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ISSN 1560-3547, Regular and Chaotic Dynamics, 2009, Vol. 14, No. 6, pp. 615­620. c Pleiades Publishing, Ltd., 2009.

RESEARCH ARTICLES

Sup erintegrable System on a Sphere with the Integral of Higher Degree
A. V. Borisov* , A. A. Kilin** , and I. S. Mamaev***
Institute of Computer Science, Udmurt State University, ul. Universitetskaya 1, Izhevsk, 426034 Russia
Received Octob er 21, 2009; accepted Novemb er 16, 2009

Abstract--We consider the motion of a material point on the surface of a sphere in the field of 2n + 1 identical Hooke centers (singularities with elastic potential) lying on a great circle. Our main result is that this system is superintegrable. The property of superintegrability for this system has been conjectured by us in [1], where the structure of a superintegral of arbitrarily high o dd degree in momemnta was outlined. We also indicate an isomorphism between this system and the one-dimensional N -particle system discussed in the recent paper [2] and show that for the latter system an analogous superintegral can be constructed. MSC2000 numbers: 70Hxx, 70H06, 70G65, 37J35, 70F10 DOI: 10.1134/S156035470906001X Key words: superintegrable systems, systems with a potential, Ho oke center

1. INTRODUCTION In the pap er we discuss the existence on an additional (sup er)integral for the system of 2n +1 identical Hooke centers arranged in a regular 2n + 1-gon inscrib ed in a great circle. Based on simulation results, we conjectured the existence of such an integral in [1]. The problem of Hooke centers attracted our attention when we were working at new constructive methods of reduction of systems of particles interacting with a p otential of the Jacobi typ e (a homogeneous function of degree = -2). This problem, viewed in the context of the theory of multi-particle systems in constant-curvature spaces, has its own interest. Some recent and classical results on the sub ject are presented in the collection of works [3] (see also [4­7]). 1 The spherical analog of a Hooke center is a singularity whose p otential is cos2 ( is the latitude), thus the p otential p ossesses central symmetry. Such a p otential naturally arises in various problems from the rigid b ody dynamics. In [8] a p otential of this typ e occurred in the course of reduction of various systems from the dynamics of axially symmetric b odies. It should b e noted that in the rigid b ody dynamics such a p otential was systematically investigated D.N. Goryachev. A similar p otential occurs from the analysis of integrable systems connected with the motion of a material p oint on the surface of a three-axial ellipsoid (Rosochatius systems). For a modern treatment of this problem see [9]. Sup erintegrability of the system of three Hooke centers on a sphere situated at the ends of three orthogonal axes has b een earlier proved in [10] (a particular case of the Rosochatius system), where the additional sup erintegral was shown to b e quadratic in momenta; the prop erty of sup erintegrability can b e extended to the n-dimensional case. A general approach to detection of such sup erintegrals has b een recently develop ed in [11]; this approach traces back to the classical methods and results from works of Richelot, Jacobi and Weierstrass (see also [12])). In the case we consider here the sup erintegral can b e of arbitrarily high odd degree in momenta (as predicted in [1]); moreover this degree is exactly 2n +1 for 2n + 1 Hooke centers. It will b e
* ** ***

E-mail: borisov@ics.org.ru E-mail: aka@ics.org.ru E-mail: mamaev@ics.org.ru

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shown in the conclusion of the pap er that this integral can easily b e "raised" to the system of particles interacting with a p otential of the Jacobi typ e in one dimension [2]. Anyway, the structure of our integral is somewhat different from the integral obtained in [2]. Moreover, though in [2] a general form of the sup erintegral is indicated, a proof that this form is valid in the case of arbitrary numb er of particles is missing. In our turn, we present an effective and straightforward method for construction of sup erintegrals in separable Hamiltonian systems. 2. A NEW INTEGRAL OF THE PROBLEM OF N HOOKEAN CENTERS ON A SPHERE Consider a system describing the motion of a material p oint on a sphere S 2 in the field of N = 2n + 1 equal Hooke centers located at the vertices of a regular N -gon on an equator of the sphere. The p otential of the Hooke center has the form (r,1 )2 , where the vector r gives the p osition of the center and gives the p osition of the material p oint. Let N Hooke centers b e located as follows: m m , cos , 0 , m = 1,... ,N . rm = sin N N Then in the spherical frame of reference the Hamiltonian of the system in question takes the form H= 1 2 p2 + p sin
2 2 N

+
m=1

a sin2 sin
2

+

m N

.

(2.1)

Using the following trigonometric relation 1 1 =2 2 N sin N
N m=1

1 sin
2

+

m N

,

(2.2)

and denoting k = aN 2 , we write the Hamiltonian 1 1 H = p2 + 2 sin2 The corresp onding equation

in the form 12 k . p + 2 2 sin N s of motion then b ecome 2kN cos N p p = , = 2, sin sin2 sin3 N 2cos 1 2 k . p + p = = p , 3 2 sin 2 sin N

(2.3)

(2.4)

It is not difficult to notice that the system (2.3) is separable. As a matter of fact, 12 k G 12 = G, = H, p + p + (2.5) 2 2 2 sin N sin2 where G is a constant of separation, which is an additional integral of motion. Let us prove the following. Prop osition 1. Al l trajectories of the system (2.3) are closed.
1 Proof. Let us introduce a new time such that d = sin2 and express the momenta p , p in terms dt of the derivative of the variables and with resp ect to the new time, that is,

p = ,

p = · sin-2 .

(2.6)

Inserting the expressions (2.6) in (2.5) and integrating, we obtain G cos N = cos(N 2G + C1 ), G-k G ctg = cos( 2G + C2 ). H -G
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Here C1 and C2 are constants of integration, which are given by the initial conditions. From Eqs. (2.7) it can b e seen that the variables and are p eriodic functions of with commensurable frequencies = N . Thus, all tra jectories of the system (2.3) are closed. The integrals H and G parametrize the Liouville tori of the system (2.3). On the tori themselves the tra jectories of the system are parametrized by means of another first integral of the system. In order to find it, we write the equations for the tra jectories lying on a given torus: d 2 G-
k sin2 N

= sin
2

d 2 H-
G sin2

.

(2.8)

Integrating Eq. (2.8), we obtain a new integral of motion G G cos N - N arccos ctg , (2.9) G-k H -G which is a multivalued function of the variables of the system (2.3). The integral (2.9) can also b e obtained by eliminating the time from Eqs. (2.7). In order to eliminate the multivaluedness, we consider as integrals the following functions J2 = sin I, J1 = cos I, G (2.10) 2 2n+2 2n+1 (H - G) (G - k). where = 2 I = arccos After a numb er of trigonometric transformation the integrals J1 and J2 assume the form
n

J1 = J2 =

(-1)m (2G)n
m=0 n

-m 2m p

ctg ctg

2(n-m)

2 2 (CNm2G ctg cos N + CNm+1 p p sin N),

(2.11)
m

(-1)
m=0

(2G)n-m p2m

2(n-m)



2 (CNmp

ctg sin N -

2 CNm+1 p

cos N),

N! k where CN = k!(N -k)! are binomial coefficients. As may b e seen from (2.11), the integrals J1 and J2 are one-valued functions of 2n +2 and 2n + 1 degree in momenta resp ectively. Besides, it is easy to notice that they are dep endent and connected to each other by 2 2 J1 + J2 G2 = 2 .

(2.12)

Thus, the hyp othesis of sup erintegrability of the system on a sphere in the field of N Hooke centers located on an equator, advanced by us in [1], has b een proved. Note that apart from the integral of an arbitrary odd degree in momenta, we have also obtained an integral of an arbitrary even degree in momenta, which is always one degree higher but is functionally dep endent with the "odd" integral. 3. We now present an variables (M , ) using spherical top around a like ALGEBRAIC FORM OF THE SUPERINTEGRAL other form of the sup erintegral expressed in terms of the Euler­Poisson an analog b etween the motion of a p oint on a sphere and the motion of a fixed p oint (see, e.g., [13]). In terns of M , the Hamiltonian of (2.1) looks 1 H = M2 + 2
N i=1

a , (ri , )2

(3.1)

where ri radius-vector of the Hooke centers. The equations of motion are determined by the Poisson bracket of the algebra e(3) {Mi ,Mj } = eij k Mk ,
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{Mi ,j } = eij k k ,
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{i ,j } = 0,

(3.2)

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which is degenerate and p ossesses two Casimir functions (M , ) = C, 2 = 1. (3.3)

For the systems on the sphere C = 0 (i.e. for problems of rigid b ody dynamics all the integrals mentioned are only partial). The additional integral of motion G in the variables M , takes the form: 12 2 G = Mn +(1 - n ) 2
N i=1

a , (ri , )2

where Mn and n are the comp onents of the vectors M and p erp endicular to the plane in which the Hooke centers lie. The second of the integrals (2.11) in the variables M , has the form
n

J2 = -C

(-1)
m=0 2m N

m

2 (2Gn )n-m 2n+1

2n+1

( â M , en )

2n

2 CNm+1

(-1) ( â M , en )
k k =1

( â rk , en ) (3.4)

Mn n
k =1

(rk , ) /(1 -

2 n )2n+1

,

where en is the unit vector of the normal to the plane in which the Hooke centers lie. If all centers are located on an equator, then n = 3 , Mn = M3 , and en = (0, 0, 1).

4. SUPERINTEGRABILITY OF THE RELATED MULTIPARTICLE SYSTEM [2] The algebraic form of the integral (3.4) obtained can b e generalized to the case of interacting particles on a straight line. As shown in our pap er [1], a natural system of dimension D with a homogeneous p otential of degree = -2 is reduced to a system of dimension D - 1 on a sphere. Thus, there exists a connection b etween the ab ove-mentioned problem of N Hooke centers and the problem of motion of a material p oint in R3 (or of three b odies on a straight line) under the action of p otential forces of sp ecial kind. The corresp onding reduction theorem has the following form. Theorem 1. The natural system with a Hamiltonian of the form 1 H = p2 + U-2 (x), 2
N

U-2 (x) =
i=1

a , (x, ri )

x, p R

3

(4.1)

admits reduction by one degree of freedom by the change of time and coordinates dt = |x|dt, = x . |x| (4.2)

The equations of motion in the new variables govern the motion of a material point on the twodimensional sphere 2 = 1 =- U-2 ( ) + , U-2 ( ) -
2

,

where the prime denotes differentiation with respect to the new time. The proof of this theorem for the case of arbitrary dimension u and p otential U-2 is given in [1]. By applying the transformation (4.2) to the ab ove-mentioned system of N Hooke centers it can b e shown that the following prop osition holds.
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Prop osition 2. The system (4.1) resulting from "raising" the system (2.1) is maximal ly superintegrable and in addition to the Hamiltonian possesses four independent first integrals of motion 1) Pn = (p, n), 2) J = 2|x|2 H - (x, p)2 , 1 3) G = (x â p, en )2 + 2
n N i=1

a(x â en )2 , (ri , x)2
-m

4) J2 = âC -C

(-1)m (2G(x, en )2 )n
m=0 2m+1 N

(x â (x â p), en )2m â
N

(4.3)

(-1)n (x â (x â p), en )
k =1 N

(x â rk , en )-

2m N

(x â p, en )(x, en )

(ri , x) /(x â en )2N .
k =1

The proof of the prop osition is a straightforward verification of conservation of the integrals (4.3) and their indep endence. Note that the existence of the integral Pn is due to the invariance of the system (4.1) under translation along the axis en p erp endicularly to the plane in which the Hooke centers lie. The integral J is the Jacobi integral existing for any natural systems with a homogeneous p otential of degree = -2, and the integrals G and J2 are generalizations of the corresp onding integrals for the problem of N Hooke centers on a sphere. 5. CONCLUSION In applied mechanics the sup erintegral found in this pap er is one of the most complicated known to date. Examples of other sup erintegrable systems and an exhaustive reference can b e found in, for example, [11, 12]. In the last decade the search for sup erintegrals has b ecome p opular in the physics community. The method we used for detection of sup erintegrals reveals an intimate relation b etween the prop erty of sup erintegrability on compact surfaces and the closedness of all the tra jectories. Interestingly, numerical simulations showed that in our problem all the tra jectories are indeed closed thereby conjecturing the existence of sup erintegrals. Closedness of tra jectories on compact surfaces of revolution in the presence of p otential forces is a classical research topic, which seemingly was initiated by Darb oux [14]. Several more fresh examples of surfaces with closed tra jectories are collected in [15]. The existence of sup erintegrals for such surfaces seems to b e even a more complicated issue to explore. Conjectures were made that for integrable systems with analytic p otential on sphere-typ e surfaces the degree of an additional sup erintegral is always b ounded (see, for example, [16]). However, it turned out that on a sphere there exists an analytic geodesic flow of arbitrarily high degree in momenta [17]. Unfortunately, the pap er [17] does not present an explicit form of the metric in terms of analytic or elementary functions, instead for the coefficients of the metric the author just formulates a b oundary problem and proves that this problem has an analytic solution. It should b e noted that in our example of integral of arbitrarily high degree, the numb er of singular p oints of the p otential increases as so does the degree of the integral. This naturally suggests the following problem: using the method develop ed in the pap er, construct a dynamical system on a sphere with analytic p otential without singularities for which a sup erintegral of essentially high degree (> 4) in momenta exists. In this connection, mention should b e made of the Gaffet system [1]. In a series of works, Gaffet explores a system on S 2 with a p otential of degree 6 in momenta. Unfortunately, the singularities of this p otential are not isolated and fill the great circles of intersection with the coordinate planes.
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ACKNOWLEDGMENTS The authors are most grateful to Andrzej Maciejewski and Maria Przybylska for valuable discussions during their visit to Izhevsk and bringing an intriguing system found by H. Yoshida to our attention. We are also thankful to Andrey Tsiganov for sending us useful pap ers. The work is partly supp orted by the grants RFBR (pro jects 09-01-12151, 09-01-00791), Federal target program "Scientific and scientific-p edagogical p ersonnel of innovative Russia" (pro jects 2009-1.5-503-004019). The work of I. S. Mamaev and A. A. Kilin was done in the framework of the program of the President of the Russian Federation for the supp ort of young scientists, Doctors of Science (pro ject code MD-5239.2008.1) and Candidates of Science (pro ject code MK-6376.2008.1), resp ectively. REFERENCES
1. Borisov, A.V., Kilin, A.A., Mamaev, I.S., Multiparticle Systems. The Algebra of Integrals and Integrable Cases, Regul. Chaotic Dyn., 2009, vol. 14, no. 1, pp. 18­41. 2. Chanu, C., Degiovanni, L., and Rastelli, G. Superintegrable Three-bo dy Systems on the Line, Journal of Math. Phys., 2008, vol. 49, 112901, 10 pp. 3. Borisov A.V. and Mamaev I.S. (Eds.), Classical Dynamics in non-Eucledian Spaces, Moscow-Izhevsk: Inst. komp. issled., RCD, 2004 (in Russian). 4. Borisov, A.V. and Mamaev, I.S., Generalized Problem of Two and Four Newtonian Centers, Celestial Mech. Dynam. Astronom., 2005, vol. 92, no. 4, pp. 371­380. 5. Borisov, A.V. and Mamaev, I.S., The Restricted Two-Bo dy Problem in Constant Curvature Spaces, Celestial Mech. Dynam. Astronom., 2006, vol. 96, no. 1, pp. 1­17. 6. Borisov, A.V., Mamaev, I.S., and Kilin, A.A., Two-Bo dy Problem on a Sphere. Reduction, Sto chasticity, Periodic Orbits, Regul. Chaotic Dyn., 2004, vol. 9, no. 3, pp. 265­280. 7. Borisov, A.V. and Mamaev, I.S., Superintegrable Systems on a Sphere, Regul. Chaotic Dyn., 2005, vol. 10, no. 3, pp. 257­266. 8. Borisov, A.V. and Mamaev, I.S., Non-linear Poisson Brackets and Isomorphisms in Dynamics, Regul. Chaotic Dyn., 1997, vol. 2, no. 3­4, pp. 72­89 (in Russian). 9. Kozlov, V.V., Some Integrable Extensions of the Jacobi's Problem of Geo desics on an Ellipsoid, Prikl. Mat. Mekh., 1995, vol. 59, no. 1, pp. 3­9 [J. Appl. Math. Mech., 1995, vol. 59, no. 1, pp. 1­7]. 10. Kozlov, V.V. and Fedorov, Y.N., Integrable Systems on a Sphere with Potentials of Elastic Interaction., Mat. Zametki, 1994, vol. 56, no. 3, pp. 74­79 [Math. Notes 56 (1994), 1994, vol. 56, no. 3­4, pp. 927­930]. 11. Tsiganov, A.V. and Grigoriev, Yu.A., On Abel's Equations and Richelot's Integrals, Rus. J. Nonlin. Dyn, 2010, vol.6 (in press). 12. Tsiganov, A.V., Leonard Euler: Addition Theorems and Superintegrable Systems, Regul. Chaotic Dyn., 2009, vol. 14, no. 3, pp. 389­406. 13. Borisov A.V. and Mamaev I.S., Rigid Body Dynamics. Hamiltonian Methods, Integrability, Chaos, Moscow-Izhevsk: Inst. komp. issled., RCD, 2005 (in Russian). 14. Darboux, G., Etude d'une question relative au mouvement d'un point sur une surface de revolution, Bul l. S.M.F., 1877, T. 5, pp. 100­113. 15. Kolokol'tsov, V.N., Geo desic Flows on Two-dimensional Manifolds with an Additional First Integral Polynomial in Velo cities, Izv. Akad. Nauk. SSSR, 1982, vol. 46, pp. 994­1010 (in Russian). 16. Bolsinov, A.V. and Fomenko, A.T., Integrable Geo desic Flows on a Sphere Generated by Goryachev­ Chaplygin and Kovalevskaya Systems in the Dynamics of a rigid bo dy, Mat. Zametki, 1994, vol. 56, pp. 139­142 [Math. Notes, 1994, vol. 56, pp. 859­861]. 17. Kiyohara, K., Two-dimensional Geo desic Flows Having First Integrals of Higher Degree, Math. Ann., 2001, vol. 320, pp. 487­505.

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