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ISSN 1560-3547, Regular and Chaotic Dynamics, 2008, Vol. 13, No. 6, pp. 557­571. c Pleiades Publishing, Ltd., 2008.

¨ JURGEN MOSER ­ 80

Chaplygin Ball over a Fixed Sphere: an Explicit Integration
A. Borisov1 * , Yu. Fedorov2 ** , and I. Mamaev1
1

***

Institute of Computer Science, Udmurt State University, ul. Universitetskaya 1, Izhevsk, 426034 Russia 2 Department de Matematica Aplicada I, ´ Universitat Politecnica de Catalunya, Barcelona, E-08028 Spain
Received July 21, 2008; accepted Octob er 7, 2008

Abstract--We consider a nonholonomic system describing the rolling of a dynamically nonsymmetric sphere over a fixed sphere without slipping. The system generalizes the classical nonholonomic Chaplygin sphere problem and it is shown to be integrable for one special ratio of radii of the spheres. After a time reparameterization the system becomes a Hamiltonian one and admits a separation of variables and reduction to Abel­Jacobi quadratures. The separating variables that we found appear to be a non-trivial generalization of ellipsoidal (spheroconic) coordinates on the Poisson sphere, which can be useful in other integrable problems. Using the quadratures we also perform an explicit integration of the problem in theta-functions of the new time. MSC2000 numbers: 37J60, 37J35, 70H45 DOI: 10.1134/S1560354708060063 Key words: Chaplygin ball, explicit integration, nonholonomic mechanics

1. INTRODUCTION One of the ma jor contributions of J. Moser to integrable Hamiltonian systems has b een an explicit description of the relation b etween their geometric, symplectic, and algebraic geometric asp ects; in particular, the geometry of quadrics, separation of variables, and flows on hyp erelliptic Jacobian varieties ([1­3]). The principal model example in this description was the celebrated Neumann problem with the ellipsoidal (or spheroconic) coordinates as separating variables. On the other hand, in classical nonholonomic mechanics there exists a series of integrable problems, which are, a priori, not Hamiltonian, but also display a rich interaction b etween their geometric and algebraic geometric prop erties. Probably, the b est known example is the Chaplygin problem on a non-homogeneous sphere rolling over a horizontal plane without slipping. In [4] S.A. Chaplygin obtained the equations of motion, proved their integrability and p erformed their reduction to quadratures by using spheroconic coordinates on the Poisson sphere as separating variables. He also actually solved the reconstruction problem by describing the motion of the sphere on the plane. Various asp ects of this famous system were studied in [4­7], and its explicit integration in terms of theta-functions was presented in [8]. Several nontrivial integrable generalizations of this problem were indicated by V.V. Kozlov [9] (the motion of the sphere in a quadratic p otential field), A.P. Markeev [10] (the sphere carries a rotator), in [11] (an extra nonholonomic constraint is added) and in [12] (the sphere touches an arbitrary numb er of symmetric spheres with fixed centers). Next, amongst others, the pap ers [5, 13, 14] considered rolling of the Chaplygin (i.e., dynamically non-symmetric) sphere over a fixed sphere, so called sphere-sphere problem. They studied the
* ** ***

E-mail: borisov@rcd.ru E-mail: Yuri.Fedorov@upc.edu E-mail: mamaev@rcd.ru

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equations of motion in the frame attached to the b ody. More generic (although more tedious) form of the equations also app eared in the works of Woronetz [15, 16], who, nevertheless, solved a series of interesting problems describing rolling of b odies of revolution or flat b odies over a sphere. A rolling of a generic convex b ody over a sphere was also discussed in the recent survey [17]. In [5, 13, 14] it was observed that the equations of motion of the Chaplygin sphere-sphere problem admit an invariant measure, but, as numerical computations show, in the general case they are not integrable (there is no analog of the linear momentum integral). However, as was also found in [13], for one sp ecial ratio of radii of the two spheres an analog of such an integral does exist, and the system in integrable by the Jacobi last multiplier theorem. Until recently, no one of the ab ove generalizations was integrated in quadratures (except a case of particular initial conditions of the Kozlov generalization, considered in [18]). In this connection it should b e noted that a Lax pair with a sp ectral parameter for the Chaplygin sphere problem or for its generalizations is still unknown and, probably does not exists. Hence, one cannot use the p owerful method of Baker­Akhieser functions to find theta-function solution of the problem. The main purp ose of this pap er is to find appropriate separating variables, which allow to reduce the equations of motion of the integrable case of the Chaplygin sphere-sphere problem to quadratures, as well as to give an explicit theta-function solution. It app ears that, in contrast to the classical Chaplygin sphere problem, the usual spheroconic coordinates on the Poisson sphere do not ensure separation of variables and that the required variables should b e introduced in a more complicated way (see formulas (3.2)). Using the quadratures, we also give a brief analysis of p ossible bifurcations and p eriodic solutions. These results are presented in Sections 3­4. In Section 5 we briefly describ e another typ e of p eriodic solutions. Section 6 provides a derivation of explicit theta-function solutions of the problem in a selfcontained form. Finally, in App endix we show how the separating variables we used can b e obtained in a systematic way, by reducing a restriction of our system to an integrable Hamiltonian system with 2 degrees of freedom and applying a classical result of Eisenhart on transformation of the Hamiltonian to a St¨ckel form. a 2. EQUATIONS OF MOTION AND FIRST INTEGRALS Let , m, I = diag(I1 , I2 , I3 ), and b denote resp ectively the angular velocity vector of the Chaplygin sphere, the mass of the sphere, its inertia tensor, and the radius. By n we denote the unit normal vector to the fixed sphere S 2 at the contact p oint P . The angular momentum M of the sphere with resp ect to P can b e written as The whole phase space of the system is thus T (S O(3) â S 2 ). By using the no slip nonholonomic constraint (which corresp onds to zero velocity in the p oint of contact), one can obtain the reduced equations of motion in the frame attached to the sphere in the following closed form (see [13]): a M = M â , n = k n â , k = , (2.2) a+b a b eing the radius of the fixed sphere. Note that the ratio k can take any p ositive or negative value dep ending on the relative p ositions of the rolling and fixed spheres, as illustrated in Fig. 1. For arbitrary k equations (2.2) p ossess three indep endent integrals F0 = n, n = 1, which, in view of (2.1), can b e written as F0 = n, n = 1, where J = I + dE. F1 = J, J - 2 J, n n, + d2 n, 2 ,
REGULAR AND CHAOTIC DYNAMICS Vol. 13 No. 6

M = I + d n â (n â ),

d = mb2 .

(2.1)

H = M, ,

F1 = M , M ,

(2.3)

H = , J - d2 n, 2 ,

(2.4)

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n r b

559

y a

a) 0 < k < 1 a n n b b a

b) k > 1

c) k < 0

Fig. 1. Rolling of the Chaplygin sphere inside/over the fixed sphere (dashed).

As shown in [14], the equations, expressed in terms of , n also have the invariant measure d dn with the density = n, n - d n, J-1 n . (2.5)

The Chaplygin sphere on the plane. Clearly, the case k = 1 corresp onds to a , that is, the fixed sphere transforms to a horizontal plane with the unit normal vector n, and we arrive at the classical integrable Chaplygin problem, when the linear (in M ) momentum integral is preserved: M , n I , n . (2.6)

Second integrable case. According to [13], the system (2.2) is also integrable in the case k = -1, which describ es rolling of a non-homogeneous ball with a spherical cavity over a fixed b sphere and the quotient of the radii of the spheres equals a = 1 (see Fig. 1 c). 2 In this case, instead of (2.6), there is the following linear integral1) F2 = AM , n , where Remark 2.1. Note that, as was shown in [19], the modification of this system obtained by imp osing the extra "no twist" constraint , n = 0 (sometimes called as rubber Chaplygin bal l) is also integrable for the ratio k = -1. In the next sections we give an explicit integration of this case under the condition F2 = 0. The procedure is similar to the integration of the problem of the Chaplygin sphere rolling on the horizontal plane made in the case of zero value of the area integral (2.6), (see [4, 6, 20]), however, analytically, it is more complicated. Integration of the most general case F2 = 0 is still an op en problem.
1)

(2.7)

A = diag(J2 + J3 - J1 , J3 + J1 - J2 , J1 + J2 - J3 ).

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A remark on reducibility to quadratures in the case d = 0. d = 0 one has M = I and the equations (2.2) take the form As was noticed in [13], by the substitution I = I â , M = AI, n = -n â . =n

Note that in the limit case

(2.8)

and the sign change t -t, the latter system transforms to the Euler­Poisson equations for the classical Euler top, M = M â , = â , (2.9) i = ai Mi , ai = ((Jj + Jk - Ji )Ji )-1 , (2.10) which p ossesses first integrals M, = g, M, AM = h, M, M = f , where A = diag(a1 , a2 , a3 ). As was indicated in several publications (see, e.g., [21]), the Euler­Poisson equations (2.9) can b e integrated by separation of variables. Namely, by an appropriate choice of the constant vector in space we can always set g = 0. Then the equations (2.9) reduce to a flow on the tangent bundle of the Poisson sphere S 2 = { , = 1}. In the spheroconic coordinates 1 , 2 on S 2 such that
2 i =

the flow is reduced to the quadratures d1 d2 + = 0, R(1 ) R(2 ) 1 d1 2 d2 + = C dt, C = const, R(1 ) R(2 ) R() = -( - a1 )( - a2 )( - a3 )(f - h)

(ai - 1 )(ai - 2 ) , (ai - aj )(ai - ak )

i = j = k = i,

(2.11)

(2.12)

The latter contain one holomorphic and one meromorphic differential on the elliptic curve E = {µ2 = R()}. Thus the quadratures give rise to a generalized Ab el­Jacobi map and, following the methods develop ed in [22], they can b e inverted to express the variables , in terms of thetafunctions of E and exp onents (see, e.g., [21, 23] for concrete formulas). Note that in the case d = 0 the substitution (2.8) seems not to b e useful. In particular, it does not transform equations (2.2) in the second integrable case k = -1 to the case of the classical Chaplygin sphere problem (k = 1). 3. REDUCTION TO QUADRATURES IN THE CASE F2 = 0 We now consider the case d = 0, but assume that the linear integral F2 in (2.7) is zero, which imp oses restrictions of the initial conditions. Then, from (2.2) with k = -1 and F2 = 0 we get where B = (J - dn n)A. This allows to express the angular velocity in terms of n, n in the following homogeneous form = Bn â n . n, Bn (3.1) n = -n â , , Bn = 0,

In view of the ab ove remark on the reduction to quadratures in the case d = 0, to p erform separation of variables it seems natural to use the spheroconic coordinates 1 , 2 given by (2.11)
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(with i replaced by ni ) in the general case too. However, this choice does not lead to success: the first integrals H, F1 have mixed terms in the derivatives 1 , 2 . It app ears that a correct choice is given by the following quasi-spheroconical coordinates z1 , z2 on the Poisson sphere n, n = 1: n2 = i where G(z1 , z2 ) = 1 - d(TrJ - 2d)(z1 + z2 ) + d(4 det J - d Tr(JA))z1 z2 , and, as in (2.10), ai = (Ai Ji )-1 . Note that when d = 0, the factor G b ecomes 1 and the relation (3.2) takes the form of (2.11), that is, z1 , z2 do b ecome the usual spheroconical coordinates on S 2 . A systematic derivation of the substitution (3.2) is presented in App endix. We note that similar quasi-spheroconical coordinates were already used in [17, 19] to integrate the "rubb er" Chaplygin sphere-sphere problem. In the ab ove coordinates z1 , z2 one has 2 = (In, J-1 n = and ni = and the expressions (3.1) yield i = nj nk Jj - Jk 2 Ji - d (1 - dAi z2 )z1 (1 - dAi z1 )z2 + , -1 -1 -1 -1 (1 - aj z1 )(1 - ak z1 )z2 (1 - aj z2 )(1 - ak z2 )z1 (3.6) (3.7) 1 2 z1 z2 G(z1 , z2 ) + - z1 - ai z2 - ai G(z1 , z2 ) ni , (3.5) det I 1 , det J G(z1 , z2 ) n, Bn = det I det A z1 z2 G(z1 , z2 ) (3.4) (3.3) 1 det I (ai - z1 )(ai - z2 ) , G(z1 , z2 ) (Ji - d)Jj Jk (ai - aj )(ai - ak ) (i, j, k) = (1, 2, 3), (3.2)

, n =

n1 n2 n3 (J1 - J2 )(J2 - J3 )(J3 - J1 ) G(z1 , z2 ) z1 z2 + , 2 det I (z1 )z2 (z2 )z1 (z ) = (a-1 z - 1)(a-1 z - 1)(a-1 z - 1). 1 2 3

Substituting them, as well as (3.2) into the integrals H, F1 in (2.4), after simplifications we get det I (z2 ) 2 (z1 ) 2 2 z1 - (z )z 2 z2 , (z1 , z2 ) (z1 )z2 21 det I (z2 ) 2 (z1 ) 2 F1 = (z1 - z2 ) 2 2 z1 - (z )z 2 z2 , 4G (z1 , z2 ) (z1 )z2 21 H = (z1 - z2 ) 4G2 where (z ) = d det Az 2 - Tr(AJ)z + 2, (z ) = (4 det J - d Tr(AJ))z - (TrJ - 2d). Next, substituting (3.6), (3.7) and (3.2) into (2.1), we also obtain Mi = nj nk z1 z2 (Jk - Jj ) + . -1 -1 -1 2 (1 - aj z1 )(1 - ak z1 )z2 (1 - aj z2 )(1 - a-1 z2 )z1 k (3.9)

(3.8)

Now, fixing the values of the integrals by setting H = h, F1 = f , then solving (3.8) with resp ect to z1 , z2 and using the relation 2 2 (z2 ) (z1 ) - (z1 ) (z2 ) = det A (z2 - z1 ) G(z1 , z2 ),
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we get z = - 2
2 z G(z1 , z2 ) 4(z )(f (z ) + h (z )) (z1 - z2 )2 det I det A G(z1 , z2 ) =- 4(z - a1 )(z - a2 )(z - a3 ) (f (z ) + h (z )), (z1 - z2 )2

= 1, 2.

After the time reparameterization dt = the ab ove relations give

1 G(z1 , z2 )

d

det J In, J-1 n d det I

(3.11)

The latter are equivalent to the following Ab el­Jacobi typ e quadratures dz1 dz2 z1 dz1 z2 dz2 + = 2d , + = 0, R(z1 ) R(z2 ) R(z1 ) R(z2 )

z2 R(z1 ) z1 R(z2 ) dz1 dz2 = = , , d z1 - z2 d z2 - z1 R(z ) = -(z - a1 )(z - a2 )(z - a3 ) (f (z ) + h (z )).

(3.12)

(3.13)

which contain 2 holomorphic differentials on the hyp erelliptic genus 2 curve = {w2 = R(z )}. Note that for d = 0 the p olynomial R(z ) b ecomes a degree 4 p olynomial, and (3.13) reduce to the quadratures (2.12) for the Euler top problem, as exp ected. It is interesting that, like in the integration of the original Chaplygin sphere problem presented in [4], the reparameterization factor in (3.11) coincides with the density (2.5) of the invariant measure. In the real motion this factor never vanishes, hence the reparameterization is non-singular. Now substituting the ab ove formulas for z1 , z2 into (3.9) and simplifying, we express the angular 2 2 momentum M in terms of z1 , z2 and the conjugated coordinates w1 , w2 : Mi = (Ji - d)2 Ji2 Ai â (aj - z1 )(aj - z2 ) 2 G(z1 , z2 ) (ak - z1 )(ak - z2 ) (3.14)

1 w1 w2 - . z1 - z2 (z1 - aj )(z1 - ak ) (z2 - aj )(z2 - ak )

In Section 6 we shall use the ab ove expressions to obtain explicit theta-function solutions for the comp onents of n, , M in terms of the new time . 4. QUALITATIVE STUDY OF THE MOTION AND BIFURCATIONS For generic constants h, f the p olynomial R(z ) in (3.13) has simple roots ai , c1 , c2 , and in the real motion the separating variables z1 , z2 evolve b etween the roots in such a way that R(z1 ), R(z2 ) remain non-negative. This corresp onds to a quasi-p eriodic motion of the sphere. In the sequel we assume that the moments of inertia I1 , I2 , I3 corresp onds to a physical rigid b ody, i.e., that the triangular inequalities Ii + Ij > Ik are satisfies. For concreteness, assume also that d < J1 < J2 < J3 . This also implies 0 < A3 < A2 < A1 and 0 < a1 < a2 < a3 . As follows from the first expression in (3.4), in the real case the factor G(z1 , z2 ) is always p ositive. Hence, the right hand sides of (3.2) are p ositive and the coordinates ni are real and satisfy n, n = 1 if and only if z1 [a1 , a2 ] and z2 [a2 , a3 ], like the usual spheroconic coordinates. Next, we have Prop osition 4.1. For the real motion, when the constants h, f are positive, and for any d, Ji satisfying the above inequalities, the roots c1 c2 never coincide and c1 < a1 and if f /h = Ji , then c1 = 2d - Ai < a1 , d Ak Aj c2 = ai .
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Proof. Set f /h = R. The roots c1 , c2 coincide with those of (z ) + (z ) and have the form Tr(AJ)( + d) ± D c1,2 = , (4.1) d det A D = (Tr(AJ))2 ( + d)2 - 8d det A( + d) + 4d det ATrJ. The condition D = 0 gives a quadratic equation on , whose determinant equals -d always a negative numb er. Hence D > 0 and c1 < c2 . Next, in view of (4.1), the condition c1 - a1 = 0 also leads to a quadratic equation with always negative determinant. Then, evaluating c1 - a1 for one , we find c1 < a1 Finally, setting in (4.1) = Ji and simplifying, we obtain the indicated ab ove for c1 , c2 .

det I det A2 , for , again . expressions

Combining the statement of Prop osition 4.1 with the p ermitted p ositions of z1 , z2 , we conclude that, dep ending on p ossible value of c2 , Then, in view of (3.2), the vector n always fills a ring R on the unit sphere S 2 = { x, x = 1} b etween the lines of its intersection with the cone
3 i=1

z1 [a1 , c2 ], z2 [a2 , a3 ],

or

z1 [a1 , a2 ], z2 [c2 , a3 ].

(4.2)

Ji - d x2 i = 0. Ji ai - c2

Perio dic solutions with bifurcations. As follows from Prop osition 4.1, the only p eriodic solutions with bifurcations can occur when the root c2 coincides with a1 , a2 or a3 2) . This happ ens under the initial conditions i = j = 0, nk = 0, (i, j, k) = (1, 2, 3), when the sphere p erforms a p eriodic circular motion with nk 0 and one has , n 0, H = Jk k , 22 F1 = Jk k , which yields f /h = Jk . Then, in view of the ab ove prop osition, c2 = ak and the p olynomial R(z ) in (3.12) has the double root ak , as exp ected. When = f /h leaves the interval [J1 , J3 ], the root c2 goes b eyond [a1 , a3 ]. Then for R(z1 ), R(z2 ) to b e b oth p ositive, one of zi must violate the condition (4.2). This implies that in the real case the quotient f /h b elongs to [J1 , J3 ], and the bifurcation diagram on the plane (h, f ) consists only of 3 lines f /h = J1 , J2 , J3 . Note that, according to the results of [4], the same situation takes place for the Chaplygin sphere on the horizontal plane. 5. A SPECIAL CASE OF PERIODIC MOTION Apart from the particular case of the motion with F2 = 0, there is another sp ecial case, when this integral takes the maximal value, that is, when An is parallel to the momentum vector M , namely M = h An, h =const. In view of (2.1), this implies J - d , n = h An and = hJ-
1

An +

d An, J-1 n 2

,

(5.1)

b eing the same as in (2.5). Substituting the expression for into the second equation in (2.2) and simplifying, we get the following closed system for n: n=h
2)

J1 + J2 + J3 - 2d (n â J-1 n). 2

(5.2)

There is another typ e of p eriodic solutions corresp onding to p eriodic windings of the 2-dimensional tori. However, the latter are not related to bifurcations and we do not consider them here. Vol. 13 No. 6 2008

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It has two indep endent integrals n, n and n, J-1 n or An, An , which implies that the factor is constant on the tra jectories and that the system has the form of the Euler top equations. As a result, in the general case the comp onents of n and are expressed in terms of elliptic functions of the original time t and their evolution is p eriodic. This situation is similar to that of the sp ecial case of the motion of the Chaplygin sphere on a horizontal plane, when the momentum vector M is vertical, and when the solutions are elliptic without change of time.

6. THETA-FUNCTION SOLUTIONS IN THE CASE F2 = 0 In order to find explicit solutions for the variables , n, we first remind some necessary basic facts on the Jacobi inversion problem and its solution. Solving the Jacobi inversion problem by Wurzelfunktionen. genus g hyp erelliptic Riemann surface , which obtained from the curve {µ2 = R()}, R() = ( - E1 ) · · · ( - E2g
+1

Consider an odd-order

)},

by adding one infinite p oint . Let us choose a canonical basis of cycles a1 , . . . , ag , b1 , . . . , bg on such that ai aj = bi bj = 0, ai bj = ij , i, j = 1, . . . , g,

where 1 2 denotes the intersection index of the cycles 1 , 2 (For real branch p oints see an example in Figure 2). Next, let 1 , . . . , g b e the conjugated basis of normalized holomorphic ¯ ¯ differentials on such that i = 2 ij , ¯ = -1 .
aj

Consider the p eriod lattice = {2 B Zg } of rank 2g in Cg = (z1 , . . . , zg ). The complex torus Jac() = Cg /0 is called the Jacobi variety (Jacobian) of the curve .

The g â g matrix of b-p eriods Bij = 0

¯ bj i g+ Z

is symmetric and has a negative definite real part.

Fig. 2. A canonical basis of cycles on the hyp erelliptic curve represented as 2-fold covering of the complex plane . The parts on the cycles on the lower sheet are shown by dashed lines.

Now consider a generic divisor of p oints P1 = (1 , µ1 ), . . . , Pg = (g , µg ) on it, and the Ab el­ Jacobi mapping with a basep oint P0
P P
0 1

P

g

+ ··· + ¯

= Z, ¯
P
0

(6.1)

= (1 , . . . , g )T , ¯ ¯ ¯

Z = (Z1 , . . . , Zg )T Cg .

Under the mapping, symmetric functions of the coordinates of the p oints P1 , . . . , Pg are 2g-fold p eriodic functions of the complex variables Z1 , . . . , Zg with the ab ove p eriod lattice 0 (Ab elian functions).
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Explicit expressions of such functions can b e obtained by means of theta-functions on the universal covering Cg = (Z1 , . . . , Zg ) of the complex torus. Recall that customary Riemann's thetafunction (Z |B ) associated with the Riemann matrix B is defined by the series3) (Z |B ) =
g

exp B M , M + M , Z ),
M Zg g

(6.2)

M, Z =
i=1

Mi Zi ,

BM, M =
i,j =1

Bij Mi Mj .

Equation (Z |B ) = 0 defines a codimension one subvariety Jac() (for g > 2 with singularities) called theta-divisor. We shall also use theta-functions with characteristics = (1 , . . . , g ), = (1 , . . . , g ), j , j R,

which are obtained from (Z |B ) by shifting the argument Z and multiplying by an exp onent4) : 1 · · · g (Z ) (Z ) = exp{ B , /2 + Z + 2 , } (Z + 2 + B ). 1 · · · g (Z + 2 K + B M ) = exp(2 ) exp{- B M , M /2 - M , Z } (Z ), = , K - , M ,

All these functions enjoy the quasi-p eriodic prop erty (6.3)

Now for a generic divisor P1 = (1 , µ1 ), . . . , Pg = (g , µg ) on , introduce the p olynomial U (, s) = (s - 1 ) · · · (s - g ), C. Theorem 6.1. (see, e.g., [21, 24]). Under the Abel mapping (6.1) with P0 = the fol lowing relations hold [ + i ](Z ) U (, Ei ) (Ei - 1 ) · · · (Ei - g ) = ki , (6.4) [](Z )
g k =1 l =k

µk (k - l )

U (, Ei ) U (, Ej ) [ + ij ](Z ) = kij , (Ei - k )(Ej - k ) [](Z ) i, j = 1, . . . , 2g - 1, i = j,

(6.5)

are half-integer theta-characteristics such that
2 i + B i = (Ei ,0)

where ki , kij are certain constants depending on the periods of only and 1 = , i = i , , , i , i Zg /Z 2 i ¯


g

(mod ), and ij = i + j (mod Z2g ).

(6.6)

2 i + B = K (mod ),
3)

4)

The expression for (Z ) we use here is different from that chosen in a series of b ooks on theta-functions by multiplication of Z by a constant factor. Here and b elow we omit B in the theta-functional notation. Vol. 13 No. 6 2008

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Apparently, relations (6.5) were first obtained in the explicit form by K¨nigsb erger ([25]). o Earlier, expressions (6.4) had b een considered by K.Weierstrass as generalizations of the Jacobi elliptic functions sn(Z ), cn(Z ), and dn(Z ). This set of remarkable relations b etween roots of certain functions on symmetric products of hyp erelliptic curves and quotients of theta-functions with halfinteger characteristics is historically referred to as Wurzelfunktionen (root functions). One can show (see, e.g., [26, 27]) that for the chosen canonical basis of cycles a1 , . . . , ag , b1 , . . . , b on , = (1/2, . . . , 1/2)T , = (g/2, (g - 1)/2, . . . , 1, 1/2)T (mod 1).
g

(6.7)

In the case g = 2, all the functions in (6.4), (6.5) are single-valued on the 16-fold covering T2 Jac() with each of the four p eriods of 0 doubled, so that T2 and Jac() are transformed to each other by the change Z 2Z. In view of (6.6), (6.7) , 1/2 1/2 0 1/2 0 1/2 = , + 2 = , , + 1 = 0 1/2 1/2 1/2 0 1/2 1/2 0 1/2 0 1/2 1/2 , + 4 = , + 5 = . + 3 = (6.8) 1/2 1/2 1/2 0 1/2 0 Explicit theta-function solution. {w = R(z )},
2

Now let b e the genus 2 curve

c1 , < c2 b eing the roots of f (z ) + h (z ). Thus we identify

R(z ) = -(z - a1 )(z - a2 )(z - a3 ) (z - c1 )(z - c2 ) ,

and denote the corresp onding half-integer characteristic i by ai and c . Next, choose the canonical basis of cycles as depicted in Fig. 2 and calculate the 2 â 2 p eriod matrix dz z dz i , 1 = , 2 = . Aij = R(z ) R(z ) aj Then the normalized holomorphic differentials on are
2 j -1

{E1 , . . . , E5 } = {a1 , a2 , a3 , c1 , c2 },

k = ¯
j =1

Ck

z
j

dz

R(z )

,

C = A-1 ,

and the quadratures (3.13) give
(z1 ,w1 ) (z2 ,w2 ) (z1 ,w1 ) (z2 ,w2 )

1 + ¯


1 = Z1 , ¯


2 + ¯


2 = Z2 , ¯

(6.9) (6.10)

Z1 = 2C11 + Z10 ,

Z2 = 2C21 + Z20 ,

Z10 , Z20 b eing constant phases. Now, comparing the last fraction in (3.2) with the expression (6.4) in Theorem 6.1, we find Si (ai - z1 )(ai - z2 ) [ + ai ](Z ) = ki , [](Z ) (ai - aj )(ai - ak ) (i, j, k) = (1, 2, 3), (6.11)

where Z = (Z1 , Z2 ) and the comp onents of Z dep end on according to (6.10). Setting here z1 = aj , z2 = ak , in view of (6.9), we get 1 = ki [ + ai ](A(aj ) + A(ak )) [](0) = ki . [](A(aj ) + A(ak )) [ + ai ](0)
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Here in the last equality we used the definition of with characteristics and the quasip eriodic prop erty (6.3). In order to express in theta-functions the factor G(z1 , z2 ) in (3.2) given by (3.3), we fist note that it does not split into a product of linear functions in z1 and z2 , hence one cannot use an analog of the formula (6.4). On the other hand, from the condition n2 + n2 + n2 1 we find that 1 2 3 G= which, in view of (6.11), gives G(z1 , z2 ) = det I 1 â det J [](Z ) J2 J3 J1 2 [ + a1 ](Z ) + 2 [ + a2 ](Z ) + 2 [ + a3 ](Z ) I1 I2 I3 However, from his expression one cannot see whether the sum under the radical is a full square or not, that is, whether the root G(z1 , z2 ) is a single-valued function of Z1 , Z2 and, therefore, of the new time . On the other hand, the study of zeros and p oles of G(z1 , z2 ) under the Ab el map (6.9), shows that G(z1 , z2 ) = hence 2 [ + c1 + c2 ](Z ) , 2 [](Z ) (6.12) det I det J J2 J3 J1 S2 + S2 + S J1 - d 1 J2 - d 2 J3 - d
2 1

,

G(z1 , z2 ) is again a quotient of theta-functions with half-integer characteristics.

As a result, using theta-function expressions (6.11), (6.12) in formulas (3.2), (3.14) and applying the Wurzelfuntion (6.5) we arrive at Theorem 6.2. Under the Abel map (6.9), [ + [ + c1 [ + aj Mi ( ) = i [ + c1 i , i = const, ni ( ) = i ai ](Z ) , + c2 ](Z ) + ak ](Z ) , + c2 ](Z ) (i, j, k) = (1, 2, 3), (6.13) (6.14)

where the characteristics are given in (6.8) and Z1 , Z2 depend linearly on as described in (6.9). One can observe that, like the classical Chaplygin sphere problem, the variables ni , Mi are meromorphic functions of the new time . Finally, given the expression (6.12) for the factor G, the original time t can b e found by integrating (3.11).

APPENDIX. SEPARATION OF VARIABLES VIA REDUCTION TO A HAMILTONIAN SYSTEM As mentioned ab ove, the substitution (3.2) is quite non-trivial and can hardly b e guessed a priori. Below we describ e how one can obtain it in a systematic way.
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A1. Reduction to a Hamiltonian System on S

2

First introduce the spheroconical coordinates u, v on the Poisson sphere n, n = 1: n2 = i (Ji - u)(Ji - v ) , (Ji - Jj )(Ji - Jk ) i = j = k = i, Ji = Ii + d. (A.1)

Then, under the substitution (3.1), the equations (2.2) with k = -1 give rise to the following Chaplygin-typ e system on S 2 d T T d T T - = u, - = -v , (A.2) dt u u dt v v 1 T = (buu u2 + buv uv + bvv v 2 ), = (au u + av v ), 2 where T is the energy integral (2.3) expressed in the spheroconical coordinates under the condition F2 = 0 and is linear homogeneous in u, v . Explicit expressions for the coefficients of T , are quite tedious, so we do not give them here. T T , Pv = , this system can b e transformed to a Hamiltonian Introducing the momenta Pu = u v form with extra terms, which p ossesses invariant measure N du dv dPu dPv with the density N= 2uv + (u + v )(2d + 1 ) + 2 - d1 (43 + 21 2 - 3 - d2 1 1 det(J - dn n)

where 1 = Ji , 2 = Ji2 , 3 = J1 J2 J3 . According to the Chaplygin theory of reducing multiplier ([4]), after the time reparameterization N (u, v ) dt = d the system (A.2) is transformed to the Lagrange form T T du dv d T d T - - = 0, = 0, u = , v = . (A.3) d u u d v v d d As a result, under the time reparameterization we obtain an integrable Hamiltonian system on the cotangent bundle T S 2 with local coordinates u, v , pu = T / u , pv = T / v . A2. Separation of Variables Equations (A.3) p ossess 2 homogeneous quadratic integrals, which come from H, F1 (2.3) and which can b e written in the form 1 H = T = (guu (u )2 + 2guv u v + gvv (v )2 ), 2 1 F1 = (Guu (u )2 + 2Guv u v + Gvv (v )2 ). 2 Explicit expressions for the coefficients guu , . . . , Gvv as functions of u, v are suppressed due to their complexity. According to the result of Eisenhart [28] (see its modern accounting in [29]), separating variables s1 , s2 can b e chosen as the roots of the equation det(G - sg) = 0, G= Gu
u v

+(2 - 22 + 4d1 )(u + v ) - 4d(u + v )2 )-1 , 1

(A.4)
u v

Gu Gv

v

,g =

gu

gu gv

v

.

Gu

v

gu

v

Note that the roots dep end only on the local coordinates u, v on the configuration space S 2 .
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The spheroconical coordinates (A.1) dep end explicitly on the 1 1 u = - (y + y 2 - 4x), v = - (y - 2 2 where ±s2 Q(s1 ) ± s1 Q(s2 ) ±Q(s1 ) x= , y= 4d(s1 - s2 ) 2d(s1 Q(s) = b1 = b1 s2 - b2 s + b2 , 3

roots s1 , s2 as follows y 2 - 4x), ± Q(s2 ) , - s2 ) (A.5)

1 1 1 (Tr(JA))2 - d det A, b3 = 2 det J - dTr(JA), (A.6) 16 2 2 1 1 b2 = det JTr(JA) - d(Tr(JA))2 - d det ATrJ + d2 det A. (A.7) 4 2 T T In the new variables s1 , s2 and the conjugated momenta p1 = , p2 = the integrals take s1 s2 the Liouville form S2 (s2 ) 2 s2 S1 (s1 ) 2 s1 S2 (s2 ) 2 S1 (s1 ) 2 H= p- p , F1 = p- p, (A.8) s1 - s2 1 s1 - s2 2 s1 - s2 1 s1 - s2 2 where 2(8x3 + 8(d - )x2 + (22 - 4d)x - 4 - d + (x)) S (x) = , (A.9) (2x - + 2d)2 and we used the notation = (J2 + J1 - J3 )(-J2 + J2 - J3 )(-J2 + J2 + J3 ) = -A1 A2 A3 , = x2 ( 2 + 8d) + 2x(4 + d 2 - 2d + 4d2 ) + (4 + d )2 ,

2 2 2 = J1 + J2 + J3 - 2J1 J2 - 2J2 J3 - 2J3 J1 = -A1 A2 - A2 A3 - A3 A1 , = J1 J2 J3 , = J1 + J2 + J3 .

(A.10)

Due to the Hamilton equations with the Hamiltonian H in (A.8), the evolution of s1 , s2 is describ ed as follows 2/ 2/ y1 y2 s = , s = , (A.11) 1 2 2s1 - e + 2d s1 - s2 2s2 - e + 2d s2 - s1 yi = Y (xi ), Y (x) = (x) · (hx - f ) 8x3 + 8(d - e)x2 + (2e2 - - 4de)x - c + (x) , (A.12)

where h, f are the constants of the integrals H, F1 . Hence, we p erformed a separation of variables, however the evolution equations (A.11) have a quite tedious form. One can show that the equation y 2 = Y (x) defines an algebraic curve of genus 2 on the plane 2 = (x, y ). According to the theory of algebraic curves (see, amongst others, [30]), any curve of C genus 2 is hyp erelliptic and can b e transformed to a canonical Weierstrass form by an appropriate birational transformation of the coordinates x, y . One of such transformations is induced by the chain of substitutions x z x= 4b3 , ( + b2 )2 - 4b1 b2 3 = -(4 det J - dTr(AJ)) z + TrJ - 2d , 2 det A det I

b1 , b2 , b3 b eing defined in (A.6), (A.7). It converts (x) in (A.12), as well as Q(x) (A.5) into full squares. After some tedious calculations, one finds that in the new variables z1 = z (x1 ), z2 = z (x2 ) the expressions (A.1) take the form (3.2), which ensures the reduction to hyp erelliptic quadratures in the canonical form (3.13).
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ACKNOWLEDGMENTS A.V.B. and I.S.M. research was partially supp orted by the Russian Foundation of Basic Research (pro jects Nos. 08-01-00651 and 07-01-92210). I.S.M. also acknowledges the supp ort from the RF Presidential Program for Supp ort of Young Scientists (MD-5239.2008.1). Yu.N.F. acknowledges the supp ort of grant BFM 2003-09504-C02-02 of Spanish Ministry of Science and Technology. REFERENCES
1. Moser, J., Various Aspects of Integrable Hamiltonian Systems, Proc. CIME Conf. (Bressanone, 1978), also published in Progress of Mathematics, Vol. 8, Boston: Birkh¨user, 1980, pp. 233­289. a 2. Moser, J., Geometry of Quadrics and Spectral Theory, The CHERN Symposium 1979, Berkely, New York: Springer Verlag , 1980, pp. 147­187. 3. Moser, J., Integrable Hamiltonian Systems and Spectral Theory, Fermi Lectures, Pisa 1981, Lezioni Fermiane, Acad. Nat. dei Lincei, Pisa, 1981. Reprinted in: Proc. 1983 Beijing Symposium on Differential Geometry and Differential Equations, ed. Liao Shantao, S.S. Chern, Science Press, Beijing, China 1986, pp. 157­229. 4. Chaplygin, S.A., On a Ball's Rolling on a Horizontal Plane, Matematicheski sbornik (Mathematical i Collection), 1903, vol. 24. [English translation: Regul. Chaotic Dyn., 2002, vol. 7, no. 2, pp. 131­148. http://ics.org.ru/eng?menu=mi pubs&abstract=312.] 5. Borisov, A.V. and Mamaev, I.S., The Rolling of Rigid Body on a Plane and Sphere. Hierarchy of Dynamics, Regul. Chaotic Dyn., 2002, vol. 7, no. 1, pp. 177­200. 6. Kilin, A.A., The Dynamics of Chaplygin ball: the Qualitative and Computer Analisis, Regul. Chaotic Dyn., 2001, vol. 6, no. 3, pp. 291--306. 7. Kharlamov, A.P., Topologicheskii analiz integriruemykh zadach dinamiki tverdogo tela (Topological Analysis of Integrable Problems of Rigid Body Dynamics), Leningrad. Univ., 1988. 8. Fedorov, Yu., A Complete Complex Solution of the Nonholonomic Chaplygin Sphere Problem, Preprint, 2007. 9. Kozlov, V.V., On the Integration Theory of Equations of Nonholonomic Mechanics, Adv. in Mech., 1985, vol. 8, no. 3, pp. 85­107 (in Russian). See also: Regul. Chaotic Dyn., 2002, vol. 7, no. 2, pp. 191­176. 0. Markeev, A.P., Integrability of a Problem on Rolling of Ball with Multiply Connected Cavity Filled by Ideal Liquid, Izv. Akad. Nauk SSSR, Mekh. Tverd. Tela, 1986, vol. 21, no. 1, pp. 64­65 (in Russian). 1. Veselov, A.P. and Veselova, L.E., Integrable Nonholonomic Systems on Lie Groups, Math. Notes, 1988, vol. 44, no. 5­6, pp. 810­819 [Mat. Zametki, 1988, vol. 44, no. 5, pp. 604­619]. 2. Fedorov, Yu.N., Motion of a Rigid Body in a Spherical Suspension, Vestnik Moskov. Univ. Ser. I Mat. Mekh., 1988, no. 5, pp. 91­93 [Mosc. Univ. Mech. Bul l., 1988, vol. 43, no. 5, pp. 54­58]. 3. Borisov, A.V. and Fedorov, Y.N., On Two Modified Integrable Problems of Dynamics, Vestnik Moskov. Univ. Ser. I Mat. Mekh., 1995, no. 6, pp. 102­105 (in Russian). 4. Yaroshchuk, V.A., New Cases of the Existence of an Integral Invariant in a Problem on the Rolling of a Rigid Body, Without Slippage, on a Fixed Surface, Vestnik Moskov. Univ. Ser. I Mat. Mekh., 1992, no. 6, pp. 26­30 (in Russian). ¨ 5. Woronetz, P., Uber die rollende Bewegung einer Kreisscheibe auf einer beliebigen Fl¨che unter der a Wirkung von gegebenen Kr¨ften, Math. Annalen., 1909, Bd. 67, S. 268­280. a ¨ 6. Woronetz, P., Uber die Bewegung eines starren K¨rpers, der ohne Gleitung auf einer beliebigen Flache o ¨ rollt, Math. Annalen., 1911, Bd. 70, S. 410­453. 7. Borisov, A.V. and Mamaev, I.S, Conservation Laws, Hierarchy of Dynamics and Explicit Integration of Nonholonomic Systems, Regul. Chaotic Dyn., 2008, vol. 13, no. 5, pp. 443­490. 8. Fedorov, Yu.N, Integration of a Generalized Problem on the Rolling of a Chaplygin Ball, in Geometry, Differential Equations and Mechanics, Moskov. Gos. Univ., Mekh.-Mat. Fak., Moscow, 1985, pp. 151­ 155. 9. Borisov, A.V. and Mamaev, I.S., Rolling of a Non-homogeneous Ball over a Sphere Without Slipping and Twisting, Regul. Chaotic Dyn., 2007, vol. 12, no. 2, pp. 153­159. 0. Borisov, A.V. and Mamaev, I.S., Isomorphism and Hamilton Representation of Some Nonholonomic Systems, Siberian Math. J., 2007, vol. 48, no. 1, pp. 26­36 [Sibirsk. Mat. Zh., 2007, vol. 48, no. 1, pp. 33­45]; arxiv.org/pdf/nlin.SI/0509036. 1. Fedorov, Yu., Classical Integrable Systems and Billiards Related to Generalized Jacobians, Acta Appl. Math., 1999, vol. 55, no. 3, pp. 251­301. 2. Clebsch, A. and Gordan, P., Theorie der abelschen Funktionen, Leipzig: Teubner, 1866. 3. Jacobi, K.G., Sur la rotation d'un corps, Gesamelte Werke , Vol. 2, 1884, pp. 139­172 4. Buchstaber, M., Enol'skii, V.Z., and Leikin, D.V., Kleinian Functions, Hyperel liptic Jacobians and Applications, Amer. Math. Soc. Transl. Ser. 2, vol. 179, Providence, USA, 1997.
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