of the vector f of constant acceleration. When f = const, the motion in the homogeneous force field is executed along a parabolic trajectory, and, moreover, the axis of the parabolic trajectory is directed along f . We pass to the parabolic coordinates according to the formulas 1 1 η = (r − q1 ). (197) ξ = (r + q1 ), 2 2 521

We obtain from this the relations q1 = ξ − η,

r = ξ + η,

q2 = 2 ξη.

In the new variables, the Hamiltonian becomes 1 1 (ξp2 + ηp2η ) + (f (ξ 2 − η 2 ) − γ), (198) H= 2(ξ + η) ξ ξ+η where pξ and pη are impulses corresponding to the coordinates ξ and η. Therefore, the system has the Liouville form and its integration is performed by using the method of separation of variables. As a result, we obtain the qudratures

dη dξ dt = (ξ + η)dτ, (199) = R(ξ), = S(η), dτ dτ where   γ R = 2ξ l + hξ + − f ξ 2 , 2   γ S = 2η − l + hη + + f η 2 . 2 In Fig. 52, we depict the bifurcation diagram for the flat Lagrange problem. l

Γ3

Γ4

Γ2 Γ1

2

1

3

h

4

Fig. 52. Bifurcation diagram for the flat Lagrange problem [6]. Let us consider the Lagrange problem in the Lobachevskii space (without loss of generality, we assume that the radius of the pseudosphere is equal to unity). Theorem 28.1. A material point in the Lagrange problem in the Lobachevskii space H3 moves in the same way as in the two-dimensional system (on the unit two-dimensional pseudosphere H2 : y 2 − x2 − z 2 = 1) with the energy 1 h = (−y˙ 2 + x˙ 2 + z˙ 2 ) + Veff , 2 522

where the effective potential energy is defined by the expression βϕ2 y 1 − γ2 + . Veff = −γ1 (y − x)2 2z 2 y2 − 1 Here, the function 1 (y − x)2 is an analog of the homogeneous field in the space of constant negative curvature. Vh = −γ2

Proof. Consider the passage to the limit in the two-center problem in the space H3 moving one of the attracting centers to infinity. We first place the attracting centers at the points with coordinates r1 = (1, 0, 0, 0) and r2 = (cosh ϕ, sinh ϕ, 0, 0). The potential energy of a material point in this problem is defined as the sum of the following two quantities: the potential energy of the center placed at the vertex of the hyperboloid and the potential energy of the center at infinity. Let the coordinates of the material point be given as r = (x1 , x2 , x3 , x4 ). Then we can calculate cosh θ1 = r1 , r = x1 , 

sinh θ1 = cosh2 θ1 − 1 = (x1 )2 − 1, cosh θ2 = r2 , r = x1 cosh ϕ − x2 sinh ϕ, 

sinh θ2 = cosh2 θ2 − 1 = (x1 cosh ϕ − x2 sinh ϕ)2 − 1. The potential energy of the point has the form x1 , V1 = −γ1 coth θ1 = −γ1 (x1 )2 − 1 x1 cosh ϕ − x2 sinh ϕ V2 = −γ2 coth θ2 = −γ2 , (x1 cosh ϕ − x2 sinh ϕ)2 − 1 V = V 1 + V2 .

(200)

Let ϕ → ∞. Neglecting the value of order e−4ϕ and taking into account that tanh ϕ ≈ 1 − 2e−2ϕ , we obtain the relations x1 − x2 tanh ϕ V2 = −γ2  (x1 − x2 tanh ϕ)2 − 1 + tanh2 ϕ x1 − x2 + 2x2 e−2ϕ = −γ2 (x1 − x2 )2 + 4x2 (x1 − x2 )e−2ϕ − 4e−2ϕ   1 2x2 −2ϕ 2 x − x2 + x2 e−2ϕ −2ϕ 1− 1 = −γ2 e + 1 e x1 − x2 x − x2 (x − x2 )2   2 −2ϕ . = −γ2 1 + 1 e (x − x2 )2 Make the change 2 exp(−2ϕ)γ2 → γ2 . Neglecting the constant value and passing to the limit as ϕ → ∞, we obtain the expression for the potential energy of the center at infinity, an analog of the homogeneous field in a curved space: 1 . (201) V2 = −γ2 1 (x − x2 )2 Let us reduce the problem to two degrees of freedom analogously to the reduction in the two-center problem. Introduce new coordinates by formulas analogous to (175) and use the Routh method in order 523

to exclude the cyclic coordinate. As a result, the problem considered reduces to the motion of the material point on the two-dimensional pseudosphere y 2 − x2 − z 2 = 1 in the reduced field βϕ2 y 1 + . − γ2 Veff = −γ1 (y − x)2 2z 2 y2 − 1

(202)

The theorem is proved. y y

ϕ x x z Fig. 53. Turn of the upper flap of the hyperboloid.

29.

Pseudosphero-Parabolic Coordinates

Make the change of variables in order to obtain the Liouville form of the Hamiltonian. For this purpose, we use the equations of coordinate surfaces for the two-center problem and analogously pass to the limit. The equation for the first coordinate surface is x2 y 2 z 2 = + . α2 − ξ 2 β 2 − ξ 2 ξ 2 We now use the substitution (see Fig. 53) x → x cosh ϕ − y sinh ϕ,

y  → y cosh ϕ − x sinh ϕ,

z  → z,

ξ  → ξ sinh ϕ.

(203)

To pass to the limit, we turn the coordinate system in such a way that the attracting center located at the point with coordinates (−α, β, 0) moves at the point with coordinates (0, 1, 0). Clearly, in this case the following relations hold: β = cosh ϕ, α = sinh ϕ. Then

(y cosh ϕ − x sinh ϕ)2 z2 (x cosh ϕ − y sinh ϕ)2 = + . sinh2 ϕ(1 − ξ 2 ) cosh2 ϕ(1 − ξ 2 tanh2 ϕ) sinh2 ϕξ 2

After simple transformations, taking into account the relations tanh ϕ ≈ 1−2e−2ϕ and coth ϕ ≈ 1+2e−2ϕ , and neglecting the value of order e−4ϕ , we obtain the following equation of the coordinate surface: 2x(y − x) =

ξ2 1 − ξ2 2 2 (y − x) − z . 1 − ξ2 ξ2

(204)

Introduce a new variable µ and, as a result, obtain the following equation for the first coordinate surface in the Lagrange problem on the pseudosphere: µ2 = 524

ξ2 , 1 − ξ2

2x(y − x) = µ2 (y − x)2 −

z2 . µ2

(205)

For the second coordinate surface, we have the equation x2 z 2 y 2 = + . β 2 + η 2 α2 + η 2 η 2 Then, using the substitution η  → η cosh ϕ, we obtain the equation (x cosh ϕ − y sinh ϕ)2 z2 (y cosh ϕ − x sinh ϕ)2 = + . cosh2 ϕ(1 + η 2 ) sinh2 ϕ(1 + coth2 ϕ)) η 2 cosh2 ϕ After analogous transformations (as for the first coordinate surface), we obtain the following equation for the second coordinate surface: 1 + η2 2 η2 2 (y − x) + z . (206) 2x(y − x) = − 1 + η2 η2 Introducing the new variable η2 , ν2 = 1 + η2 we finally obtain z2 (207) 2x(y − x) = −ν 2 (y − x)2 + 2 . ν Obviously, the equation for the third coordinate surface is the equation of the hyperboloid: x2 − y 2 + z 2 = −1. We see that the new coordinates are defined in such a way that the following relations hold: ν ≤ 1,

0 ≤ µ < ∞.

(208)

The converse relations for the Cartesian coordinates are defined by the relations 1 µ2 − ν 2

, 2 (1 − ν 2 )(1 + µ2 ) 1 µν z= , 2 (1 − ν 2 )(1 + µ2 )

x=

30.

2 + µ2 − ν 2 y= , 2 (1 − ν 2 )(1 + µ2 ) 1 y−x= . 2 (1 − ν )(1 + µ2 )

(209)

Integrals of the System

We write the Hamiltonian in the variables µ and ν:     % 1 1 (1 − ν 2 )(1 + µ2 ) 1 $ 2 2 2 2 2 2 − γ2 (µ + ν ) . (210) + (1 + µ )pµ + (1 − ν )pν − γ1 H= µ2 + ν 2 2 1 + µ2 1 − ν 2 Since the Hamiltonian has the Liouville form in pseudosphero-parabolic coordinates, we can write the additional integral 2γ1 1 (1 + µ2 )3 2 1 (1 − ν 2 )3 2 p + p + + γ2 (µ2 − ν 2 + 1) (211) 2 µ2 + ν 2 µ 2 µ2 + ν 2 ν µ2 + ν 2 or the following two Liouville integrals, which are dependent (they will be needed in order to reduce the problem to qudratures):  2 1 1 2 µ˙ − 1 1 1 1 − ν 2 1 + µ2 − γ1 − γ2 µ2 + h , I1 = 2 1 + µ2 1 + µ2 1 + µ2 (212)  2 1 1 ν˙ 2 − 1 1 1 1 − ν 2 1 + µ2 − γ1 − γ2 ν 2 − h . I2 = 2 2 2 1−ν 1−ν 1 − ν2 Since I1 + I2 = 0, we introduce the notation l = I1 = −I2 . l=

525

The regularization is performed by introducing the new independent variable τ by the relation   1 1 1 dt = √ dτ. − 2γ2 1 − ν 2 1 + µ2 The problem reduces to qudratures [44]

dν = R(ν), dτ where

dµ = S(µ), dτ

(213)

R(ν) = −l(1 − ν 2 ) + h + K + ν 2 (1 − ν 2 ), S(µ) = l(1 + µ2 ) − h + K + µ2 (1 + µ2 ). 31.

Bifurcation Diagrams

Theorem 31.1. The bifurcation set is Γ1 : (l − 1)2 + 4h − 4K = 0, Γ3 : l = h + K,

Γ2 : (l − 1)2 + 4h + 4K = 0,

Γ4 : l = h − K,

Γ5 : h = −K,

K=

γ1 . γ2

(214)

Fig. 54. Regions of possible motions for the Lagrange problem on the pseudosphere. The bifurcation diagram and the classification of regions of possible motion are presented in Figs. 54 and 55. The tangent points of the hyperbolas Γ2 and Γ1 with the lines Γ3 and Γ4 , respectively, have the coordinates (−1, −(K + 1)), (−K, 1), (K − 1, −1). When we have a single field (γ1 = 0), the lines Γ3 and Γ4 and the parabolas Γ1 and Γ2 merge (Fig. 55). Correspondingly, the regions between these curves disappear (3, 4, 7, 8). Then the region where the 526

motion is unbounded is located on the line l = h. Therefore, even for a negligible change of the integrals h and l, the material point “falls” either on region 1 or on region 4 in which the field of the infinitely distant center twists the trajectory of the moving material point, for example, the trajectory of the point moving along the axis Ox (see Fig. 56).

l

Ã2 Ã1

Ã3

1 Ã4

8

h

2

3

7 6

4

5

Ã5

Fig. 55. Bifurcation diagram for the Lagrange problem on the pseudosphere (the regions in which the motion is impossible are shaded). A motion is said to be bounded with respect to one of the variables if either sup ν 2 < 1 or sup µ2 < ∞. t≥t0

t≥t0

Assertion 31.1. For h < −K, the motion is bounded with respect to the variable ν and is unbounded with respect to the variable µ. For h ≥ −K, the motion is unbounded with respect to both variables. Therefore, the motion is bounded with respect to the variable ν in regions 2, 3, 4, and 5. Let the function S have two roots µ1 < µ2 . A motion is said to be partially bounded if sup µ1 < ∞. t≥t0

Assertion 31.2. The motion is partially bounded in regions 4 and 7. Proof. In region 4, we have 0 ≤ µ ≤ µ1 < µ2 < ∞, ν 2 ≤ ν22 < 1, i.e., the conditions sup µ1 < ∞ and t≥t0

sup ν < ∞ hold. In region 7, for the coordinate µ, we have the same relation, and, for the coordinate ν,

t≥t0

we have ν 2 ≤ 1. That is, the conditions sup µ1 < ∞ and sup ν < ∞ also hold. Therefore, by definition, t≥t0

t≥t0

there are bounded trajectories in these regions.

527

Fig. 56. Bifurcation diagram for the problem of motion of a particle in a “homogeneous field” on the pseudosphere. 32.

Description of Noncompact Bifurcations and Motions on the Configuration Space

We see from Fig. 55 that there arise the following bifurcations: 2 → ∅, 8 → 1,

2 → 3, 5 → 6,

3 → 4, 4 → 7,

4 → 5, 3 → 8,

6 → 7, 2 → 1,

7 → 8, ∅ → 1.

Case 2 → ∅. For region 2, we have two cylindrical surfaces. Each such cylinder is obtained by gluing two semicylinders by their base: S 1 × [0, ∞) (the circle is multiplied by the ray). For µ = 0, the velocity changes sign so that the coordinate µ is nonnegative. As a result, we obtain the cylinder S 1 × R1 ; when the point (h, l) crosses the hyperbola Γ1 , the cylinder is contracted into its axis and then disappears. This bifurcation corresponds to periodic motion in the compact case if we add the infinitely distant point (An ). Case 2 → 3. In this case, we have the following bifurcation. The coordinate ν passes through zero value, and the two cylinders deform into one analogously to the surgery in the two-center problem under the bifurcation 8 → 7 (i.e., bifurcation of type Bn ). The limit motion has the trajectories of the following two types: the material point moves along the ray emanating from the attraction center in direction of the “homogeneous” field; the second trajectory is asymptotic, i.e., the material point asymptotically tends to this ray going to infinity. Case 3 → 4. In region 4, we have two connected components: one torus and a cylindrical surface. In the first case, the trajectory is bounded by two parabolas µ = µ1 and ν = ν2 . If the influence of an additional traction is large, then the character of motion is oscillatory, i.e., the material point bending the attraction center and then oscillating between the branches of the parabola ν = ν2 , moves to the side of increasing value of µ up to µ1 . Having attained this boundary, the point begins to move in the reverse (with respect to µ) direction along the oscillatory trajectory, approaching the center once again, then bends it, and so on. If the summand stipulated by the potential energy of the homogeneous field is small (as compared with the energy), then the trajectory changes its shape taking the quasielliptic type. This is stipulated by the smallness of influence of the homogeneous field as compared with the attraction force of the Newtonian center at the vertex of the hyperboloid. If in the process of motion, the material point attains the intersection point of two parabolas µ = µ1 and ν = ν2 , then the velocity of the material point is zero here, and further motion will be executed along the trajectory already passed but in the reverse direction. If the trajectory goes near the intersection point of the parabolas, then there is always a change of direction. Along with boundary trajectories, in this region, there exist unbounded trajectories 528

Fig. 57

(the Hamiltonian flow is on the cylindrical surface). The unboundedness of the trajectory in this case is stipulated by the distantness of the attracting center and, in this connection, the increase of influence of the homogeneous field (constant traction). Under the bifurcation, the root of the equation S = 0 is multiple. On the boundary of regions 3 and 4, we have three trajectories. There arises the periodic trajectory µ = const = µ1 = µ2 . The motion of the material point is executed along the bounded arc of this parabola, along the part lying inside a certain parabola ν = ν1 . Oscillating, the material point moves forward and backward along the arc µ = µ1 = µ2 between two points with the coordinates µ = µ1 = µ2 and ν = ν2 . A snake-type bounded trajectory asymptotically tends to this trajectory from below. Upon approaching the multiple root µ = µ1 = µ2 , the loops of the trajectory approach one another more and more tightly. One more trajectory goes above the arc µ = µ1 = µ2 ; it first oscillates near the asymptotic trajectory and then goes to infinity. The bifurcation is presented in Fig. 57. Case 4 → 5. When the point (h, l) passes the bifurcation line separating the regions of possible motion 4 and 5 (h < −K), the Liouville torus degenerates in the phase space. In this case, we have a bifurcation of type A (in the compact case). The limit motion is as follows: the material point moves along the segment between the attracting center and the vertex of the parabola ν = ν2 . As is seen, in region 5, the trajectories do not bend the attracting center. In this region, there exist self-returning trajectories, when the material point occurs at the intersection point of parabolas bounding the motion, the roots of the functions R and S. At these points, the velocity is equal to zero, but there is no equilibrium, since the forces are not zero; therefore, the point goes in the backward direction along the same trajectory. Also, there exist trajectories of oscillatory character: the material point oscillates between the branches of the parabola ν = ν2 and goes to infinity. Case 6 → 7. When the point (h, l) crosses the line Γ4 , two tori are born; in this case, we have a bifurcation of type A. The limit motion is determined by the trajectory passing through the center; the component of velocity satisfies µ˙ = 0 for this motion, and the coordinate satisfies µ = 0. Case 7 → 8. In region 7, we have the following connected components: two tori and two cylinders. Under the bifurcation, when the point (h, l) passes from region 7 to region 8, each of the tori is tangent to a cylinder, and, as a result, one cylinder is obtained (the surgery is analogous to that depicted in Fig. 57). In region 8, we also have two connected components, two cylinders. In region 8, motion is possible on the whole configuration space, i.e., ν 2 ≤ 1, 0 ≤ µ < ∞. In region 7, when the point goes away from the attracting center, µ > µ2 , the motion is executed along cylinders; in this case, the “homogeneous field” plays the main role; correspondingly, if the initial position of the point is near the attracting center, i.e., if µ < µ1 , then the influence of the attracting center is greater, and the motion is executed along the torus; on one torus the component of the velocity satisfies ν˙ > 0, and on another torus, it satisfies ν˙ < 0. Case 8 → 1. When the point (h, l) crosses the line Γ3 , under the condition that h > −K, both cylinders deform into one. In this case, this is a bifurcation of type Bn . The limit motion is asymptotic. On the 529

configuration space, the material point asymptotically tends to a ray emanating from the Newtonian center (ν = 0, ν˙ = 0). Finally, consider the bifurcations arising when the point (h, l) crosses the line h = −K (Γ5 ). When the point (h, l) crosses the line h = −K separating the regions 3 and 8 and 5 and 6, one cylinder deforms into two cylinders; this is a bifurcation of type Bn . When for the point (h, l) we have the transition 2 → 1, conversely, two cylinders deform into one. When the point (h, l) crosses the line h = −K separating the regions 4 and 7, one torus deforms into two tori (atom B) and one cylinder deforms into two cylinders; this is a bifurcation of type Bn . Acknowledgments. In conclusion, the author expresses her gratitude to Academician A. T. Fomenko, the chief of the chair of Differential Geometry and Applications, Department of Mechanics and Mathematics, Moscow State University, and A. A. Tuzhilin, A. O. Ivanova, A. I. Shafarevich, and A. A. Oshemcov, the collaborators of the chair, for valuable discussions and support. REFERENCES 1. V. M. Alekseev, “Generalized spatial two-fixed-centers problem. Classification of motions,” Bull. Inst. Teor. Astronom., 10, No. 4, 241–272 (1965). 2. V. I. Arnol’d, Mathematical Methods of Classical Mechanics [in Russian], Nauka, Moscow (1974). 3. V. I. Arnol’d, V. V. Kozlov, and A. I. Neishtadt, “Mathematical aspects of classical and celestial mechanics,” In: Progress in Science and Technology, Series on Contemporary problems in Mathematics, Fundamental Directions [in Russian], Vol. 3, All-Union Institute for Scientific and Technical Information (VINITI), Acad. Nauk SSSR, Moscow (1985), pp. 5–304. 4. Yu. A. Arkhangel’skii, Analytical Dynamics of a Rigid Body [in Russian], Nauka, Moscow (1977). 5. N. K. Badalyan, “On the shape of trajectories in the two-fixed-centers problem,” In: Proc. of AllUnion Mathematical Session [in Russian], Vol. 2, Akad. Nauk SSSR, Moscow (1937), pp. 239–241. 6. V. V. Beletskii, Essays on the Motion of Space Bodies [in Russian], Nauka, Moscow (1972). 7. A. V. Bolsinov, S. V. Matveev, and A. T. Fomenko, “Topological classification of integrable Hamiltonian systems with two degrees of freedom,” Usp. Mat. Nauk, 45, No. 2, p. 49, (1990). 8. A. V. Bolsinov and A. T. Fomenko, Integrable Hamiltonian Systems [in Russian], Vols. 1, 2, Izdatel’skii Dom “Udmurtskii Universitet,” Izhevsk (1999). 9. A. V. Bolsiniv and A. T. Fomenko, “Trajectory equivalence of integrable Hamiltonian systems with two degrees of freedom,” Mat. Sb., 185, No. 4, 27–80 (1994); 185, No. 5, 27–78 (1994). 10. A. V. Bolsinov, P. Richter, and A. T. Fomenko, “Circular molecule method and the topology of the Kowalewski top,” Mat. Sb., 191, No. 2, 1–42 (2000). 11. M. Born, Vorlesungen u ¨ber Atommechanik (1934). 12. A. V. Brailov and A. T. Fomenko, “Topology of integral manifolds of completely integrable Hamiltonian systems,” Mat. Sb., 133, No. 3, 375–385 (1987). 13. C. Charlier Celestial Mechanics [Russian translation], Nauka, Moscow (1966). 14. N. A. Chernikov, “The Kepler problem in the Lobachevskii space and its solution,” Acta Phys. Polonica, B23, 115–119 (1992). 15. A. G. Duboshin, Celestial Mechanics, Analytical and Qualitative Methods [in Russian], Nauka, Moscow (1964). 16. A. T. Fomenko, “Topology of surfaces of constant energy of integrable Hamiltonian systems and obstructions to integrability,” Izv. Akad. Nauk SSSR, Ser. Mat., 50, 1276–1307 (1986). 17. A. T. Fomenko, “Topological classification of all integrable Hamiltonian differential equations of general type with two degrees of freedom,” In: The Geometry of Hamiltonian Systems, Proc. Workshop Held June 5–16, 1989, Berkeley, USA, Springer-Verlag (1991), pp. 131–339. 18. A. T. Fomenko and Kh. Zieshang, “On typical topological properties of integrable Hamiltonian systems,” Izv. Akad. Nauk SSSR, Ser. Mat., 52, No. 2, 378–407 (1988). 530

19. A. T. Fomenko and Kh. Zieshang, “A topological invariant and a criterion of topological equivalence of integrable Hamiltonian systems with two degrees of freedom,” Izv. Akad. Nauk SSSR, Ser. Mat., 54, No. 3, 546–575 (1990). 20. Ya. I. Granovskii, A. S. Zhedanov, and I. M. Lutsenko, “Quadratic algebras in a curved spaces, I, Oscillator,” Teor. Mat. Fiz., 91, No. 2, 207–216 (1992). 21. Ya. I. Granovskii, A. S. Zhedanov, and I. M. Lutsenko, “Quadratic algebras in a curved space, II, The Kepler problem,” Teoret. Mat. Fiz., 91, No. 3, 396–410 (1992). 22. E. A. Grebenikov, V. G. Demin, and E. P. Aksenov, “General solution of the problem on the motion of an artificial Earth sattelite in the normal attraction field of Earth,” In: Artificial Earth Sattelite [in Russian], No. 8 (1961). 23. P. W. Higgs, “Dynamical symmetries in a spherical geometry. I,” J. Phys. A, 12, No. 3, 309–323 (1979). 24. M. P. Kharlamov, Topological Analysis of Integrable Problems in the Dynamics of a Rigid Body [in Russian], LGU, Leningrad (1988). 25. V. V. Kozlov, Symmetry, Topology, and Resonance in the Hamiltonian Mechanics [in Russian], Udmurtskii Univ., Izhevsk (1995). 26. V. V. Kozlov, “On dynamics in spaces of constant curvature,” Vestnik Moskov. Univ. Ser. 1, Mat. Mekh., No. 2, 28–35 (1994). 27. V. V. Kozlov and N. N. Kolesnikov, “On integrability of Hamiltonian systems,” Vestnik Moskov. Univ. Ser. 1, Mat. Mekh., No. 6, 88–91 (1979). 28. V. V. Kozlov and O. A. Harin, “Kepler’s problem in constant curvature spaces,” Celestial. Mech. Dynam. Astronom., 54, 393–399 (1992). 29. T. Levi-Chivita, “Sur la r´esolution qualitative du probl`eme restreint des trois corps,” Acta Math., 30, 305–327 (1906). 30. N. I. Lobachevskii, “New origins of geometry with a complete theory of parallels,” In: Complete Collection of Works [in Russian], Vol. 2, GITTL, Moscow–Leningrad (1949). 31. Yu. Moser, “Certain aspects of integrable Hamiltonian systems,” Usp. Mat. Nauk, 36, No. 5, 109–151 (1981). 32. A. A. Oshemkov, “Description of isoenergetic surfaces of integrable Hamiltonian systems with two degrees of freedom,” Trudy Semin. Vector Tenzor Anal., No. 23, 122–132 (1988). 33. A. A. Oshemkov, “Calculation of Fomenko invariants for main cases of dynamics of a rigid body,” Trudy Semin. Vector Tenzor Anal., No. 25, Part 2, 23–109 (1993). 34. A. M. Perelomov, Integrable systems of classical mechanics and Lie algebras [in Russian], Nauka, Moscow (1990). 35. S. Smale, “Topology and mechanics,” Invent. Math. 10, 305–331 (1970). 36. A. V. Shepetilov, “Reduction of the two-body problem with central interaction on simply connected spaces of constant sectional curvature,” J. Phys. A, 31, 6279–6291 (1998). 37. E. Schr¨ odinger, “A method for finding quantum-mechanical eigenvalues and eigenfunctions,” In: Selected Works in Quantum Mechanics [in Russian], Nauka, Moscow (1976), pp. 239–247. ¨ 38. H. J. Tallquist, “ Uber die Bewegung eines Punktes, welcher von zwei festen Zentren nach dem Newtonischen Yesetze angezogen wird,” Acta Soc. Sci. Fenn. A, NS, 1, No. 5 (1927). 39. V. V. Trofimov and A. T. Fomenko, Algebra and Geometry of Integrable Hamiltonian Systems of Differential Equations [in Russian], Faktorial, “Prosperus,” Moscow (1995). 40. T. G. Vozmischeva, “The two-center and Lagrange problem in the Lobachevskii space,” In: Proc. Int. Conf. “Geometry, Integrability, and Quantization”, Bulgaria (1999). 41. T. G. Vozmischeva and I. S. Mamaev, “Classification of motions for generalization of Euler and Lagrange problems on Lobachevskii plane,” In: Int. Conf. “Geometrization of Physics III,” Kazan’ State University, Kazan’ (1997). 42. T. G. Vozmischeva T. G., Classification of motions for generalization of the two-center problem on a sphere,” Celestial Mech. Dynam. Astronom., 77, 37–48 (2000). 531

43. T. G. Vozmishcheva, “Some integrable problems of celestial mechanics in spaces of constant curvature,” In: Progress in Science and Technology, Series on Contemporary Mathematics and Its Applications, Thematical Surveys, Dynamical Systems-12 [in Russian], Vol. 88, All-Russian Institute for Scientific and Technical Information (VINITI), Ross. Akad. Nauk, Moscow (2001), pp. 145-198. 44. T. G. Vozmischeva, “The Lagrange and two-center problems in the Lobachevskii space,” Celestial Mech. Dynam. Astronom., 84, No. 1, 65–85 (2002). 45. T. G. Vozmishcheva, “Classification of motions in the generalization of the Euler problem to a sphere,” In: Mathematical Collection [in Russian], Udmurtskii Univ. (1998), pp. 34–40. 46. T. G. Vozmishcheva and A. A. Oshemkov, “Topological analysis of the two-center problem on a two-dimensional sphere,” Mat. Sb., 193, No. 8, 3–38 (2002). 47. T. G. Vozmishcheva, “Topological analysis of integrable problems of celestial mechanics on a sphere and pseudosphere,” In: Progress in Science and Tehnology, Series on Contemporary Problems in Mathematics, Thematical Surveys [in Russian], All-Russian Institute for Scientific and Technical Information (VINITI), Ross. Akad. Nauk, Moscow (in press). 48. E. T. Whittaker and G. N. Watson, A Course of Modern Analysis, Cambridge Univ. Press, Cambridge (1932). 49. N. E. Zhukovskii, “On the motion of a material pseudospherical figure on the surface of a pseudosphere,” In: Complete Collection of Works [in Russian], Vol. 1, Moscow (1937), pp. 490–535. T. G. Vozmishcheva Izhevsk State Technical University E-mail: [email protected]

532