Äîêóìåíò âçÿò èç êýøà ïîèñêîâîé ìàøèíû. Àäðåñ îðèãèíàëüíîãî äîêóìåíòà : http://ics.org.ru/upload/iblock/261/143-superintegrable-systems-on-a-sphere-_ru.pdf Äàòà èçìåíåíèÿ: Wed Oct 28 19:31:05 2015 Äàòà èíäåêñèðîâàíèÿ: Sun Apr 10 00:04:01 2016 Êîäèðîâêà: Ïîèñêîâûå ñëîâà: m 17

A. V. BORISOV, I. S. MAMAEV
Institute of Computer Science Udmurt State University 1 Universitetskaya str., 426034 Izhevsk, Russia E-mail: b orisov@rcd.ru, mamaev@rcd.ru

SUPERINTEGRABLE SYSTEMS ON A SPHERE
Received October 25, 2004; accepted February 15, 2005

DOI: 10.1070/RD2005v010n03ABEH000314

We consider various generalizations of the Kepler problem to three-dimensional sphere S 3 (a compact space of constant curvature). In particular, these generalizations include addition of a spherical analogue of the magnetic monopole (the Poincar´ ­ Appell system) and addition of a more complicated field which is a generalization of the MICZ-system. The e mentioned systems are integrable superintegrable, and there exists the vector integral which is analogous to the Laplace ­ Runge ­ Lenz vector. We offer a classification of the motions and consider a tra jectory isomorphism between planar and spatial motions. The presented results can be easily extended to Lobachevsky space L3 .

To the memory of Henri Poincar´ e, on the 150th anniversary of his birth

Contents
1. The Kepler problem in R3 . . . . . . . . . . . . . . . . . 2. The MICZ-system in R3 . App ell's problem . . . . . . . . 3. The Kepler problem on three-dimensional sphere S 3 (and 4. Generalization of the Poincar´ and App ell problems to S 3 e 5. Generalized MICZ-mo del . . . . . . . . . . . . . . . . . . 6. Tra jectory isomorphism for central p otential systems on S References . . . . . . . . . . . . . . . . . . . . . . . . . . . . ......... ......... on Lobachevsky ......... ......... 2 and R2 . . . . ......... ... ... space ... ... ... ... . . L . . . . . . 3) . . . . ... ... .. ... ... ... ... . . . . . . . . . . . . . . 257 258 259 261 263 264 266

1. The Kepler problem in R

3

Consider the Kepler problem: a mass p oint (of unit mass, without losing in generality) moves in the Newtonian field of a fixed center; the intensity of the gravitational interaction is constant. In this problem, in addition to the integral of energy
3

H0 = 1 2 and the vector integral of angular momentum

qi + U 2
i=1

(1.1)

M = q â q,
Mathematics Sub ject Classification: 37N05, 70F10 Key words and phrases: spaces of constant curvature, Kepler problem, integrability

(1.2)



REGULAR AND CHAOTIC DYNAMICS, V. 10,

3, 2005

257


A. V. BORISOV, I. S. MAMAEV

the equations in three-dimensional Euclidean space R 3 = {q1 , q2 , q3 }, qi = U , ¨ qi U = -r,
2 2 2 r 2 = q1 + q2 + q3 ,

= const,

(1.3)

have one more remarkable vector integral, which is due to certain hidden symmetry of the Kepler problem. This vector integral is called the Laplace ­ Runge ­ Lenz vector A = (A 1 , A2 , A3 ). It exists only in the case of Newtonian p otential (of all the central p otentials) and can b e written as follows: A = M â q + r q. (1.4)

Intro ducing the momenta p = q , we can rewrite equations (1.3) and integrals (1.1), (1.2), and (1.4) in the canonical form: H0 H0 , q =- , p , q R3 . (1.5) p= q p The Poisson brackets for the comp onents of the integrals M and A are {Mi , Mj } =
ij k

Mk ,

{Mi , Aj } =

ij k

Ak ,

{Ai , Aj } = -2h

ij k

Mk ,

(1.6)

where h is the constant of energy (1.1), h = 1 p 2 - r , and ij k is the Levi-Civita symb ol. Dep ending 2 on the value of h, the algebra of integrals (1.6) is either so(4) (when h < 0) or so(3, 1) (when h > 0). Note that, since (M , A) = 0, A is always in the plane of the tra jectory. Besides, the vector's direction coincides with the direction of the ellipse's ma jor axis, while its absolute value is prop ortional to the eccentricity. The algebra of integrals (1.6) is an algebra, under which the Kepler problem is invariant. Invariance under a global group of transformations (i. e., for example, under group S O (4) for h < 0) was studied by V. A. Fok [9], G. Gy¨ orgyi [11] and J. Moser [19]. The latter work contains the most general result, which shows that even in the n-dimensional case, the constant energy surface (for h < 0) after suitable regularization is top ologically equivalent to the bundle of unit vectors tangent to n-dimensional sphere S n . Note also that the principal dynamical effect of a redundant algebra of integrals (1.6) is the fact that the tra jectories of system (1.3) are closed in the configurational and phase spaces.

2. The MICZ-system in R3 . App ell's problem
Consider other generalization of the Kepler problem, for which an analogue of integral (1.4) exists. To this end, in the phase space T R3 we define a noncanonical Poisson bracket {qi , qj } = 0, and a Hamiltonian H1 = 1 2
3

{qi , pj } = ij , µ2 p2 - r + 2 , i 2r

{pi , pk } = -µ

qk ij k 3 r

(2.1)

, µ = const.

(2.2)

i=1

Remark. This system (the differential equations of motion) can as well be obtained with the standard canonical bracket, but in this case the Hamiltonian would contain terms linear in momenta.

Equations (2.1), (2.2) define the MICZ-system (McIntosh ­ Cisneros ­ Zwanziger); it describ es a particle's motion in the asymptotic field of a self-dual monop ole [8]. It was formally studied by Zwanziger [21], McIntosh and Cisneros [18] without any relevant physical interpretation (see also [7]).


258

REGULAR AND CHAOTIC DYNAMICS, V. 10,

3, 2005


SUPERINTEGRABLE SYSTEMS ON A SPHERE

Consider some sp ecial cases of (2.1), (2.2). The Kepler problem can b e obtained if we put µ = 0. Putting = 0 and µ = 0 in the Hamiltonian (2.2), not in the bracket (2.1), we have the classical integrable Poincar´ problem of a particle moving in the field of a magnetic monop ole. As it was shown e by Poincar´ the particle's tra jectories in this case are geo desics of a circular cone. e, P. App ell considered a more general problem of a particle moving in the field of a Newtonian center and in the field of a magnetic monop ole, assuming that the center and the monop ole coincide [1]. This problem o ccurs if µ = 0 in (2.2) (in the bracket (2.1), however, µ = 0). In this case, the tra jectory is a conic section in the involute of the circular cone, while the integral of areas is preserved during the motion. On the cone itself the tra jectories are, generally, not closed. The vector integral of angular momentum M exists for all the ab ove problems, while there is no analogue of the integral A (1.4) in the Poincar´ and App ell problems, but it exists for the system e (2.1), (2.2). Indeed [3], the vector functions q M = q âp +µr, A= 1 |2H1 | q p âM - r (2.3) (2.4)

form the algebra of integrals of (2.1), (2.2), which is isomorphic to so(4) for H 1 < 0 and to so(3, 1) for H1 > 0. Again the tra jectories are conic sections, and, since (M , q /r ) = -µ, b elong to the circular cone with cone angle = arccos µ/|M | and the axis of symmetry, defined by M . Various generalizations of the Laplace ­ Runge ­ Lenz integrals to dynamical systems in the Euclidean space were studied in [16].

3. The Kepler problem on three-dimensional sphere S 3 (and on Lobachevsky space L3 )
Consider analogues of the Kepler problem in the simple non-Euclidean spaces (where curvature is constant), i. e. three-dimensional sphere S 3 and Lobachevsky space. We study the spherical case in more detail. However, all the results (after appropriate revision) can b e extended to Lobachevsky space. Let the three-dimensional sphere S 3 b e emb edded into the four-dimensional Euclidean space R 4 = = {q0 , q1 , q2 , q3 } and given by equation
2 2 2 2 q0 + q 1 + q 2 + q 3 = R 2 ,

(3.1)

where R is the sphere's radius. Spherical co ordinates on S 3 are given by relations: q0 = R cos , q2 = R sin sin cos , q1 = R sin cos , q3 = R sin sin sin . (3.2)

Consider the motion of a particle in the field of a Newtonian center, placed at one of the p oles, = 0, of the three-dimensional sphere. It is well known [17, 12, 10, 13, 15, 20] that the analogue of Newtonian p otential on a sphere is U = - ctg = - q0 , |q |
2 2 2 q 2 = q1 + q2 + q3 ,

q = (q1 , q2 , q3 ).

(3.3)

Recall that the p otential (3.3) can b e obtained either by extending Bertrand's theorem to sphere [14, 17] or as a solution of the Laplace ­ Beltrami equation on sphere S 3 (see b elow, (4.8)), this solution b eing invariant under group S O (3) and having a singularity at the p ole = 0.


REGULAR AND CHAOTIC DYNAMICS, V. 10,

3, 2005

259


A. V. BORISOV, I. S. MAMAEV

Using the indep endent co ordinates q = (q 1 , q2 , q3 ) we can write the Lagrangian of the system as L = 1 (q 2 + q 2
-2 0

(q , q )2 ) - U (q ),

(3.4)

where q0 = ± R2 - q 2 . After intro ducing the momentum p = L = q + q (q , q ) R2 - q
2

q,

(3.5)

the equations of motion can b e represented in the canonical Hamiltonian form with Hamiltonian H = 1 p 2 - 1 2 (p , q )2 + V (q ). 2 2R These equations have a vector integral of angular momentum M =p âq =q âq (it exists as well for every "central" p otential V that dep ends only on |q | = analogue of the Laplace ­ Runge ­ Lenz vector integral A = q0 p â M + R
2

(3.6)

(3.7)
2 2 q1 + q2 + q3 ) and an

q . |q |

(3.8)

The comp onents of Mi and Ai commute in the following way: {Mi , Mj } = -
ij k

Mk ,
2

{Ai Aj } = 2(R h - M )

{Mi , Aj } = -
2 ij k

ij k

Ak ,

Mk .

(3.9)

(This algebra was discussed in several pap ers [12, 10].) The Casimir functions of the nonlinear Poisson structure (3.9) are F1 = (M , A), F2 = A2 - 2h M 2 + (M 2 )2 , (3.10)

so its symplectic leaf is four-dimensional (b ecause the rank of (3.9) is four). Here, = 1/R 2 is the curvature of the space. For real motions for the Kepler problem F1 = 0, F2 = 2 R 4 . (3.11)

The compactness of the symplectic leaf (3.11) is defined by the curvature of space, , and the value of the constant of energy: 1. when = 0: h < 0 -- compact, h 2. when > 0: always compact. 3. when < 0: h < 0 and h2 > 2 -- the leaf (3.11) is disconnected, one comp onent is compact, while the other is noncompact; h > - -- the leaf is connected, but noncompact. The tra jectories of the Kepler problem on sphere (and pseudosphere) are conic sections, the generalization of Kepler's laws to this case was done in [14, 13, 5]. In the pap er [6], bifurcational analysis of the Kepler problem on S 3 and L3 was p erformed, and the action-angle variables were intro duced (see also [4]).


0 -- noncompact.

260

REGULAR AND CHAOTIC DYNAMICS, V. 10,

3, 2005


SUPERINTEGRABLE SYSTEMS ON A SPHERE

4. Generalization of the Poincar´ and App ell problems to S e

3

First, we obtain the Hamiltonian form of the equations of a particle's motion on three-dimensional sphere S 3 under generalized p otential forces. Consider the Lagrangian L= 1 q +q 2
-2 0

(q , q )

2

- q , W (q ) - U (q ),

(4.1)

where W = W (q ) = (W1 , W2 , W3 ) is the vector p otential. Intro ducing the canonical momentum p = L = q - W + q we obtain the Hamiltonian H = 1 (p + W )2 - 1 2 (p + W , q )2 + U (q ) 2 2R (4.3) (q , q ) R2 - q
2

q,

(4.2)

and the canonical Poisson bracket ({q i , pj } = ij ). But for a numb er of reasons, often it is more convenient to study Hamiltonian equations written in terms of slightly mo dified momentum p = = p + W , which form the following noncanonical Poisson brackets {pi , pj } = Wi Wj - = Bij , qj qi {qi , pj } = ij , {qi , qj } = 0, (4.4)

where B = rot W . In this case the Hamiltonian (4.3) simplifies to: H = 1 p 2 - 12 (p , q )2 + U (q ). 2 R (4.5)

In the case of three-dimensional sphere, an analogue of the vector p otential of a magnetic monop ole can b e obtained in the following way. Electromagnetic field tensor in vacuum satisfies the Maxwell equations F + F + F = 0 1 ( -g F ) = 0 -g


, = 0, 1, 2, 3 = x ,


(4.6)

where g is the metric of space-time and g = det g . On S 3 , the metric of space-time in the spherical co ordinates (3.2) is dS 2 = c2 dt2 - R2 d 2 + sin2 (d2 + sin2 d 2 ) . (4.7)

Let i, j , k stand only for the spatial indices, and g denotes the spatial metric, taken with the negative sign. We will search for the solution, similar to the magnetic monop ole in flat space, in the form F0i = 0, g F ij = ij k k f . From (4.6) we find the equation for the unknown function f
k

ik g g i f = 0,

coinciding with the Laplace ­ Beltrami equation. The solution, invariant under group S O (3) (i. e. indep endent of , ), satisfies the equation 1 sin2 sin2 f =0 (4.8)



REGULAR AND CHAOTIC DYNAMICS, V. 10,

3, 2005

261


A. V. BORISOV, I. S. MAMAEV

and lo oks as follows: f = ctg ,
3

= const.

(4.9)

Remark. For Lobachevsky space L a similar reasoning yields f = cth , = const.

The vector p otential of the magnetic monop ole W can b e found from equation F In terms of spherical co ordinates ( , , ), it reads: W = 0, W = 0, W = R cos .

ij

= i Wj - j Wi .

In terms of variables q0 , q , it can b e written as W= while for B = rot W , we have 0, q q3 q q1 , - 1 2 2 2 2 |q | q2 + q3 |q | q2 + q B = - 3 q . |q | , (4.10)

2 3

(4.11)

Consider a particle, moving on S 3 in the field of a Newtonian center and in the field of a magnetic monop ole, the center and the monop ole b eing placed at the p ole = 0. This is a spherical analogue of the App ell problem. The Hamiltonian of the problem is either (4.3) or (4.5) with W (q ) and U (q ) defined, resp ectively, by (4.10) and (3.3). Hamiltonian equations always admit the integral of angular momentum q q =q âq - . (4.12) M =p âq - |q | |q | For simplification, we put R = 1 and write the Lagrangian in the spherical co ordinates (3.2) L = 1 2 + sin2 2 + sin2 sin2 2 then M2 = M3 = 0, = 0,
2

+ cos - U ( ).

(4.13)

Supp ose that the angular momentum vector (4.12) is collinear to the axis q 1 in the space q1 , q2 , q3 ; sin2 = cos = const. 0 (4.14)

The latter relation is a generalization of Kepler's second law. Indeed, if we cho ose a p oint on q the invariant surface in S 3 given by M , = const, this p oint b eing at the (angular) distance of 2 from the particle, then the great-circle arc, joining the origin of co ordinates with the chosen p oint sweeps equal areas in equal time intervals. (Indeed, the time rate of change of the area is
dS = dt
2

|q |

sin d
0

d = 2 sin2 .) dt

Taking into account the integral of energy E = h and (4.14) , we obtain = 2(h - Uc ( )), 2 tg2 Uc ( ) = U ( ) + 1 2 sin2
0

(4.15)

and the explicit expression for the tra jectory can b e found from equation d sin where h = h cos2 2 , U = U cos2 0 . 0


2

2(h - U ) -

2 sin2 sin2

= d ,
0

(4.16)

262

REGULAR AND CHAOTIC DYNAMICS, V. 10,

3, 2005


SUPERINTEGRABLE SYSTEMS ON A SPHERE

When = 0, corresp onding quadratures for the analogue of the Poincar´ problem are obtained e from (4.16). In this case the tra jectories are geo desics on the invariant cone defined by (M , q /|q |) = = const. As it was noted ab ove, in the case of the App ell problem analogue the tra jectories are conic sections (i. e. ellipses, hyp erb olas, parab olas) on the plane development of the cone. Generally, these are not closed on the cone, but there is a p ossibility to "adjust" the p otential (3.3) so that the tra jectories will b e always closed in the presence of a monop ole. In this way the Euclidean MICZ-mo del can b e generalized to the spherical case, for which the analogue Laplace ­ Runge ­ Lenz integral exists.

5. Generalized MICZ-mo del
Consider a motion on S 3 in the field of a monop ole and in the field with the p otential µ , U ( ) = - ctg + 1 2 sin2 The tra jectory in this case is defined by (4.16). If µ = 2 , the tra jectory is d sin2 2h + 2 ctg -
2

, µ = const.

(5.1)

(5.2)

= d .
sin2

(5.3)

The tra jectory (5.3) is closed, and the gnomonic pro jection gives us the conic section tg = p 1 + e cos( - 0 ) (5.4)

with the following fo cal parameter and eccentricity: 2 , cos2 0 22 cos2
2

p= For (4.15) we have

e=

1+

0

h-

2 2 cos2

.
0

2 = 2h + 2 ctg -

2 2 = 2h + 2 ctg - c 2 = f ( , c, h), 2 cos 0 · sin sin 2

(5.5)

where c = 2 /cos2 0 . Let us plot the bifurcational diagram of the problem's solutions on the parameter plane (c2 , h). To this end, recall that for the critical values (c , h ) lying on the bifurcation curves f (0 , c , h ) = f0 (0 , c , h ) = 0. As a result, we have two curves (Fig. 1): I. 2h = c2 - 2 ; c2 I I. c2 = 0. Thus, in the plane defined by the ab ove the line c2 = 4 and b elow h = 1 c2 , the particle moves only 2 half-spheres.
3, 2005


Besides, since c = 2 / cos2 0 , the inequality c2 > 4 also holds. constants h, c2 (see Fig. 1), the domain of allowable values h, c 2 lies the hyp erb ola defined by I. For a p oint of this plane ab ove the line in the upp er half-sphere. Otherwise, the particle can move in b oth

REGULAR AND CHAOTIC DYNAMICS, V. 10,

263


A. V. BORISOV, I. S. MAMAEV

Fig. 1

It is easy to formulate an analogue of Kepler's third law, coinciding with the traditional law for a curved space [13]. Indeed, since sin 2 = c, we have sin2 d . dt = c Therefore, for the p erio d of motion we obtain
2

T=1 c
0

p2 sin ( )d = c
2



-

d = 2 + (1 + e cos )2 p

h+

2 1 + h2 ·

1+ h

2 2

-1/2

.

(5.6)

This dep endence of the orbital p erio d on energy can b e easily transformed into the dep endence on (angular) length of the semi-ma jor axis T= where tg a = - .
h

- tg a +

1 + tg2 a(1 + tg2 a)

-1/2

,

(5.7)

As in the Kepler problem (on R and S 2 ), closedness of the tra jectories is closely connected with some hidden symmetry of the problem, i. e. with existence of a vector integral of the Laplace ­ Runge ­ Lenz typ e. For the system (5.1), (5.2) this vector can b e written as A = q0 p â M + R
2

q . |q |

(5.8)

The Poisson brackets for the comp onents of A and the comp onents of the integral of angular momentum (4.12) are: {Mi , Mj } = -
ij k

Mk ,

{Mi , Aj } = -
2

ij k

Ak , (5.9)

{Ai , Aj } = 2

ij k

R2 h - M 2 + 1 2

Mk .

As b efore, we can sp ecify the conditions (in terms of the curvature of the space and the value of the integral of energy), under which the symplectic leaf (4.2) is compact.

6. Tra jectory isomorphism for central p otential systems on S and R2

2

For U = U (r ), the equations (1.3) define a system with central p otential on R 3 . If U = U ( ) in the Lagrangian (3.4), then we have a system with central p otential on S 3 . These systems, resp ectively,


264

REGULAR AND CHAOTIC DYNAMICS, V. 10,

3, 2005


SUPERINTEGRABLE SYSTEMS ON A SPHERE

have flat (R2 ) and spherical (S 2 ) invariant manifolds. Using the central (gnomonic) pro jection (from the center of the sphere tangent to the plane at the attracting center) and some suitable change of time, we can transform these two-dimensional systems into each other. Following Serret and App ell [20, 2], consider a system in R 2 with the following equations of motion (in p olar co ordinates): d Tp = ; d Tp = R; (6.1) dt dt where Tp is the kinetic energy of a p oint in the plane, Tp = 1 2 + 2 2 , 2 while R, stand for certain generalized forces (generally, they can b e non-p otential). Let us p erform the transformation of co ordinates (the gnomonic pro jection from Fig. 2), forces and time: = tg , = ,
2

(6.2)

dt = cos
2

-2

d

R = cos , The result is a system on S 2 : d d where = Ts = ,

= cos .

(6.3)

d d

Ts

= ,

(6.4)

Fig. 2. The gnomonic pro jection

on the sphere,

d dTs , = , while Ts is the kinetic energy of a p oint d d

Ts = 1 + sin2 2 It is easy to see that

2

.

(6.5)

Statement. There exists a trajectory isomorphism between the Lagrangian system in R 2 , with central potential L = 1 (2 + 2 2 ) + U (), 2 and the Lagrangian system on S 2 , with central potential of the form L = 1 ( 2 + sin2 2 ) + U (tg ). 2 To prove that, it is sufficient to put in (6.1)­(6.5) = = 0, R = - U , = - U = - U · = R2 . cos

These transformations easily bring the plane Kepler problem to its analogue on sphere. Note that under the transformation (6.3) a p otential field of forces can b e transformed into a non-p otential field, and vice versa. If we consider the inverse transformation (6.3) as a transformation from sphere to plane, we have to admit negative values of . In case of negative the tra jectory on the sphere crosses its equator, while in the plane (, )the tra jectory jumps from + to -. If, instead of = tg , we consider = ctg , then the tra jectories in the plane ( , ) are continuous.


REGULAR AND CHAOTIC DYNAMICS, V. 10,

3, 2005

265


A. V. BORISOV, I. S. MAMAEV

It is also easy to show that the describ ed isomorphism can b e extended to the generalized p otential systems discussed in Sections 4, 5. Note also that the transformation (6.3), applied to the Kepler problem on S 2 , can b e used to generalize the Bohlin (Levi-Civita) regularization. It can b e shown that on a fixed energy level, the spherical Kepler problem is reduced to the harmonic oscil lator problem . Another interesting prop erty of the Kepler system in R 2 (due to Hamilton) is that the velo city ho dograph for a moving p oint is a circle with a displaced center. A similar vector can b e sp ecified for the Kepler problem on S 2 : x , = 1 + x 2 /R2 where x = ( cos , sin ) is the radius-vector of a p oint under the gnomonic pro jection (6.3). The tra jectory of can b e easily shown to b e a circle with a displaced center. This work was supp orted by RFBR (04-05-64367), CRDF (RU-M1-2583-MO-04), INTAS (0480-7297) and the program "State Supp ort for Leading Scientific Scho ols" (136.2003.1). The authors thank Bruno Cordani for a copy of his b o ok [7].

References
[1] P. Appel l. Annales scientifical da Academia Polutechtnica. do Porto. V. IV. 1909. See also: Appell P. Traut´ de m´ e ecanique rationnelle. Paris, Gauthier ­ Villars. 1909. [2] P. Appel l. Sur les lois de forces centrales faisant d´ ecrire a leur point d'application une conique quelles que ` soient les conditions initiales. American Journal of Mathematics. 1891. V. 13. P. 153­158 . [3] L. Bates. Symmetry preserving deformations of the Kepler problem. Rep. Math. Phys. 1988. V. 26. P. 413­428 . [4] A. V. Borisov, I. S. Mamaev. Poisson Structures and Lie Algebras in Hamiltonian Mechanics. Izhevsk: SPC "RCD". 1999. (In Russian) [5] N. A. Chernikov. The Kepler problem in the Lobachevsky space and its solution. Acta Phys. Polonica. 1992. V. 23. P. 115­119. [6] V. A. Cherno¨ ivan, I. S. Mamaev. The restricted twobody problem and the Kepler problem in the constant curvature spaces. Reg. & Chaot. Dyn. 1999. V. 4. 2. P. 112­124 . [7] B. Cordani. The Kepler Problem. Birkh¨ aser. 2003. [8] L. G. Feher. Dynamical O(4) symmetry in the asymptotic field of a Prasad ­ Sommerfield monopole. J. Phys. A. 1941. V. 19. P. 83­89. [9] V. A. Fok. Hydrogen atom and non-Euclidean geometry. Izv. AN SSSR, OMEN. 1935. V. 2. P. 169­179 . (In Russian) [10] Ya. I. Granovsky, A. S. Zhedanov, I. M. Lutsenko. Quadratic algebras and dynamics in curved space. I. Oscil lator . II. The Kepler problem. Theor. and Math. 2; 3. P. 207­216; 396­410 . (In Phys. 1992. V. 91. Russian)
¡ ¡

[11] G. Gy¨ gyi. or Kepler's equations, Fock variables, Bacry's generators and Dirac brackets. Nuovo Cimento. 1968. V. 53A. P. 717­735. [12] P. W. Higgs. Dynamical symmetries in a spherical geometry. I. J. Phys. A. 1979. V. 12. 3. P. 309­323 . [13] W. Kil ling. Die Mechanik in den Nicht-Euklidischen Raumformen. J. Reine Angew. Math. 1885. V. 98. P. 1­48. [14] V. V. Kozlov. On dynamics in constant curvature 2. spaces. Vestnik MGU, ser. math. mech. 1994. P. 28­35. (In Russian) [15] V. V. Kozlov, A. O. Harin. Kepler's problem in constant curvature spaces. Celestial Mech. and Dynamical Astronomy. 1992. V. 54. P. 393­399 . [16] P. G. L. Leach, G. P. Fleassas. Generalizations of the Laplace ­ Runge ­ Lenz Vector. Journal of Nonlinear Mathematical Physics. 2003. V. 10. 3. P. 340­423 . ¨ [17] H. Liebmann. Uber die Zantalbewegung in der nichteuklidiche Geometrie. Leipzig Ber. 1903. V. 55. P. 146­153. [18] H. McIntosh, A. Cisneros. Degeneracy in the presence of a magnetic monopole. J. Math. Phys. 1970. V. 11. P. 896­916. [19] J. Moser. Regularization of Kepler's problem and the averaging method on a manifold. Comm. Pure Appl. Math. 1970. V. 23. P. 609­636 . [20] P. Serret. Th´ eorie nouvelle g´ ´ eometrique et m´ ecanique des lignes a double courbure. Paris: Librave de Mallet ­ Bachelier. 1860. [21] D. Zwanziger. Exactly soluble nonrelativistic model of particles with both electric and magnetic charges. Phys. Rev. 1968. V. 176. P. 1480­1488 .
¡ ¡ ¡



266

REGULAR AND CHAOTIC DYNAMICS, V. 10,

3, 2005