New variables of separation for particular case of the Kowalevski top

0.17 MB·English

by  A V Tsiganov

New variables of separation for particular case of the
Kowalevski top.
A V Tsiganov
St.Petersburg State University, St.Petersburg, Russia
e–mail: [email protected]
0
Abstract
1
0 Wediscussthepolynomial bi-Hamiltonian structuresfor theKowalevski top in special case of
2
zerosquareintegral. Anexplicitproceduretofindvariablesofseparation andseparationrelations

n is considered in detail.
a
J PACS: 45.10.Na, 45.40.Cc
0 MSC: 70H20; 70H06; 37K10
3
To V.V. Kozlov on the occasion of his 60th birthday

]
I
S
.
n 1 Introduction
i
l
n
During a century the only cases of integrability of the Euler-Poissonequations were the isotropic case
[
and the cases of Euler (1758) and Lagrange (1788). In 1888 S. Kowalevski found a new highly non-

3 trivial case of integrability [8]. In modern terms, this is an integrable system on the e(3) algebra with
v quadratic and quartic (in angular momenta) integrals of motion.
9
Furthermore, by using a mysterious change of variables, she showed that equations of motion
9
for the new case of integrability are linearized on the abelian variety by means of the Jacobi-Abel
5
4 theorem about the inversion of a system of abelian integrals [8]. At the moment no separation which
. is alternative to her original separation of variables is known for this system , even though there is a
1
0 large body of literature dedicated to the problem, including the detailed geometric description of the
0 invariant surfaces on which the motion evolves, see books [1, 2] and references within.
1 Inthispaperwediscussthedirectmethodoffindingvariablesofseparationwithoutanyadditional
:
v information (ingenious and at times obscure change of variables, Lax matrices, r-matrices, links with
i soliton equations etc). For example, we apply the machinery of bi-Hamiltonian geometry to the
X
Kowalevski top at zero level of the cyclic integral of motion, which is a particular case of the generic
r
Kowalevski top. The rational Poisson bivector associated with famous Kowalevski variables may be
a
found in [24]. Here we specially do not consider Kowalevski variables in order to get only the new
variables of separation and the new underlying polynomial Poissonstructures.
Theotheraimistheconstructionofdifferentvariablesofseparationlyingonthedistinctalgebraic
curves [26]. Relations between such distinct curves give us a lot of new examples of reductions of
Abelian integrals and, therefore, they may be the source of new ideas in the number theory, algebraic
geometry and modern cryptography [1, 3, 10].
In Section 2 we construct new compatible Poisson bivectors for the Kowalevski top. In Section
3 we find the new corresponding variables of separation and the separated relations. Finally, some
concluding remarks can be found in the last Section.
1
2 The bi-hamiltonian structure
2.1 Description of the model
According to [8], the Kowalevskitop is a dynamical system with the following integrals of motion
H = J2+J2+2J2+c x , c R, (2.1)
1 1 2 3 1 1 1 ∈
H = (J2+J2)2 2 x (J2 J2)+2x J J c +(x2+x2)c2.
2 1 2 − 1 1 − 2 2 1 2 1 1 2 1
(cid:16) (cid:17)
Here J arethe components ofthe angularmomentum in the moving frameof coordinatesattachedto
i
the principalaxes ofinertia. The positionofa rigidbody is fixed by the components x of the Poisson
i
vector, which are the cosines between the axes of the body frame and the field up to a constant.
Using the Hamiltonfunction H and the Lie-Poissonbracket .,. on the Euclideanalgebrae (3)
1 ∗
{ }
the customary Euler-Poissonequations may be rewritten in the hamiltonian form
J˙ = J ,H , x˙ = x ,H , where f,g = Pdf,dg . (2.2)
i i 1 i i 1
{ } { } { } h i
In coordinates z =(x ,x ,x ,J ,J ,J ) on e (3) the Lie-Poissonbivector P is the following antisym-
1 2 3 1 2 3 ∗
metric matrix
0 0 0 0 x x
3 2
−
0 0 x 0 x
3 1
 ∗ − 
0 x x 0
P = ∗ ∗ 2 − 1 .
 ∗ ∗ ∗ 0 J3 −J2 
 0 J1 
 ∗ ∗ ∗ ∗ 
 0 
 ∗ ∗ ∗ ∗ ∗ 
 
It has two Casimir elements
3 3
PdC =0, C = x2 x2, C = x,J x J . (2.3)
1,2 1 | | ≡ k 2 h i≡ k k
k=1 k=1
X X
After fixing values of the Casimir elements
C =a2, C =b
1 2
onegetsagenericfour-dimensionalsymplecticleaf ,whichistopologicallyequivalenttothe cotan-
ab
O
gentbundleT 2 ofthesphere 2 withradiusa. However,the symplecticstructureof isdifferent
∗ ab
S S O
from the standard symplectic structure on T 2 by the magnetic term proportionalto b [13].
∗
S
The Kowalevskitop is an integrable system on the phase space because the two independent
ab
O
integrals of motion H (2.1) are in the involution
1,2
H ,H = PdH ,dH =0. (2.4)
1 2 1 2
{ } h i
In mechanics the Casimir function C is a norm of the unit Poisson vector such as a = 1, whereas
1
second Casimir function C is called a square or cyclic integral of motion [2, 27].
2
Remark 1 In original Kowalevski work the first step in the separation of variables method is the
complexification: she considers
z =J +iJ , z =J iJ
1 1 2 2 1 2
−
as independent complex variables. Next she makes her famous change of variables
R(z ,z ) R(z ,z )R(z ,z )
1 2 1 1 2 2
s = ± .
1,2 2(z z )2
p1− 2
ThefourthdegreepolynomialsR(z ,z )wewillnotspecifyhere. Itbringsthesystem(2.2)totheform
i k
( 1)k(s s )s˙ = P(s ) , k =1,2,
1 2 k k
− −
p
2
where
K K
P(s)=4 (s H)2 s (s H)2+c2a2 +c2b (2.5)
− − 4 − 1 − 4 1
(cid:18) (cid:19)(cid:20) (cid:18) (cid:19) (cid:21)
Here the pairs s ,η = P(s ) , can be regarded as coordinates of points on the Kowalevski curve
k k k
of genus two (cid:16) p (cid:17)
: η2 P(s)=0. (2.6)
kow
C −
At zero level of the cyclic integral of motion C =0 the Kowalevskicurve has the same genus 2.
2
We address the problem of separation of variables for the Hamilton-Jacobi equation as well. At
C =0thesymplecticleaves arecompletelysymplectomorphictoT 2 [13]. Wewillonlyconsider
2 a0 ∗
O S
such symplectic leaves and, therefore, all the formulae below hold true up to C =0.
2
For this Kowalevskitop on the two-dimensionalsphere we want to calculate different variables of
separation and, according to the general usage of the bi-hamiltonian geometry, firstly we have to find
the second dynamical Poissonbivector P equipped with some necessary properties [5, 9].
′
2.2 Dynamical Poisson bivectors
According to [21, 22, 23, 25] let us suppose that the desired second Poisson bivector P is the Lie
′
derivative of P along some unknown Liouville vector field X
P′ = X(P). (2.7)
L
In addition it has to satisfy the following equations
[P ,P ] [ (P), (P)]=0, (2.8)
′ ′ X X
≡ L L
and
H ,H = P dH ,dH =0, (2.9)
1 2 ′ ′ 1 2
{ } h i
where [.,.] is the Schouten bracket.
The first assumption (2.7) guarantees that this dynamical bivector P is compatible with the
′
given kinematic Poisson bivector P, i.e. [P,P ] = 0. In geometry such bivector P is said to be the
′ ′
2-coboundary associated with the Liouville vector field X in the Poisson-Lichnerowicz cohomology
defined by P.
The second condition (2.8) means that P is the Poisson bivector, i.e. that the Jacobi identity
′
is true. The third equation (2.9) relates P with the given integrable system. In the wake of this
′
agreement the foliation defined by the H is the bi-Lagrangianfoliation [5, 9].
1,2
The system of equations (2.8-2.9) has infinitely many solutions with respect to X [20, 23]. So, in
order to get some particular solution we have to narrow the search space. In this paper we suppose
that
P′dC1,2 =0, (2.10)
and that the components X of the Liouville vector field X = X ∂ are non-homogeneous polyno-
j j j
mials in momenta J
k
N m P
Xj = gjNkm(x1,x2,x3)J1kJ2m−k
m=0k=0
X X
with unknown coefficients g(x ,x ,x ) [21, 22, 25]. Here we explicitly use the restriction C = 0, i.e.
1 2 3 2
that J = (x J +x J )/x .
3 1 1 2 2 3
−
Upon substituting this polynomial ans¨atze into the equations (2.8,2.9-2.10) and demanding that
all the coefficients at powers of J vanish one gets the over determined system of algebro-differential
k
equations. Such systems are solved on personal computer by using modern software in a few seconds.
So, the only real problem is the classification and the analysis of the received computer results.
3
The first three nontrivial solutions arise only in the cubic case N = 3. Components of the first
real vector field X(1) are equal to
x2+x2 x J x J J x2+x2 x J x J J
X(1) = 1 2 1 1− 2 2 3, X(1) = 1 2 1 1− 2 2 3, X(1) =0,
1 −p (cid:16)2x1x3 (cid:17) 2 p (cid:16)2x2x3 (cid:17) 3
(x2+x2)J3 J2J c x J
X(1) = x2+x2 1 2 1 2 3 + 1 3 3 , (2.11)
4 −q 1 2(cid:18) 6x22x23 − 2x1x3(cid:19) 4 x21+x22
(x2+x2)J3 J2J c (x J x J )
X(1) = x2+x2 1 2 2 1 3 + 1p 1 2− 2 1 ,
5 −q 1 2(cid:18) 6x21x23 − 2x2x3(cid:19) 4 x21+x22
X(1) = x2+x2(x21+x22)J33 + c1 x21+x22J1 . p
6 − 1 2 6x2x2 4x
1 2 p 3
q
The components of the second real vector field X(2) read as
2(x2+x2) 2x (x2+x2)
X(2) = 1 2 J J , X(2) = 1 1 2 J J , X(2) =0,
1 x 1 3 2 − x x 1 3 3
3 2 3
(x2+x2)2 x x x2+x2 c x J
X(2) = 1 2 J3 J + 1 3J J2+ 1 2 J J J + 1 2 2 ,
4 3x2x2 1 − 1 3x2 3 3 x x 1 2 3 2
2 3 (cid:18) 2 (cid:19) 2 3
(x2+x2)2 (2x2 x2)x 2(x2+x2)
X(2) = 1 2 J3 J 1− 2 3 J 1 2 J J2 (2.12)
5 3x2x2 2 − 2− 3x2x 3− x x 1 3
1 3 (cid:18) 1 2 1 2 (cid:19)
x2+x2 c (2x J x J )
+ 1 2 J J J 1 1 2− 2 1
1 2 3
x x − 2
1 3
2x2+x2 c x (x J x J )
X(2) = 1 2 J3+ 1 2 1 2− 2 1
6 3x2 3 2x
2 3
The components of the third vector field X(3) are the complex functions on initial variables
ix (x +ix )2 2x (x +ix )
X(3) = 2 1 2 J2+ 2 1 2 J J , i=√ 1,
1 − x2 2 x 1 2 −
1 1
i(x +ix )2
X(3) = 1 2 J2 2(x +ix )J J , X(3) =0, (2.13)
2 x 2 − 1 2 1 2 3
1
1 2(2x +ix )x 4x ix (x J x J )
X(3) = (J iJ )J2 J3+ 1 2 3 J3+ 2J3+ 2 1 1− 3 3 J2
4 1− 2 2 − 3 1 3x2 3 3x 2 x2 2
1 1 1
+ c x J ,
1 3 3
2i 1 2ix x2(2x ix ) ix2x
X(3) = J3 (J iJ )J J J3 3J3+ 2 1− 2 J3 2 3J2J
5 3 1 − 1− 2 1 2− 3 2 − 3x 3 3x3 2 − x13 2 3
1 1
+ ic x J ,
1 3 3
2(x +ix )x2 2x3
X(3) = 1 2 3− 1 J3 (J2+J2)J c (x +ix )J .
6 3 x3 3 − 1 2 3− 1 1 2 3
1
The quartic ansa¨tze yields a lot of solutions, which will be classified and studied in future.
Let us show the simplest part of these real and complex Poissonbrackets explicitly
x2+x2J
x ,x =ε x , x ,x (1) =ε 1 2 3 x ,
i j ijk k i j ijk k
{ } { } x
p 3
2(x2+x2)J
x ,x (2) = ε 1 2 3 x , x ,x (3) =2iε (ix J x J +x J )x .
i j ijk k i j ijk 3 3 1 2 2 1 k
{ } − x { } −
3
Here ε is the totally skew-symmetric tensor. Other brackets are appreciably more tedious expres-
ijk
sions. The complex Poisson structure may be rewritten in the lucid form by using the 2 2 Lax
×
matrices [7, 18] and the bi-hamiltonian structure associated with the reflection equation algebra [24].
ItiseasytoprovethatthecorrespondingPoissonbivectorsP(1), P(2) andP(3) havethefollowing
properties
[P(1),P(2)]=0, [P(1),P(3)]=0, [P(2),P(3)]=0 (2.14)
6 6
4
with respectto the Schouten brackets. It means that P(1) and P(2) are compatible bivectors,whereas
the complex bivector P(3) is incompatible with the real bivectors.
Remark 2 ForanybivectorsP andP there arealotofvectorfields X,suchasP = (P). Above
′ ′ X
L
we put X = 0 in order to restrict this freedom. It may be the origin of some non-symmetry and
3
irregularity in expressions (2.11,2.12) and (2.13).
Remark 3 There are two rational Poisson bivectors P for the Kowalevski top. The first bivector is
′
associated with the Kowalevskivariables of separation and the underlying reflection equation algebra
[24]. The second bivector is related with the Lax matrix of Reyman-Semenov-Tian-Shansky and the
linear r-matrix algebra [19]. The components of the corresponding vector fields X are logarithmic
functions in momenta.
To sum up, using the applicable polynomial ansa¨tze for the Liouville vector field X we got two
compatiblerealcubicbivectorsP(1,2) = (P)andonecomplexcubicbivectorP(3) = (P)for
LX(1,2) LX(3)
theKowalevskitop. AlthoughthesebivectorsaredefinedbyarbitraryvalueofC ,theyarecompatible
2
with the initial Poisson bivector P only for C = 0. The application of this Poisson bivectors will be
2
given in the next section.
3 Calculation of the variables of separation and the separation
relations
A system of canonical variables (q ,...,q ,p ,...,p )
1 n 1 n
q ,q = p ,p =0, q ,p =δ (3.1)
i k i k i k ik
{ } { } { }
will be called separated if there are n relations of the form
∂Φ
i
Φ (q ,p ,H ,...,H )=0 , i=1,...,n , with det =0, (3.2)
i i i 1 n
∂H 6
(cid:20) j(cid:21)
binding together each pair (q ,p ) and H ,...,H .
i i 1 n
ThereasonforthisdefinitionisthatthestationaryHamilton-JacobiequationsfortheHamiltonians
H =α can be collectively solved by the additively separated complete integral
i i
n
W(q ,...,q ;α ,...,α )= W (q ;α ,...,α ), (3.3)
1 n 1 n i i 1 n
i=1
X
where W are found by the quadratures as the solutions of the ordinary differential equations.
i
Theintegralsofmotion(H ,...,H )aretheSt¨ackelseparableintegralsiftheseparationrelations
1 n
(3.2) are given by affine equations in H , that is,
j
n
S (q ,p )H U (q ,p )=0 , i=1,...,n , (3.4)
ij i i j i i i
−
j=1
X
withaninvertiblematrixS. Thefunctions S andU dependonlyononepair(q ,p )ofthecanonical
ij i i i
variables of separation, it means that
S ,q = S ,p = S ,S =0, i=j, (3.5)
ik j ik j ik jm
{ } { } { } 6
and similar to U
U ,q = U ,p = U ,U =0, i=j. (3.6)
i j i j i j
{ } { } { } 6
In this case S is called the St¨ackel matrix, and U the St¨ackel potential.
Remark 4 We have to point outthat the definition of the St¨ackelseparabilitydepends on the choice
of H . Indeed, if (H ,...,H ) are St¨ackel-separable, then H = H (H ,...,H ) will not, in general,
i 1 n i i 1 n
fulfill the affine relations of the form (3.4).
b b
5
Remark 5 Themethodoftheseparationofvariablesforalongtimeservedanimportant,buttechni-
cal role in solving Liouville integrable systems of classicalmechanics. A new, and much more exciting
application of the method came with the development of quantum integrable systems. Because of
the fact that the quantization of the action variables seems to be a rather formidable task, quantum
separationofvariablesbecame aninevitable refuge. In fact, itcanbe successfullyperformedfor many
families of integrable systems with affine separated relations (3.4) only [16, 17].
So, our second step is the calculation of the canonical variables of separation (q ,p ) and of the
i i
separationrelationsΦ (3.2). Accordingto[5,9],thecoordinatesofseparationq areeigenvaluesofthe
i i
recursion operator,which are the so called Darboux-Nijenhuis variables. In order to get the recursion
operator N = P P 1 we have to find restrictions P,P of the Poisson bivectors P and P onto the
′ − ′ ′
symplectic leaves.
We can avobidbthe procedure of restriction usingbthbe n n control matrix F defined by
×
n
P dH=P FdH , or P dH =P F dH , i=1,...,n. (3.7)
′ ′ i ij j
j=1
(cid:0) (cid:1) X
The bi-involutivity of the integrals of motion (2.4,2.9) is equivalent to the existence of F, whereas
the imposed condition (2.10) ensures that F is a non-degenerate matrix. In this case eigenvalues
of this matrix coincide with the Darboux-Nijenhuis variables and we can easily calculate the desired
coordinates of separation q .
i
Moreover,fortheSt¨ackelseparablesystemsthesuitablenormalizedlefteigenvectorsofthecontrol
matrix F form the St¨ackel matrix S
F =S 1diag(q ,...,q )S
− 1 n
which would allow us to get separated relations (3.4).
So, the main problems are the finding of the conjugated momenta p and the construction of the
i
separation relations φ (3.2) for the generic non-St¨ackel separable systems. Below we show how we
j
can solve these problems using the same control matrix F and some additional useful observations.
3.1 The real compatible Poisson bivectors
For the first Poisson bivector P(1) (2.11) the entries of the control matrix F(1) read as
(2x2+2x2+x2)(J2+J2) 1
F(1) = 1 2 3 1 2 F(1) =
11 4x2 x2+x2 12 −8 x2+x2
3 1 2 1 2
F(1) = (2x21+2x22p+x23)(J12+J22) c1 x21+x22px1(J12−J22)+2x2J1J2 c21 x21+x22
21 2x23 x21+x22 − p (cid:0) x23 (cid:1) − p 2
J2+J2
F(1) = 1 2p
22 −2 x2+x2
1 2
The eigenvalues q pof this matrix are the required variables of separation q
1,2 1,2
det(F(1) λI) = (λ q )(λ q )
1 2
− − −
= λ2 x21+x22(J12+J22)λ c1 2x1(J12−J22)+4x2J1J2+c1x23 .
− 2x2 − (cid:16) 16x2 (cid:17)
p 3 3
The matrix of normalized eigenvectors of F(1) does not form the St¨ackel matrix, because property
(3.5) is missed, and the underlying separation relations differ from the St¨ackel affine equations (3.4)
in H .
1,2
6
For the second Poisson bivector P(2) (2.12) the entries of the control matrix F(2) are equal to
J2+J2 c x (x J x J )2 1
F(2) = 1 2 − 1 1 + 1 2− 2 1 , F(2) = ,
11 − 2 x2 12 4
3
F(2) = (J2+J2)2 1+ 2(x21+x22) +c (x2+x2) 2 x1(J12−J22)+2x2J1J2 +c ,
21 − 1 2 x2 1 1 2  (cid:16) x2 (cid:17) 1
(cid:18) 3 (cid:19) 3
J2+J2+2J2+c x  
F(2) = 1 2 3 1 1 .
22 2
The eigenvalues f of the matrix F(2) are the roots of the equation
1,2
det(F(2) λI) = (λ f )(λ f )
1 2
− − −
(x J x J x J )(x J x J +x J )
= λ2 c x 2 1− 1 2− 3 3 2 1− 1 2 3 3 λ
− 1 1− x2
(cid:18) 3 (cid:19)
(2(x J x J )J c x x )2
2 1 1 2 3 1 2 3
− − .
− 4x2
3
Remark 6 Accordingto[5,9]thecompatibilityofP(1,2) (2.14)ensuresthatintheDarboux-Nijenhuis
variables q,p the corresponding restrictions of P(1,2) look like
0 0 q 0 0 0 f 0
1 b 1
0 0 0 q 0 0 0 f
P(1) = 2 , P(2) = 2 .
q 0 0 0 f 0 0 0
1 1
− −
 0 q2 0 0   0 f2 0 0 
b  −  b  − 
   
where f are the functions on q,p such as
1,2
q ,f = p ,f =0, i=j.
i j i j
{ } { } 6
So, f is the function only on q and p and similar f is the function on q and p .
1 1 1 2 2 2
We can find these functions f using the Poisson bracket. Namely, it is easy to see that the
1,2
recurrence chain
φ = f (q ,p ),q , φ = φ ,q , ..., φ = φ ,q (3.8)
1 1 1 1 1 2 1 1 i i 1 1
{ } { } { − }
breaks down on the third step φ = 0. It means that f is the second order polynomial in momenta
3 1
p and, therefore, we can define this unknown momenta in the following way
1
2x 4(x J x J )q +c x2+x2J
φ1 3 2 1− 1 2 1 1 1 2 2
p = = (3.9)
1 φ2 (4 x2+x(cid:16)2(J2+J2)q +c x (Jp2 J2)+2x(cid:17) J J
1 2 1 2 1 1 1 1 − 2 2 1 2
p (cid:16) (cid:17)
up to canonical transformations p p +g(q ).
1 1 1
→
The similar calculation for the function f (q ,p ) yields the definition of the second momenta
2 2 2
2x 4(x J x J )q +c x2+x2J
3 2 1− 1 2 2 1 1 2 2
p = . (3.10)
2 (4 x2+x(cid:16)2(J2+J2)q +c x (Jp2 J2)+2x(cid:17) J J
1 2 1 2 2 1 1 1 − 2 2 1 2
p (cid:16) (cid:17)
In fact we have to substitute q instead of q only.
2 1
So,onegetsthe canonicaltransformationfromthe initialphysicalvariables(x,J)tothevariables
of separation (q,p) using a pair of compatible bivectors P(1,2) and the corresponding control matrices
F(1,2).
In these separated variables entries of the matrix S of normalized eigenvectorsof F(1) depend on
the pair of variables (q ,p ) and the Hamilton function
i i
c2
S = 2H 4q2 1 p2, S =1, i=1,2.
i1 − 1− i − 4 i 1,2
(cid:18) (cid:19)
7
In this case S may be called the generalized St¨ackel matrix. The separated relations (3.4) look like
(c2 16q )2p4
S H +H H2 1− i i a2(c2 16q2) =0, i=1,2.
i1 1 2− 1 − 64 − 1− i
(cid:18) (cid:19)
and, therefore, the generalized St¨ackel potential U depends on the Hamilton function too.
i
So, we can say that the variablesof separation(q ,p ) lie on the algebraichyperelliptic curve of
i i
C
genus three defined by
(c2 16q2)p2 (c2 16q2)p2
: Φ(q,p) = 1− H H 1− H + H
1 2 1 2
C 8 − − 8 −
(cid:18) (cid:19)(cid:18) (cid:19)
a2(c2 16q2)=0. p p (3.11)
− 1−
This equation is invariant with respect to involution (q,p) ( q,p). Factorization with respect to
→ −
this involution give rise to elliptic curve
(c2 16z)p2 (c2 16z)p2
: Φ(z,p) = 1− H H 1− H + H
1 2 1 2
E 8 − − 8 −
(cid:18) (cid:19)(cid:18) (cid:19)
a2(c2 16z)=0, zp=q2 p (3.12)
− 1−
Due to the standard formalism we have to calculate differential on this curve
dz
Ω= , Z(z,p)=p(c2 16z) 8H p2(c2 16z) .
Z(z,p) 1− 1− 1−
(cid:16) (cid:17)
Then it’s easy to prove that
q˙ q˙ (c2 16q2)p2q˙ (c2 16q2)p2q˙ 1
1 + 2 =0, 1− 1 1 1 + 1− 2 2 2 = (3.13)
Z(q2,p ) Z(q2,p ) Z(q2,p ) Z(q2,p ) −4
1 1 2 2 1 1 2 2
and
q1 dq q2 dq
+ =β ,
Z(q2,p) Z(q2,p) 1
Z Z
q1 (c2 16q2)p2dq q2 (c2 16q2)p2dq t
1− + 1− = +β
Z(q2,p) Z(q2,p) −4 2
Z Z
where p has to be obtained from (3.11).
Remark 7 The equationsofmotion arelinearizedonanabelianvariety,whichis roughlyspiking the
complexified of the corresponding Liouville real torus. So, even though q are the real variables of
1,2
separation we have to solve the Jacobi inversion problem over the complex field, see more detailed
discussion in [1, 4].
Remark 8 We have to point out the Kowalevski separation of variables leading to hyperelliptic
quadratures, whereas in the new variables of separation q equations of motion are integrable by
1,2
quadratures in terms of elliptic functions.
ThethirdpartoftheJacobimethodconsistsoftheconstructionofnewintegrablesystemsstarting
withknownvariablesofseparationandsomeotherseparatedrelations[6]. Ifwesubstituteourvariables
of separation (q,p) into the following deformation of (3.11)
Φ(d)(p,q)=Φ(p,q) 8d q 16d q2 =0, d ,d R, (3.14)
1 2 1 2
− − ∈
we get the following generalization of the initial Hamilton function
d d
H(d) =J2+J2+2J2+c x2+ 1 + 2 (J2+J2). (3.15)
1 1 2 3 1 1 x2+x2 x2 1 2
1 2 3
Herethemainproblemishowtogetthe Hamiltonianptobeinterestingtophysics. Forexample,inour
case we obtained the naturalHamiltonian atd =0 only. For this systemonly the integralsofmotion
2
have been known [27].
8
The apparentproblemis that generalizedequations (3.14) have not involution (q,p) ( q,p) at
→ −
d = 0. Thereby equations of motion are related with the hyperelliptic curve of genus three [11, 12]
1
6
instead of elliptic curve. Nevertheless, at d =0, d =0 we have the same equations (3.13) as above
1 2
6
q˙ q˙
1 2
+ = 0,
Z(q2,p ) Z(q2,p )
1 1 2 2
(3.16)
(c2 16q2)p2q˙ (c2 16q2)p2q˙ 1
1− 1 1 1 + 1− 2 2 2 =
Z(q2,p ) Z(q2,p ) −4
1 1 2 2
where p satisfy to the deformed equations (3.14).
1,2
3.2 The complex Poisson bivector
For the cubic in momenta Poissonbivector P(3) (2.13) the control matrix is equal to
1
2(J2+J2+2J2)+c (x +ix )
F(3) = 1 2 3 1 1 2 −2
2(J2+J2)2 2c (x ix )(J +iJ )2 0 !
1 2 − 1 1− 2 1 2
and the Darboux-Nijenhuis coordinates λ are the roots of the characteristic polynomial
1,2
F(3)
det(F λI)=(λ λ )(λ λ )=λ2 F(3)λ+ 21 . (3.17)
− − 1 − 2 − 11 2
As above we can get the conjugated momenta µ by using compatible with P(3) bivector of fourth
1,2
order in momenta J . However we can do it without such calculations as well.
k
It is easy to see that in this case matrix S of normalized eigenvectors of F(3) is the standard
St¨ackel matrix
S = 2λ , S =1, i=1,2,
i1 i i,2
−
and, therefore, the St¨ackel potentials
U = (S H +H )
1,2 i1 1 2
−
are some functions on (λ ,µ ) and (λ ,µ ), respectively.
1 1 2 2
In fact notion of the St¨ackel potentials allows us to find the unknown conjugated momenta µ
1,2
using the Poisson brackets only. Namely, the following recurrence chain of the Poissonbrackets
φ = λ ,U , φ = λ ,φ ,..., φ = λ ,φ (3.18)
1 1 1 2 1 1 i 1 i 1
{ } { } { − }
is a quasi-periodic chain
φ =16λ φ .
3 1 1
It means that the St¨ackel potential U is a trigonometric function on momenta µ and, therefore, we
1 1
can determine this desired momenta
µ =ϕ(λ )ln 16λ φ +φ
1 1 1 1 2
(cid:16)p (cid:17)
up to canonical transformations µ µ +g(λ ). Here the function ϕ(λ ) is easily calculated from
1 1 1 1
→
λ ,µ =1.
1 1
{ }
These variables of separation(λ ,µ ) lie on the hyperelliptic curve of genus three
i i
a4c4
: Φ(λ,µ)=e4i√λµ+ 1 e 4i√λµ+λ2 2H λ+H =0. (3.19)
− 1 2
C 16 −
According to [10], ethis cureve are related with an elliptic curve and equations of motion for the
C E
Kowalevskitop are linearized on the corresponding abelian variety.
One main difference is thaet the variables of separation λ1,2 aree complex functions on the initial
variables (x,J), whereas q are real functions on them. It will be important when we express initial
1,2
real variables via real or complex variables of separationafter solving of the Jacobi inversion problem
over the complex field.
The other difference is that the affine relations of separations (3.19) allows us to study quantum
counterpartoftheKowalevskitop[7,17]. Forrealvariablesofseparationtheprocedureofquantization
is unknown.
9
Remark 9 In framework of the Sklyanin formalism [16] variables of separation are the poles of the
Baker-Akhiezer function with suitable normalization. In [7, 18] we find such variables of separation
u for the Kowalevski-Goryachev-Chaplygingyrostat
1,2
c
H =J2+J2+2J2+ρJ +c x +c (x2 x2)+c x x + 4 (3.20)
1 1 2 3 3 1 1 2 1− 2 3 1 2 x2
3
using 2 2 Lax matbrix, its Baker-Akhiezer vector-function and the reflection equation algebra.
×
It iseasyto provethat the Darboux-Nijenhuisvariablesλ (3.17)arerelatedwiththe polesu
1,2 1,2
of the Baker-Akhiezer function by the following point transformation
λ =u2 , (3.21)
1,2 1,2
which gives rise to a ramified two-sheeted covering of , see [10].
C
e
4 Conclusion
Starting with the integrals of motion for the Kowalevski top we found three polynomial in momenta
Poisson bivectors, which are compatible with the canonical Poisson bivector on the cotangent bundle
T 2 of two-dimensional sphere.
∗
S
Then in framework of the bi-hamiltonian geometry we get new real variables of separation (q,p)
for the Kowalevski top on the sphere and reproduce known complex variables (λ,µ). These variables
are related by the canonical transformation
λ =λ (q ,q ,p ,p ), µ =µ (q ,q ,p ,p ),
1,2 1,2 1 2 1 2 1,2 1,2 1 2 1 2
which may be rewritten as a quasi-point canonical transformation [14]
λ =λ (q ,q ,H ,H ),
1,2 1,2 1 2 1 2
which relates two hyperelliptic curves of genus three. We can assume that it is no rational cover and
that these Jacobians are non-isogeneous in Richelot sense [15]. Similar transformations relate these
curveswiththe Kowalevskicurveofgenustwo. Furtherinquiryofsuchrelationsbetweenhyperelliptic
curves goes beyond the scope of this paper, see discussion in [1, 3, 10].
The proposed approach may be useful for the investigationof other integrable systems with inte-
gralsofmotionhigherorderinmomenta,forinstance,thesearchofanotherrealvariablesofseparation
for the Kowalevski-Goryachev-Chaplygingyrostat and its various generalizations [27].
5 Acknowledgement
We would like to thank Yu.N. Fedorov and A.V. Bolsinov for helpful discussions.
References
[1] M. Audin, Spinning Tops, a Course on Integrable Systems, (Cambridge Studies in Advanced
Mathematics vol. 51, Cambridge: Cambridge University Press, 1996.
[2] A.V. Borisov, I.S. Mamaev, Rigid Body Dynamics. Hamiltonian Methods, Integrability, Chaos,
Moscow-Izhevsk,RCD, 2005.
[3] Computational aspects of algebraic curves, edited by Tanush Shaska. Lecture Notes Series on
Computing, v.13, World Scientific Publishing Co. Pte. Ltd., Hackensack,NJ, 2005.
[4] B. A. Dubrovin, Riemann Surfaces and Nonlinear Equations, AMS, 2002.
[5] G. Falqui, M. Pedroni, Separation of variables for bi-Hamiltonian systems, Math. Phys. Anal.
Geom., v.6, p.139-179,2003.
10