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ISSN 1560-3547, Regular and Chaotic Dynamics, 2014, Vol. 19, No. 3, pp. 415­434. c Pleiades Publishing, Ltd., 2014.

Sup erintegrable Generalizations of the Kepler and Ho ok Problems
Ivan A. Bizyaev1* , Alexey V. Borisov1,
2, 3 **

, and Ivan S. Mamaev1,

4 ***

1 Udmurt State University, Universitetskaya 1, Izhevsk, 426034 Russia 2 A. A. Blagonravov Mechanical Engineering Research Institute of RAS, Bardina str. 4, Moscow, 117334, Russia 3 National Research Nuclear University "MEPhI", Kashirskoye shosse 31, Moscow, 115409, Russia 4 Institute of Mathematics and Mechanics of the Ural Branch of RAS, S. Kovalevskaja str. 16, Ekaterinburg, 620990, Russia

Received March 27, 2014; accepted May 13, 2014

Abstract--In this paper we consider superintegrable systems which are an immediate generalization of the Kepler and Ho ok problems, both in two-dimensional spaces -- the plane R2 and the sphere S 2 -- and in three-dimensional spaces R3 and S 3 . Using the central pro jection and the reduction pro cedure proposed in [21], we show an interrelation between the superintegrable systems found previously and show new ones. In all cases the superintegrals are presented in explicit form. MSC2010 numbers: 70H06, 70G10, 37J35 DOI: 10.1134/S1560354714030095 Keywords: superintegrable systems, Kepler and Ho ok problems, isomorphism, central pro jection, reduction, highest degree polynomial superintegrals

Contents
INTRODUCTION 1 A MATERIAL POINT ON THE PLANE R 1.1 The Motion of a Point on a Plane in a Potential Field Satisfying the Bertrand Condition 1.2 The Bertrand Method for Finding Quadratic Integrals 1.3 Systems with a Redundant Set of Quadratic Integrals 1.4 Systems with a Polynomial Superintegral of the Highest Degrees
2

416 418 418 419 420 421 424 424 426 427 427 429
3

2

CENTRAL PROJECTION AND SUPERINTEGRABLE SYSTEMS ON S 2.1 The Tra jectory Isomorphism of Systems on a Plane and a Sphere 2.2 Superintegrable Systems on S 2 REDUCTION AND SUPERINTEGRABLE SYSTEMS IN R3 3.1 Reduction of the System in R3 to a System on a Sphere 3.2 Superintegrable Potentials in R3 CENTRAL PROJECTION AND SUPERINTEGRABLE SYSTEMS IN S
*

2

3

4
** ***

430

E-mail: bizaev 90@mail.ru E-mail: borisov@rcd.ru E-mail: mamaev@rcd.ru

415


416 5 CONCLUSION ACKNOWLEDGMENTS REFERENCES

BIZYAEV et al. 432 432 432

INTRODUCTION This pap er is a combination of a review and research work, in which we try to systematize the well-known methods of proving the equivalence of sup erintegrable systems on a plane and a sphere. This is necessary due to an increasing numb er of works in this area, in each of which new sup erintegrable systems are found. In reality, most of them can b e obtained by fairly simple transformations for the well-known systems. Here we do not intend to cover all well-known results in this direction, we only show a p ossible way to systematize them. 1. In this pap er we consider sup erintegrable systems which are an immediate generalization of the well-known Kepler and Hook problems, b oth in two-dimensional spaces -- the plane R2 and the sphere S 2 -- and in three-dimensional spaces -- R3 and S 3 . As b efore, these systems p ossess a maximal p ossible redundant set of integrals, however, the system tra jectories themselves are no longer curves of the second order. Recent interest in the sup erintegrable systems of classical mechanics has focused on their quantum-mechanical analogs. A detailed discussion of results in this vein and an extensive list of references can b e found in the review [51]. In this connection we note a somewhat one-sided, basically algebraic, approach to sup erintegrability, develop ed in [51] (and other works of these authors), which is essentially based on the Poisson structure of the systems under consideration. Obviously, this is the reason why the list of references given by the authors of the review does not include interesting results on the dynamics and geometry of sup erintegrable systems. In the classical p eriod associated with the names of J. Darb oux, G. Halphen, and J. Bertrand, sup erintegrable systems were studied as systems with closed tra jectories. Despite a large b ody of literature in this area, interrelations b etween results are, as a rule, not established. As a result, the same systems have b een presented several times as new ones. Using different analytical methods, the authors often derive a very cumb ersome (or implicit) form of integrals, although in one of J. Bertrand's early works these integrals were found in a much more elegant and general form. In this pap er we supplement the review [51], raising geometrical questions of sup erintegrable systems and showing interrelations b etween them. First of all (see Section 1) we dwell on the classical work of J. Bertrand (1857) [12] (see also the review pap er of V. A. Steklov [62]) analyzing the conditions for the existence of p olynomial (in velocities) integrals of motion of the system of equations x = X (x, y ), ¨ y = Y (x, y ). ¨ (0.1)

The most completed results were obtained by J. Bertrand for the case of quadratic integrals. Afterwards they were re-discovered many times, see, e.g., [9, 10, 58]. Their application to the case of p otential forces most naturally leads to sup erintegrable systems with a set of quadratic integrals. A slight modification of the p otentials obtained in this way leads to generalizations, when one of the integrals (sup erintegral) turns out to b e p olynomial, but its degree can turn out arbitrarily large. Developing a previously prop osed method, we present here a sup erintegral in the most natural form. Apparently this method of algebraization of an integral of a high degree, which corresp onds to a constant phase displacement in the separation of variables, was first prop osed in [21] for the proof of sup erintegrability of the system of n equal Hookean centers on a sphere, located on the equator. We note that this approach does not require the presence of a Poisson structure, and these results can b e generalized to the case of a priori non-Hamiltonian (for example, nonholonomic) systems. 2. One of the natural generalizations of the system (0.1) are natural Hamiltonian systems of the form H = H0 (p, q )+ U (q ),
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where p, q are canonically conjugate coordinates and momenta, and H0 (p, q ) is a Hamiltonian of the geodesic flow of some (Riemannian) metric. Any integral of motion that is p olynomial in degree and momenta can b e represented as F = P0 (p, q )+ P
(n) (n) (n-1)

(p, q ),

where P0 (p, q ) is a homogeneous p olynomial of degree n (terms of high degrees), and (n) P (n-1) (p, q ) are the remaining terms of a lower degree. It is well known that P0 (p, q ) is an integral of the geodesic flow with Hamiltonian H0 (p, q ), which implies that superintegrable systems (0.2) can arise only in the case of a superintegrable metric. Such metrics (different from the axisymmetric ones) in the case of a quadratic set of first integrals are the sub ject of the work of G. Koenigs [47], where nontrivial metrics are presented; the corresp onding sup erintegrable generalizations of the form (0.2) are discussed in [42, 43]. Here we consider the simplest case of a sup erintegrable metric (which is different from a flat metric), namely, the metric of the sphere S 2 and S 3 . As is well known, the metrics of the surfaces of constant curvature are geodesically equivalent to flat metrics, which also determines the interrelation b etween the corresp onding p otential sup erintegrable systems (see Section 2). Transformations establishing b oth the tra jectory isomorphism b etween the systems on S n and Rn and the tra jectory isomorphism in the case of different systems of the form (0.1) on Rn were p ointed out by P. App el [2, 5]. These transformations are natural pro jective symmetries of such systems, therefore, in the search for and investigation of integrable and sup erintegrable cases, they should b e regarded as nonequivalent only if they are not related to each other by any of these transformations. In particular, the pap er [28] shows the tra jectory isomorphism of integrable systems describing the motion of a particle in the field of two fixed centers on the plane R2 and the sphere S 2 . Remark. The transformations Rn mapping the geodesics into geodesics form a pro jective group whose prop erties were describ ed in detail by S. Lie [49, Chapter 26]. 3. Analogs of the Kepler and Hook laws on S 2 , S 3 (and their noncompact analogs -- the plane and the Lobachevsky space H 2 , H 3 ) were presented in the works of P. Serret, N. I. Lobachevsky, E. Schr¨ inger, W. Killing and were studied from different standp oints in many present-day works. od A more detailed historical comment can b e found in [1, 26] and in the monograph [23] (which unfortunately has not b een translated into English) and the collected pap ers [24], where all necessary historical sources are given in detail. Sup erintegrable systems on S n , which consist of Hookean centers (or the Rosochatius term [59]) were studied in [48, 52]. We note that the terms analogous to the Hookean centers arise when a reduction in linear integrals in rigid b ody dynamics is p erformed (Goryachev's term) [25]. The most general case of arrangement of Hookean centers on S 2 , which leads to a sup erintegrable system was found in [20] (2009). As shown in [21], in this case an additional sup erintegral has an arbitrary degree in momenta. A year earlier a system on the plane R2 was investigated in [32] with a p otential that in the p olar coordinates r , has the form , = const. (0.3) U= 2 2 r sin A sup erintegral was found for = 5 and a hyp othesis of sup erintegrability for an arbitrary N was advanced. It turns out that the systems [20] and [32] are connected (see Section 2), so that actually the problem of the sup erintegral > 5 has b een solved in [21]. In [63] (2009) the closedness of the system tra jectories on the plane R2 was shown with a generalization of the p otential (0.3) of the form: b 1 a + , Q. V = 2r2 + 2 2 2 r sin cos An explicit form of the sup erintegral for this system was found in [35]. For Kepler's analog of this p otential, b 1 a k1 + +2 , V= 2 2 r r sin cos the sup erintegrability was shown in [56].
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Remark. It is interesting that the authors pap ers [63, 64] as a "dramatic change of the see that in these works, as opp osed to [21, 32], of sup erintegrable p otentials and no integrals ignored by the authors).

of the review [51] assess the contribution of the situation in sup erintegrable systems", whereas we only an additional parameter is added to the family are found in explicit form (the pap er [32] is fully

We also note an interesting system in R3 , which was found in 2008 [65] and which generalizes the Kepler problem: a2 a3 a1 V = + 2 + 2 + 2. r x y z Below (see Sections 3, 4) this system is also studied from different sides using various transformations b etween R3 and S 3 , such as App el's transformation, and the reduction procedure (and the inverse reconstruction procedure) b etween R3 and S 2 , which is always applicable to p otentials having the homogeneity degree (-2) and which is describ ed in detail in our review [20]. We p oint to the interrelation b etween the previously found sup erintegrable systems, show new sup erintegrable systems and present the corresp onding integrals in explicit form. Remark. Sup erintegrable systems on R2 and S 2 are closely related to the problem of closedness of the geodesics on two-dimensional compact surfaces. This issue was discussed in the b ooks [13, 18]. One of the results of Guillemin [36] states that there are very many "closed" metrics on the sphere. But this is only an existence theorem and does not show explicit integrals, which are evidently not p olynomial in velocities. We also mention the pap er of Kiyohara [46], who p ointed out a metric on the sphere with an additional (only one) integral having any degree in velocities. In this case, it follows from [46] that there exists another (not necessarily p olynomial in velocities) integral and all geodesics turns out to b e closed. However, all these results, which are top ological in nature, have little to do with the sub ject matter of this pap er, since we explicitly find additional integrals and the systems considered by us have singularities, in contrast to the geodesic flows. Remark. In addition to the studies on the motion of a p oint in Euclidean space and in spaces of constant curvature, one should mention the works on the sup erintegrability of various modifications of the Toda lattices [4] and the Moser ­ Calogero lattices [32, 33] (which are not discussed here). 1. A MATERIAL POINT ON THE PLANE R Consider the dynamical system x = X (x, y ), ¨ y = Y (x, y ), ¨ (1.1) which describ es the motion of a material p oint on a plane, under the action of forces dep ending only on the coordinates. One of the b est-known cases when this system admits a quadratic integral is the case of a p otential external force: X (x, y ) = - U (x, y ) , x Y (x, y ) = - U (x, y ) . y (1.2)
2

1.1. The Motion of a Point on a Plane in a Potential Field Satisfying the Bertrand Condition

As is well known, the energy of the system is preserved in this case: E= 12 ( + y 2 )+ U (x, y ), x 2

Another well-known quadratic integral that exists under sufficiently general assumptions is the Bertrand integral [12] FB = 1 x (y x - xy )2 + . 2 y
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Furthermore, it is necessary that the combination yX - xY b e a homogeneous function of degree (-2): yX - xY = 1 x2 x . y (1.4)

As we can see from (1.2) and (1.4), the conditions for the integrals E and FB to exist are not the same, therefore, in the system (1.1) these integrals may arise separately. The Bertrand integral (1.3) was rediscovered later many times for various particular cases of the systems (1.1) (see, e.g., [9, 10, 58]). We shall assume that the right-hand side of Eqs. (1.1) satisfies b oth the conditions (1.2) and the conditions (1.4). In this case it is convenient to pass to the p olar coordinates r , : x = r cos , H , pr y = r sin , p H = , 2 r p (1.5) and to rewrite the equations of motion in Hamiltonian form r = pr = pr = =

p2 H H U U =- , p = - =- , - 3 r r r 1 2 p2 pr + 2 + U (r, ), H= 2 r

where the Hamiltonian H is an energy integral expressed in terms of canonical momenta. From the condition (1.4) we find that U (r, ) = v (r )+ u() , r2 (1.6)

where v (r ), u() are arbitrary functions and the Bertrand integral (1.3) can b e written as FB = 12 p + u(). 2 (1.7) the canonical Poisson bracket, i.e., able by the Liouville ­ Arnold theorem the form (1.6) were used in celestial theory, see [21] for a detailed historical

The integrals H and FB are in involution relative to {H, FB } = 0, and the system (1.1) turns out to b e also integr (in a compact or noncompact version) [7, 8]. Potentials of mechanics to explain observations that do not fit the Newton comment. In what follows we consider the following problems:

-- for the system (1.5) find potentials of the form (1.6) at which there exists another additional (i.e., independent with H and FB ) integral of motion that is quadratic in velocities (superintegral); -- generalize the resulting potentials for the case where the superintegral is of a higher degree (than the second). 1.2. The Bertrand Method for Finding Quadratic Integrals Before we proceed to analysis of the system (1.5), we briefly review the results of Bertrand [12]. The quadratic integral of Eqs. (1.1) of the most general form is represented as F
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(2)

= F0

(2)

+ K (x, y ),
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where F0 is a homogeneous quadratic integral of the geodesic flow on a plane, which is a combination of three linear integrals x = const, y = const, y x - xy = const F0
(2)

= P (x, y ) 2 + Q(x, y ) y + R(x, y ) x x y

2

= a(y x - xy )+ (b1 x + b2 y)(y x - xy )+ c1 x2 + c2 y 2 + c3 xy, P = ay 2 + b1 y + c1 , R = ax2 - b2 x + c2 , Q = -2axy - b1 x + b2 y + c3 .

(1.9)

Remark. The leading terms in any system of such form define the integral of the corresp onding geodesic flow. In (1.8) there are no terms linear in velocities, since they in themselves form a linear integral of motion. The function K (x, y ) in the integral (1.8) must satisfy the relations: 2PX + QY + K = 0, x 2RY + QX + K = 0. y (1.10)

It is convenient to formulate the relations (1.10) in terms of differential forms: Theorem 1 (Bertrand [12]). The system of equations (1.1) possesses a quadratic integral if and only if there exist quadratic polynomials P , Q, R, given by the relations (1.9), such that the 1-form = (2PX + QY )dx +(2RY + QX )dy is exact. In the examples considered b elow, it is generally sufficient to verify the condition of closedness of the form (1.11) (2RY + QX ) - (2PX + QY ) = 0. x y (1.12) (1.11)

If the functions X (x, y ), Y (x, y ) are defined, this relation reduces to a system of linear equations for the coefficients of the p olynomials P , Q, R. The numb er of free parameters in its solution coincides with the numb er of indep endent quadratic integrals (in particular, it is p ossible that the system admits no integral). If we fix a certain form of p olynomials P , Q, R, then the relation (1.12) reduces to a system of linear partial differential equations of first order with unknown functions X (x, y ), Y (x, y ). We also note that cases are p ossible when the closedness condition (1.12) turns out to b e insufficient for the existence of single-valued integrals [3]. Thus, we see that the Poisson structure plays no role for finding quadratic integrals (and for the analysis of sup erintegrability in general) of the system (1.1). Remark. When the force (X , Y ) is p otential, the relation (1.12) is often called the Bertrand ­ Darb oux equation, see, e.g., [51].

1.3. Systems with a Redundant Set of Quadratic Integrals Now, using the Bertrand theorem, we consider, for the system (1.5) with p otential (1.6), the first problem formulated ab ove (see Section 2.1): find the functions u(), v (r ) for which there exists an additional quadratic integral independent with H and FB .
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Using freedom in the choice of the origin of the angle and the conditions of indep endence with H and FB , the sup erintegral of the system according to (1.8), (1.9) can b e represented as x Fs = (b1 x + b2 y)(y x - xy )+ c( 2 - y 2 )+ K (x, y ). Rewriting for this integral the 1-form (1.11) in p olar coordinates, we obtain = 2c cos 2(U d - Ur dr )+ 2rb0 cos( + 0 )U d U dr + rUr d , +(b0 r sin( + 0 )+ c sin 2) r b2 b0 = b2 + b2 , 0 = arctg , 1 2 b1 (1.13)

(1.14)

where we use the notation Ur = U , U = U . r The condition of closedness of this form takes the form: b0 sin( + 0 )G1 +2c sin 2G2 = 0, 1 G1 = -rv (r ) - 2v (r )+ 2 (u ()+3 ctg( + 0 )u () - 2u()), (1.15) r 1 G2 = v (r ) - rv (r )+ 3 (u ()+6 tg 2u () - 8u()). r The unknowns are here the constants b0 , 0 , c and the functions v (r ), u(). This system has the following nontrivial solutions: 1) c = 0, v (r ) =
k1 r

, u() =

k2 +k3 cos(+0 ) sin2 (+0 )

, k1 ,k2 ,k3 = const, and the integral has the form p2 + 2(k2 + k3 cos( + 0 )) k3 + k1 r + ; 2 r sin ( + 0 )

s = sin( + 0 )pr p + 2) b0 = 0, v (r ) = k1 r 2 , u() = s = sin 2

cos( + 0 ) r

k2 +k3 cos 2 sin2 2

, k1 ,k2 ,k3 = const, and
2

p2 pr p 1 k2 + k3 cos 2 + k1 r - cos 2 p2 - 2 - r r 2 2r r 2 sin2 2

+

k3 . r2

Remark. Here and b elow the index s is used to denote the sup erintegral. We note that by an appropriate change of the coordinate one can always ensure that in the first solution 0 = 0, Hence, in what follows we set 0 = 0. Thus, we obtain the following assertion. Prop osition 1. The Hamiltonian system (1.5) in the potential fields: U1 (r, ) = k1 k2 + k3 cos , + r r 2 sin2 U2 (r, ) = k1 r 2 + k2 + k3 cos 2 r 2 sin2 2

admits a redundant set of quadratic integrals. 1.4. Systems with a Polynomial Sup er We now consider p ossible generalizations of the case where the additional integral Fs is p olynomial we rescale the time [20] 2dt = r 2 d in the system equations of motion on the level set of first integrals r 2 pr dr = =r d 2
2

integral of the Highest Degrees sup erintegrable systems found ab ove to the in velocities of a higher degree. To this end, (1.5) with p otential (1.6) and represent the H = h and FB = f in the form p d = = d 2 f - u(). (1.16)

h-

f - v (r ), r2

If this system admits a (single-valued in ) integral for any values of the constants h, f , then the initial system (1.5) p ossesses a redundant (sup erintegrable) set of integrals.
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Choose a natural generalization of the functions u() in the form: k2 + k3 cos , u() = sin2

(1.17)

where is an arbitrary constant, which we define b elow. The first of Eqs. in (1.16) can b e integrated as follows cos + k3 = 2f 1+
2 k3 k2 sin( - 2 4f f

f + c1 ),

(1.18)

where c1 is the integration constant. If we take the Newtonian p otential v (r ) = we obtain 1 k1 + =- r 2f

k1 r

for the radial comp onent v (r ), then from (1.16)

k2 h + 12 sin( f + c2 ), f 2f

(1.19)

and if we take the Hookean p otential v (r ) = k1 r 2 , we find 2f (1.20) h - 2 = - h2 - 4fk1 sin(2 f + c2 ). r From these relations we can see that the "frequencies" of motions in each of the variables r and turn out to b e commensurate if one assumes that is a rational numb er. (The frequencies have b een put in quotes since the tra jectories of the system (1.16) turn out to b e b ounded not for all parameter values). In this case the system (1.5) p ossesses another integral of motion that is p olynomial in the momenta and single-valued relative to the variables r , sin , cos . We note that for that is not an integer this integral has no physical interpretation (at least within the framework of classical mechanics), since the p otential U (r, ) is no single-valued function on a plane. In order to construct an integral explicitly for arbitrary = m , where m, n N , we differentin ate (1.18) with resp ect to and, using (1.16), we obtain the relation p sin = - 2f 1+
2 k3 k2 cos( - 2 4f f

f + c1 ).
i( f +c1 )

Hence, appropriately redefining the constant c1 c1 , we obtain 2fp sin + i(2f cos + k3 ) = e- Similarly, from (1.19) and (1.20) we find that for v (r ) = 2fpr - i and for v (r ) = k1 r
2 k1 r

.

2f + k1 r

= ei(

f +c2 )

,

2f 2f pr + i h - 2 r r

= e-

i(2 f +c2 )

.

From these relations, eliminating time , we obtain the following result. Prop osition 2. If we choose as v (r ) either Hookean k1 r 2 or Newtonian kr1 potential and = m , n the system (1.16) admits a complex-valued integral (single-valued on the n-sheeted covering of the plane)
(m,n) s

=

2fpr - i

2f + k1 r
m

m

n

2fp sin + i(2f cos + k3 )
2n

, ,

for for

v=

k1 , r

(m,n) s

=

2f 2f pr + i h - 2 r r

2fp sin + i(2f cos + k3 )
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Remark. The multivaluedness of these integrals, which are p olynomial in pr , p , arises due to the presence of the functions cos , sin . To and f In ord which express these integrals in terms of the initial variables, it is necessary to substitute for h the expressions for the Hamiltonian H (1.5) and the Bertrand integral (1.7), resp ectively. er to obtain real p olynomial integrals from these relations, we note that the p olynomials of (m,n) consists can b e represented as the sup erintegral s P (a, b, f )+ i fQ(a, b, f ), n - even, ( fa + ib)n = fP (a, b, f )+ iQ(a, b, f ), n - odd, d Q are real p olynomials in their arguments. Since four first integrals are dep endent in is sufficient to take one of these p olynomials: P or Q. to represent the p otential in the form presented in [53, 63], it is necessary to rewrite by (1.17), as u() = k2 + k3 4sin
2 2

where P an this case, it In order u(), given

+

k2 - k3 4cos
2 2

.

(1.21)

The ab ove systems may b e regarded as sup erintegrable generalizations of the Kepler and Hook problems. Another well-known sup erintegrable system is the anisotropic oscillator with commensurate frequencies. In Cartesian coordinates its p otential can b e written as V0 (x, y ) = m2 x2 + n2 y 2 , m, n N .

The p otential admits two generalizations [40] which are analogous to the previous ones. In the former case k1 k2 V (x, y ) = m2 x2 + n2 y 2 + 2 + 2 , m, n N, x y and the quadratic integrals have the form 1 k1 F1 = p2 + m2 x2 + 2 , x 2 x 1 k2 F2 = p2 + n2 y 2 + 2 . y 2 y 1 2 k2 p+ - n2 y 2 y y2
n 2

The sup erintegral is also represented in complex form: m 1 2 k1 (m,n) 22 p+ = 2mxpx + i -m x 2ny py - i s 2 x x2 The second system has the p otential V (x, y ) = m2 x2 + n2 y 2 k2 + k1 x + 2 , 4 y

.

m, n N .

The quadratic integrals and the sup erintegral are represented as F1 =
(m,n) s

12 p + m2 x2 + k1 x, 2x
2 n

=



2mpx + i(k1 +2m x)

1 2 n2 2 k2 p + y + 2. 2y 4 y 1 2 k2 n2 n ypy + i p+ -y 2 y y2 4 2 F2 =

m 2

.

Remark. Another sup erintegrable system with a sup erintegral that is p olynomial in momenta (of degree 2k + 2) is found in [31]. This system is a generalization of one of Drach systems: H = px py - k1 y 2+1 y - k2 +2 . x2+3 x x2 + y 2 with homogeneity degree (-2).

As can b e seen, its p otential field is homogeneous in r =
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2. CENTRAL PROJECTION AND SUPERINTEGRABLE SYSTEMS ON S 2.1. The Tra jectory Isomorphism of Systems on a Plane and a Sphere

2

As noted ab ove (see the Introduction), one of the central problems in the theory of integrable systems is that of equivalence of various integrable cases. There are examples where the isomorphism is so nontrivial that it can b e found only after a detailed study of the corresp onding systems [16, 17, 28, 29, 61]. On the other hand, there are, as a rule, some standard transformations (symmetries). The integrable and sup erintegrable systems are regarded as equivalent if they can b e mapp ed using these transformations. In this section we consider the central (gnomonic) pro jection of the twodimensional sphere S 2 onto the plane R2 (see Fig. 1), which relates the Kepler and Hook problems in the flat space and their sup erintegrable analogs in the space of constant curvature [6, 60]. We recall that in this case the geodesics on the sphere and the plane are mapp ed into each other, therefore (after appropriately rescaling the time) the problem of motion of a material p oint on the plane R2 in the field of forces dep ending on the coordinates, see (1.1), b ecomes an analogous problem on the sphere [2, 26]. We consider the unit sphere S 2 , which has b een emb edded in a standard way in the threedimensional Euclidean space R3 = {x, y , z }: x2 + y 2 + z 2 = 1. The plane R2 will b e given by the equation z = 1 (Fig. 1). If we define the spherical coordinates (, ) on S 2 and the two-dimensional coordinates on the plane R2 , then the central pro jection is given by the relation r = tg . (2.1) The relation of this transformation to the particle dynamics on a plane and a sphere is describ ed by the following natural assertion. Theorem 2 (App el [6]). Suppose we are given a system describing the motion of a particle on a plane in a field of forces depending only on the coordinates T (p r
) .

T (p) ( = Fr p) (r, ), - r
(p)

T (p

)

. ( = Fp) (r, )

(2.2)

1 r T = ( 2 + r 2 2 ). 2 Then the transformation (2.1) and the change of time d = cos2 dt lead to the system on the sphere T (s
) .

T (s) Fr (tg , ) (s) = F (, ) = , - cos2 T
(s)

(p)

T (s =

)

. ( = Fs) (r, ) =

F (tg , ) cos2 (2.3)

(p)

=

1 (( )2 +sin2 ( )2 ), 2

d , d

=

d , d

which describes the motion of a material point on S 2 , in a field of forces which also depend only on the coordinates. The systems (2.2) and (2.3) are trajectory isomorphic. Remark. This theorem can b e naturally generalized to the multidimensional case. If we assume that the forces acting on the material p oint are p otential, i.e.,
( Fr p) =

U (p) , r

( Fp) =

U (p) ,

F

(s)

=

U (s) ,

( Fs) =

U (s) ,
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SUPERINTEGRABLE GENERALIZATIONS OF THE KEPLER AND HOOK PROBLEMS 425

where U (p) and U (s) are the p otentials on the plane and the sphere, resp ectively, then, as is seen from (2.3), this transformation does not necessarily imply that a p otential force on the sphere will arise from the p otential force on the plane: for this to happ en, the following equality must hold: U (s) U (p) = , 1 U (p) U (s) = , cos2 U
(p)

=U

(p)

(r, ) |r

=tg

.

(2.4)

If we take as p otential U (p) (r, ) on the plane (1.6), then it turns out that the system (2.4) admits a solution for the p otential U (s) (, ) on a sphere of the following form: U
(s)

(, ) = v (tg )+

u() . sin2

(2.5) on a sphere of pro jective For example, and a sphere

Remark. We recall that the most general into the geodesics on a plane consist of the transformations [5]. The p otentials of a more in [28] the tra jectory isomorphism of the prob is shown.

transformations mapping the geodesics central pro jection (2.1) and the group general form are connected in this case. lem of two Newtonian centers on a plane

z R
2

(r, j)
2

S

(q, j) q x, y

Fig. 1. Central pro jection.

It follows that the system on the plane (1.5) with p otential (1.6) is tra jectory isomorphic to the system on the sphere, which is written as: p H H = , = , = p = 2 p p sin H H cos 2 U U p = - =- , 3 p - = - , sin u() 1 p2 2 U (, ) = v (tg )+ . + 2 + U (, ), 2 sin sin2

p = 1 H= p 2

(2.6)

In addition to the energy integral, this system admits a quadratic integral (an analog of the Bertrand integral) for any functions of v and u: FB = and hence is integrable.
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(2.7)


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2.2. Sup erintegrable Systems on S

2

Based on the App el theorem, one can find sup erintegrable systems on a sphere that are equivalent to those presented ab ove in the case of a plane. Indeed, for the system (2.6) we choose a p otential in accordance with (2.5) in the form U
(s)

(, ) = v ( )+

k2 + k3 cos . sin2 sin2

(2.8)

By analogy with the case of a plane (Section 1.4) we obtain the following result. Prop osition 3. Let v ( ) = k1 ctg or v ( ) = k1 tg2 and = m , where m, n N, then the n system (2.6) admits an additional integral independent with H and FB :
(m,n) s (m,n) s

=

2fp - i(2f ctg + k1 )

m

n

2fp sin + i(2f cos + k3 )
m

, for v = k1 ctg ,
2n

=

2fp ctg + i h- ctg2 f -

f sin2

2fp sin +i(2f cos +k3 )

, for v = k1 tg2 . (2.9)

Remark. The sup erintegral

(m,n) s

with v ( ) = k1 tg2 has b een found in the recent work [55].

Remark. The authors of [30] show the sup erintegrability of the system on S 2 in a p otential field of the form U (, ) = 1 sin2 k1 k2 + sin2 cos2 , (2.10)

which, as (1.21) implies, is a particular case of the p otential (2.8). In the same work a similarity of the tra jectories of the system (2.10) with the Lissa jous figures is revealed. We note that in this case only the case n = 1 has a physical meaning: the p otential (2.8) and (m,1) are single-valued functions on S 2 . Moreover, since the sphere is compact, all the integral s tra jectories are closed for these p otentials. As shown in [21], the p otential (2.8) turns out to b e related to the problem of particle motion on a sphere in the field = 2n + 1 of equal Hookean centers located at the vertices of a regular k-gon inscrib ed into one of the large circles. To show this, we use the generalization of the p otential of elastic interaction on a sphere: U
(s)

( ) = k1 tg2 .

It may b e assumed that the center of attraction is located at the p ole of the sphere. If one places k Hookean centers on the equator, then the p otential can b e represented up to a constant in the form: 1 sin2
k i=1

1 . sin ( + i )
2

In the case where the centers are located at the vertices of a regular -gon: i = 2i , i = 1,... ,

using the well-known trigonometric identity, see [38, 54, p. 645]:
i=1

2 1 = , sin2 ( + i ) sin2
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SUPERINTEGRABLE GENERALIZATIONS OF THE KEPLER AND HOOK PROBLEMS 427

we find U
(s)

( ) =

k1 2 , sin2 sin2

which is seen to b e a particular case (2.8). The typical tra jectories on S 2 for the generalization of the Kepler and Hook problems are shown in Fig. 2. We note an interesting observation (which has, as yet, no theoretical explanation): if = n N, then the numb er of p oints of self-intersection of the generic tra jectory for the generalization of the Kepler problem is equal to n - 1 (see Fig. 2). In the case of generalization of the Hook problem the numb er of p oints of self-intersection dep ends on the evenness (see Fig. 3). 3. REDUCTION AND SUPERINTEGRABLE SYSTEMS IN R
3

In this section we show one of the p ossible ways of generalization of the previously found sup erintegrable systems to the three-dimensional space. In principle, to construct a sup erintegrable system, one can take some integrable (in particular, Stackel) system and try to vary arbitrary ¨ functions app earing as parameters so that the tra jectories on the tori are closed. As a rule, this is not so easy to achieve in real examples, therefore, we consider here a construction allowing one to construct from the given sup erintegrable system a sup erintegrable system of a larger dimension. To this end, we consider in more detail a procedure of reconstruction for reduction that was prop osed in [20]. 3.1. Reduction of the System in R3 to a System on a Sphere Introduce on R3 the spherical coordinate system: x = r sin sin , y = r sin cos , z = r cos , the Hamiltonian of the particle moving in a p otential field has the form 1 p2 1 p2 1 + V (r, , ). (3.1) + H = p2 + r 2 2 sin2 2 2r 2r As in the previous sections, we shall assume that the p otential field which is a sup erp osition of the central field and the field homogeneous in r , with homogeneity degree (-2), takes the form V (r, , ) = w(r )+ U (, ) . r2 (3.2)

In this case, as on the plane, there is an analog of the Bertrand integral (1.7): G= 1 2 p2 + p
2 2

sin

+ U (, ).

(3.3)

The integral (3.3) for v (r ) = 0 was found in [2], and the integral in the field (3.2) was found in [20]. Prop osition 4. In the Hamiltonian system (3.1) with potential (3.2) after rescaling the time: r
-2

dt = d

a system of equations decouples which defines the evolution of the variables p , p , , and coincides with the system (2.6) governing the motion of a material point on S 2 , with the Hamiltonian H = G. Thus, for the Hamiltonian system (3.1) with p otential (3.2) to b e Liouville integrable, we need one additional integral. In contrast to the previous systems with two degrees of freedom, the numb er of integrals in the redundant set can b e different in this case and is either four or five. We consider only maximal ly superintegrable systems, which p ossess five first integrals.
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428
for different and the initial angles (0), (0); but constant

REGULAR AND CHAOTIC DYNAMICS
k2 sin2 sin2 k2 sin2 sin2

Fig. 2. Typical tra jectories of the system on S 2 in a p otential field: U = k1 ctg + k1 = 1, k2 = 1 and the initial momenta p (0) = 6, p (0) = 5 2

BIZYAEV et al.

Vol. 19 for different and the initial angles (0), (0); but constant

No. 3

2014

Fig. 3. Typical tra jectories of the system on S 2 in a p otential field: V = k1 tg + k1 = 1, k2 = 1 and the initial momenta p (0) = 6, p (0) = 5 2


SUPERINTEGRABLE GENERALIZATIONS OF THE KEPLER AND HOOK PROBLEMS 429

Remark. In [11, 57] a sup erintegrable p otential on S2 is presented. In the spherical coordinates ~~ (, ) this p otential has the from U
(s)

sin2 ~~ (, ) = 1 - sin

~ ~ cos2 4 sin2 cos2 ~ + 2~ ~ ~ ~ cos2 (1 - sin2 cos2 )(cos2

~ sin2 ~ . - ~ - sin2 sin2 )2 sin2 cos2 ~ ~ ~ ~

This p otential is a particular case of the p otential (2.8) considered earlier. To show this, we rewrite the initial p otential in terms of the redundant variables : ~ ~ ~ ~ ~ 1 = sin sin , 2 = sin cos , 3 = cos U
(s)

( ) =

2 + 2 1 3

2 2 +

22 41 3 2 2 (1 - 3 )2

+

2 2

and introduce the new local coordinates (, ) as follows: 1 = sin cos , 2 = cos , 3 = sin sin . In the new coordinates the p otential takes the form U
(s)

(, ) =

- . + cos2 sin2 sin2 2

As we can see, it reduces to (2.8) with v ( ) = k1 tg2 , = 2. 3.2. Sup erintegrable Potentials in R
3

We now show that if we take the Newtonian or Hookean p otential as the function w(r ) in the p otential (3.2), and if we take as U (, ) the sup erintegrable p otential on the sphere S 2 of the form: U (, ) = v ( )+ k3 + k4 cos , sin2 sin2 v ( ) = k2 , cos2 = m , n m, n N , (3.4)

then the initial system (3.1) will b e maximally sup erintegrable. Indeed, the Liouville integrability of this system is obvious: in addition to H and G there is also the Bertrand integral on the sphere (2.7), and H , G, FB are in involution. For maximal sup erintegrability it remains to define two integrals. To do so, we p erform a procedure similar to that p erformed previously, namely, we restrict the equations of motion to the level set of first integrals H = h, G = g, FB = f . After rescaling the time 2dt = r 2 d the equations for r and take the form: r 2 pr dr = =r d 2
2

h-

g - w(r ), r2

p d = = d 2

g - v ( ) -

f . sin2

(3.5)

As can b e seen, only for v ( ) defined ab ove (3.4), the system (3.5) is a particular case of the equations on the plane (1.16). Hence we have the following: Prop osition 5. If we choose either Hookean k1 r 2 or Newtonian system (3.5) admits a complex-valued integral:
(2,1) s k1 r

potential as w(r ), then the k1 , r

=

2gpr - i

(2,1) s

=

2g 2g pr + i h - 2 r r

2g + k1 r
2

2

2gp sin 2 +2i(g cos 2 + f - k2 ) , for 2gp sin 2 +2i(g cos 2 + f - k2 )
2

w=

, for

w = k1 r 2 . (3.6)

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Applying the ab ove procedure (see Section 1.4), we can obtain a real integral that is p olynomial in the momenta. In the case w = kr1 this integral has degree 5 (deg = 5), and in the case w = k1 r 2 degree 6 (deg = 6). Further, in order to obtain the remaining integral, we rescale the time 2 dt = r 2 sin2 d and represent the equations for and as sin2 pr d = = sin2 d 2 p f d = = 2 - v ( ), d sin 2 k3 + k4 cos u() = . sin2 g- f - u(), (3.7)

Using the procedure for reducing the system in R3 to the system on the sphere, we can show that the second sup erintegral (2.9) admits an immediate generalization: Prop osition 6. The system (3.7) admits a complex-valued superintegral:
(m,n) s

=

2fp ctg + i g - k2 - ctg2 f -

f sin2

m

2n

2fp sin + i(2f cos + k4 )

. (3.8)

Thus, the following theorem holds. Theorem 3. The Hamiltonian system in R3 with the Hamiltonian H= p2 1 2 p2 pr + 2 + 2 2 2 r r sin + w(r )+ 1 k3 + k4 cos k2 , + 2 cos2 r sin2 sin2
k1 r

(3.9)

where = m , m, n N , and w(r ) is either Hookean k1 r 2 or Newtonian n superintegrable (i.e., it admits 5 integrals of motion).

potential, is maximal ly

Remark. The authors of [41], using the method of separation of variables, show the sup erintegrability of a more general system with a Hamiltonian that has no explicit physical meaning: p2 1 1 p2 1 1 + w(r )+ 2 + H = p2 + r 2 2 sin2 2 2r 2r r k1 r k3 + k4 cos k2 + 2 cos sin2 sin2 ,

where the function w(r ) can b e chosen to b e either the Newtonian or Hookean p otential: w(r ) = or w(r ) = k1 r 2 , , Q.

One of the tra jectories for the system (3.9) with w(r ) = k1 r 2 is shown in Fig. 4. We note an unexp ected feature, namely, that although the generic tra jectory is a spatial curve, it has several p oints of self-intersection (three in this case). It would b e interesting to ascertain whether this prop erty is preserved for larger dimensions. 4. CENTRAL PROJECTION AND SUPERINTEGRABLE SYSTEMS IN S
3

Using the construction describ ed in Section 2.1, we show how the sup erintegrable systems on R3 can b e carried over to the three-dimensional sphere S 2 (in the general case to the Lobachevsky space). We assume that the three-dimensional sphere S 3 is emb edded in the four-dimensional Euclidean space R4 and is given by the equation x2 + x2 + x2 + x2 = 1. 1 2 3 4 We parameterize S 3 by the spherical coordinates (, , ): x1 = sin sin sin , x2 = sin sin cos , x3 = sin cos , x4 = cos .
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SUPERINTEGRABLE GENERALIZATIONS OF THE KEPLER AND HOOK PROBLEMS 431

k2 Fig. 4. The tra jectory of the system in R3 in the p otential field V = k1 r 2 + r2 cos2 + k3 +k4sicos sin2 n2 k1 = 1, k2 = 1 , k3 = 1 , k4 = 1 , r (0) = 1, (0) = , (0) = , pr (0) = 1, p (0) = 2, p (0) = 1. 6 2 4 3 5



for = 3,

In this case the Hamiltonian of the particle moving in a p otential field has the form p2 1 1 p2 1 + + V (, , ). H = p2 + 2 2 2 2 sin 2 sin sin2 Further, we restrict our attention to the p otential of the form: V (, , ) = ( )+ U (, ) , sin2 U (, ) = k2 k3 + k4 cos , + 2 ( ) cos sin2 sin2
2 2

(4.1)

=

m n

m, n N.

(4.2)

In this case, as in S 2 , there is an analog of the Bertrand integral (2.7): G= 1 2 p2 + p sin + U (, ). (4.3)

It turns out that there is a relation b etween the systems in the space R3 considered ab ove and the systems on the sphere S 3 : Prop osition 7. The Hamiltonian system (3.1) in R3 with the potential U (r, ) , r2 wil l, as a result of central projection and after rescaling the time V = w(r )+ r = tg , pr = p , d = cos2 dt,

become the Hamiltonian system (4.1) on S 3 with the potential V = w(tg )+ As a result, we arrive at the following theorem:
REGULAR AND CHAOTIC DYNAMICS Vol. 19 No. 3 2014

U (, ) . sin2


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Theorem 4. The system of equations (4.1) in the potential field (4.2), in which ( ) is either Hookean k1 tg2 or Newtonian k1 ctg potential, possesses a redundant set of five integrals. It is easy to see that in this case the Bertrand integral FB and and the remaining integral takes the form:
(2,1) s (2,1) s (m,n) s

(3.8) remain unchanged

= = â

2gp - i(2g cot + k1 )

2

2gp sin 2 +2i(g cos 2 + f - k2 ) , for v = k1 ctg , g sin2
2 2

2gp ctg + i h - ctg2 g -

2gp sin 2 +2i(g cos 2 + f - k2 )

, for v = k1 tg2 , (4.4)

where g = G, f = FB . Remark. The validity of Theorem 4 for = 1 was obviously first shown in [37]. 5. CONCLUSION In conclusion we formulate a numb er of problems from the viewp oint of classical mechanics (without touching up on the quantum case). 1. Carry out the results obtained to the noncompact case of a plane and of the Lobachevsky space H 2 , H 3 . 2. Construct the algebra of integrals. This topic is outlined in [19, 34] and is of interest from the viewp oint of the general theory of Poisson geometry and Poisson structures. 3. Give a bifurcation (top ological) analysis of the found systems, which allows one to find systems in the space of integrals and parameters for which the tra jectories are compact. Another problem is that of stability of these tra jectories, for integrable systems with two degrees of freedom the top ological methods of stability analysis were develop ed in [14]. 4. Another problem that requires a more detailed analysis is that of preservation of p otentiality (Hamiltonian prop erty) under pro jective transformations. ACKNOWLEDGMENTS The authors sincerely thank A. V. Bolsinov and A. V. Tsiganov for fruitful discussions and useful comments. The work of A. V. Borisov was done within the framework of the State assignment of the Udmurt State University "Regular and Chaotic Dynamics". The work of I. S. Mamaev was supp orted by the grant of the RFBR 13-01-12462-ofi m, and the work of I. A. Bizyaev was supp orted by the grant of the RFBR 14-01-00395-a. REFERENCES
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