On the generalized integrable Chaplygin system

0.16 MB·English

by  A V Tsiganov

On the generalized integrable Chaplygin system.
A V Tsiganov
St.Petersburg State University, St.Petersburg, Russia
e–mail: [email protected]
Abstract
0 We discuss two polynomial bi-Hamiltonian structures for the generalized integrable
1 ChaplyginsystemonthesphereS2withanadditionalintegraloffourthorderinmomenta.
0 An explicit procedure to find thevariables of separation, theseparation relations and the
2
transformation of the corresponding algebraic curvesof genus two is considered in detail.

n
PACS: 45.10.Na, 45.40.Cc
a
J MSC: 70H20; 70H06; 37K10

0
1
1 Introduction

]
I
S We address the problemof the separationofvariables for the Hamilton-Jacobiequation within
. thetheoreticalschemeofbi-hamiltoniangeometry. Themainaimistheconstructionofdifferent
n
variablesofseparationforthe givenintegrablesystemwithoutanyadditionalinformation(Lax
i
l matrices, r-matrices, links with soliton equations etc.)
n
[ Different variables of separationmay be useful in different perturbation theories,aswell as

distinctproceduresofquantizationandvariousmethodsofqualitativeanalysisetc. Frommath-
1
ematicalpointofviewthenotionofdifferentvariablesofseparationallowsustostudyrelations
v
7 between distinct algebraiccurves associatedwith the correspondingseparatedequations. Such
0 relations give us a lot of new examples of reductions of Abelian integrals and, therefore, they
5 may be a source of new ideas in number theory, algebraic geometry and modern cryptography.
1
The paper is organized as follows. In Section II, the necessary aspects of bi-hamiltonian
.
1 geometry are briefly reviewed. Then, we discuss a possible application of these methods to
0 calculation of the polynomial bi-hamiltonian structures for the given Chaplygin system. In
0
Section III, the problem of finding variables of separation and corresponding separation rela-
1
tions is treated and solved. Some applications of real and complex variables of separation are
:
v discussed in the final Section.
i
X
r 2 The bi-hamiltonian structure of the Chaplygin system
a
In order to get variables of separation according to general usage of bi-hamiltonian geometry
firstly we have to calculate the bi-hamiltonian structure for the given integrable system with
integralsofmotionH ,...,H onthe PoissonmanifoldM withthekinematic Poissonbivector
1 n
P and the Casimir functions C ,...,C [3, 8, 13, 14, 17].
1 k
Let us consider the generalized Chaplygin system [1, 5, 10] with the following integrals of
motion
c
H = J2+J2+2J2+c (x2 x2)+ 4, c ,c R, (2.1)
1 1 2 3 2 1− 2 x2 2 4 ∈
3
2
c
H = J2+J2+ 4 +2c x2(J2 J2)+c2x4.
2 1 2 x2 2 3 1 − 2 2 3
(cid:18) 3(cid:19)
1
InthisstandardcoordinatesJ =(J ,J ,J )andx=(x ,x ,x )ontheEuclideanalgebrae (3)
1 2 3 1 2 3 ∗
the Poisson bivector looks like the following antisymmetric 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 =a2 x2, C = x,J x J . (2.2)
1,2 1 ≡ k 2 h i≡ k k
k=1 k=1
X X
At C = x,J =0 this integrals of motion are in involution
2
h i
H ,H = PdH ,dH =0.
1 2 1 2
{ } h i
andthe correspondingsymplectic leavesareequivalentto cotangentbundle T 2 ofthe sphere
∗
S
with the radius a = x. We will consider such symplectic leaves only, and, therefore, all the
| |
formulae below hold true up to C =0.
2
Followingto [12,13,14,17]wewillsupposethatthe desiredsecondPoissonbivectoristhe
Lie derivative of P along some unknown Liouville vector field X
P = (P) (2.3)
′ X
L
which has to satisfy to the equations
H1,H2 ′ = P′dH1,dH2 =0, [P′,P′]=[ X(P), X(P)]=0, (2.4)
{ } h i L L
where [.,.] means the Schouten bracket.
Obviously enough, in their full generality equations (2.4) are too difficult to be solved
because it has infinitely many solutions [11, 14]. In order to get some particular solutions we
will use the additional assumption
P dC =0, (2.5)
′ 1,2
and polynomial in momenta J ans¨atze for the components Xj, j = 1,...,6,of the Liouville
vector field X = Xj∂ [17, 12, 13].
j
Substitutingthisans¨atzeintotheequations(2.4-2.5)anddemandingthatallthecoefficients
P
at powers of J vanish one gets the over determined system of algebro-differential equations on
functions of x which can be easily solved in the modern computer algebra systems.
For the linear ansa¨tze one gets the trivial solutions P = P only. The quadratic ansa¨tze
′
yields first non-trivial solution
W c J
3 2 1
X = W ,W ,W ,0,0,0 0,0,0,x ,0, x , W =J x,
a2x 1 2 3 − x 3 − 1 ×
3 3
(cid:16) (cid:17) (cid:16) (cid:17)
whereW is a crossproductofJ andx. This Liuville vectorfieldgivesriseto the followingreal
Poisson bivector
0 J x2J3 J1W1 J2W1 0 0 0 0 0 0 0
03 −x1xJ33 J1xW232 J2xW232 0 0 0 x3 0 x1
 ∗ x3 x23 x23   ∗ 0 x 0 −x1x2 
0 J1W3 J2W3 0 ∗ ∗ − 2 x3
P′ = ∗∗∗ ∗∗∗ ∗∗ x∗023 x0032 00 −c2 ∗∗ ∗∗ ∗∗ 0∗ xx20J32 x23Jx21+xxxx232321Jx22J1 ,
 0   0 
 ∗ ∗ ∗ ∗ ∗   ∗ ∗ ∗ ∗ ∗ 
   (2.6)
2
For the cubic ansa¨tze we get one real Poissonbivector compatible with (2.6) and two new
hermitian conjugated bivectors P and P . This new and more cumbersome solution reads as
′ ′∗
P =x (W +ix J ), P = x (W +ix J ), P =x (W +ix J )
1′2 3 3 3 3 1′,3 − 2 3 3 3 2′,3 1 3 3 3
x2(c x4 c )
P1′4 = J2W1−ix3J1J3−√c2x3(x2J3−W1)+ 2 2x33− 4
3
ix (ix2 2x x )(c x4+c )
P1′5 = −(J1+iJ2)W1− 23(J12−J22)+i√c2(x2J3−W1)x3− 3− 122x3 2 3 4
3
ix (T ic (ix2 4x x )) ic x
P1′6 = −iJ3W1+ 2 − 2 23− 1 2 −√c2(W1−x2J3)(x1−ix2)+ 24x22
3
x (ix2+2x x )(c x4 c )
P2′4 = −i(J1+iJ2)W2− 23(J12−J22)+√c2x3(x1J3+W2)− 3 122x3 2 3− 4
3
x2(c x4+c )
P2′5 = −J1W2−ix1J2J3−i√c2x3(x1J3+W2)− 1 2x33 4 (2.7)
3
ix (T ic (ix2+4x x ) ic x
P2′6 = −iJ3W2− 1 − 2 2 3 1 2 −√c2(x1J3+W2)(x1−ix2)− x421
3
ix (J2+J2) ix (c x4 c )
P3′4 = J2W3+ 2 12 2 +√c2x3W3+ 2 22x32− 4
3
ix (J2+J2) ix (c x4+c )
P3′5 = −J1W3− 1 12 2 −i√c2x3W3+ 1 22x32 4
3
P3′6 = −iJ3W3−√c2W3(x1−ix2)−ic2x1x2x3
i(J2+J2)J (x ix )c
P4′5 = 1 2 2 3 +√c2 c2x23(x1+ix2)+ 1−x2 2 4 −x3J3(J1−iJ2)
(cid:18) 3 (cid:19)
ix (J +2iJ )(c x4+c ) ix (J +2iJ )(c x4 c )
1 1 2 2 3 4 2 2 1 2 3− 4
− 2x3 − 2x3
3 3
iJ T c x
2 4 2
P4′6 = 2 −√c2 c2x2x3(x1+ix2)−(x1−ix2) J2J3− x3
(cid:18) (cid:16) 3 (cid:17)(cid:19)
ic J iJ c
x x 2c J + 4 1 2(x2+2x2) c 4
− 1 2 2 2 x4 − 2 3 2 2− x4
(cid:18) 3 (cid:19) (cid:18) 3(cid:19)
iJ T c x
1 4 1
P5′6 = − 2 +√c2 c1x1x3(x1+ix2)−(x1−ix2) J1J3− x3
(cid:18) (cid:16) 3 (cid:17)(cid:19)
ic J iJ c
+ x x 2c J 4 2 1(x2+2x2) c + 4 .
1 2 2 1− x4 − 2 3 1 2 x4
(cid:18) 3 (cid:19) (cid:18) 3(cid:19)
Here T = J2 +J2 +2J2 and i = √ 1. This cubic Poisson structure may be rewritten in
1 2 3 −
lucid formby using 2 2Laxmatrices forthe Chaplyginsystem[5, 10]andthe bi-hamiltonian
×
structure associated with the reflection equation algebra [15].
Thequarticansa¨tzeyieldsalotofsolutions,whichwillbeclassifiedandstudiedatafuture
date.
Tosumup, usingapplicableansa¨tzefortheLiouvillevectorfieldX wegetarealrelatively
simple quadratic bivector (2.6) and a more complicated complex cubic bivector (2.7). Modern
computer software allows to do it on a personal computer wasting only few seconds. The
application of this Poisson bivectors will be given in the next section.
3
3 Variables of separation and separation relations
Thesecondstepinthebi-hamiltonianmethodofseparationofvariablesiscalculationofcanon-
ical variables of separation (q ,...,q ,p ,...,p ) and separation relations of the form
1 n 1 n
∂φ
i
φ (q ,p ,H ,...,H )=0 , i=1,...,n , with det =0. (3.1)
i i i 1 n
∂H 6
(cid:20) j(cid:21)
The reason for this definition is that the stationary Hamilton-Jacobi equations for the Hamil-
tonians H can be collectively solved by the additively separated complete integral
i
n
W(q ,...,q ;α ,...,α )= W (q ;α ,...,α ), (3.2)
1 n 1 n i i 1 n
i=1
X
where W are found by quadratures as solutions of ordinary differential equations.
i
According to [3, 13, 17], separatedcoordinatesq are the eigenvaluesof the controlmatrix
j
F defined by
P′dH=P FdH .
Its eigenvalues coincide with the Darboux-Nijenh(cid:0)uis co(cid:1)ordinates (eigenvalues of the recursion
operator) on the corresponding symplectic leaves. Using control matrix F we can avoid the
procedure of restriction of the bivectors P and P on symplectic leaves, that is a necessary
′
intermediate calculation for the construction of the recursion operator [3].
Recall that independent integrals of motion (H ,...,H ) are St¨ackel separable if the cor-
1 n
responding separation relations are given by the affine equations in H , that is,
j
n
S (p ,q )H U (p ,q )=0 , i=1,...,n , (3.3)
ij i i j i i i
−
j=1
X
with S being an invertible matrix. Functions S and U depend only on one pair (p ,q ) of
ij i i i
canonical variables of separation, thus it means that
S ,q = S ,p = S ,S =0, i=j, (3.4)
ik j ik j ik jm
{ } { } { } 6
and
U ,q = U ,p = U ,U =0, i=j. (3.5)
i j i j i j
{ } { } { } 6
In this case S is called a St¨ackel matrix, and U - a St¨ackel potential.
For St¨ackel separable systems the suitable normalized left eigenvectors of control matrix
F form the St¨ackel matrix S [3]
F =S 1diag(q ,...,q )S.
− 1 n
However, we have to notice that the definition of St¨ackel separability depends on the choice
of H . Indeed, if (H ,...,H ) are St¨ackel-separable, then H = H (H ,...,H ) will not, in
i 1 n i i 1 n
general, fulfill relations of the form (3.3).
So,themainproblemsarethefindingoftheconjugatedmbomentbap ,suchthat p ,q =δ
i i j ij
{ }
and the constructionof separationrelationsφ (3.1) for generic non-St¨ackelseparable systems.
j
We show below how we can solve these problems using the same control matrix F with the
addition of some useful observations.
3.1 The quadratic Poisson bivector
According to [3], the bi-involutivity of integrals of motion
H ,H = H ,H =0
1 2 1 2 ′
{ } { }
4
is equivalent to the existence of the non-degenerate control matrix F such that is
2
P dH=P FdH , or P dH =P F dH , i=1,2. (3.6)
′ ′ i ij j
j=1
(cid:0) (cid:1) X
For the quadratic in momenta Poisson bivector P (2.6) control matrix F reads as
′
1 J2+J2 c 1
1 2 +c 4
2 x2 2− x4 4x2
 (cid:18) 3 3(cid:19) 3 
F = (3.7)
 (J12+J22)2 +2c (J2 J2)+c2x2 c24 1 J12+J22 +c + c4 
 x2 2 1 − 2 2 3− x6 2 x2 2 x4 
 3 3 (cid:18) 3 3(cid:19) 
 
The eigenvalues of this matrix are the required variables of separation q
1,2
J2+J2 c J2
det(F λI)=(λ q )(λ q )=λ2 1 2 +c λ+ 2 2. (3.8)
− − 1 − 2 − x2 2 x2
(cid:18) 3 (cid:19) 3
IfthecorrespondingseparatedrelationsareaffineequationsinH thenthesuitablenormalized
1,2
left eigenvectors of F form the St¨ackel matrix S
F =S 1diag(q ,q )S.
− 1 2
In our case the matrix of normalized eigenvectors
s s 2c
S = 1 2 , s = 4 2 (J2+J2)2+2c x2(J2 J2)+c2x4 (3.9)
1 1 1,2 − x2 ± 1 2 2 3 1 − 2 2 3
(cid:18) (cid:19) 3 q
does not form the standard St¨ackel matrix because
q ,s =0, and s ,s =0.
i j 1 2
{ }6 { }6
Itmeansthattheunderlyingseparationrelationsdo notformthe St¨ackelaffineequations(3.3)
inH . SubstitutingfunctionsontheintegralsofmotionH =f (H ,H )intotheequation
1,2 1,2 1,2 1 2
(3.6)
P dH=P FdH , b
′
we can try to get a new control matrix F which(cid:0)satisfi(cid:1)es the St¨ackel properties (3.4). In the
b b b
Chaplygin case it is a very simple calculation which yields to the following results
b
H =H , H =H H2,
1 1 2 2− 1
and b b
J2+J2+J2 c (x2 x2+x2) 1
F = 1 2 3 + 2 1− 2 3 , F = ,
11 x2 x2 12 4x2
3 3 3
Fb = 4J32(J12+J22+J32) +2c J2 J2 b(x21−x22)(J12+J22+2J32)
21 − x2 2 1 − 2 − x2
3 (cid:18) 3 (cid:19)
b c22(x21−x22+x23)(x21−x22−x23),
− x2
3
J2 c (x2 x2 x2)
F = 3 2 1− 2− 3 ,
22 −x2 − x2
3 3
so that b
S = s1 s2 , s =2 H c4 2 (J2+J2)2+2c x2(J2 J2)+c2x4 .
1 1 1,2 1− x2 ± 1 2 2 3 1 − 2 2 3
(cid:18) (cid:19) (cid:18) 3(cid:19) q
b b
b b b
5
The St¨ackel conditions (3.4) are fulfilled and, therefore, s is a function on the separated
1
coordinate q defined by (3.8) and the unknown conjugated momenta p . It is easy to see that
1 1
the recurrence chain
b
φ = s ,q , φ = φ ,q , ..., φ = φ ,q (3.10)
1 1 1 2 1 1 i i 1 1
{ } { } { − }
breaks down on the third step φ =0. It means that s is the second order polynomial in the
3 1
b
momenta p and, therefore, we can define this unknown momenta in the following way
1
φ x (x J x Jb)q +c x J
1 3 2 1 1 2 1 2 1 2
p = = − (3.11)
1 φ − 2 (J2+J2)q c J2
2 1 2 1− 2 2
up to the canonical transformations p p +g(q ).
1 1 1
→
The similar calculation for s yields definition of the second momenta
2
x (x J x J )q +c x J
3 2 1 1 2 2 2 1 2
p2 =b− 2 (J2−+J2)q c J2 . (3.12)
1 2 2− 2 2
So, one gets canonical transformation from initial physical variables x,J to the variables of
separation p,q (3.8,3.11,3.18).
The inverse mapping looks like
q q q q 2 q q (q c )(c q )(p p )
1 2 1 2 1 2 1 2 2 2 1 2
J = q +q c x , J = x , J = − − −
1 1 2 2 3 2 3 3
− − c c q q
r 2 r 2 p 1− 2
(3.13)
where x = a2 x2 x2 and
3 − 1− 2
p
q q p q p q c (p p ) (q c )(c q ) p q p q
1 2 1 1 2 2 2 1 2 1 2 2 2 1 1 2 2
x =2 − − − , x = 2 − − − .
1 2
r c2 q1−q2 − s c2 q1−q2
Using the elements of the St¨ackel matrix
s =2 8(c q )q p2 a2(c 2q )
1,2 2− 1,2 1,2 1,2− 2− 1,2
(cid:16) (cid:17)
we can easily derive the required affine St¨ackel separated relations
b
s2
s H +H = k 2c (c 2q ), k=1,2.
k 1 2 4 2 k
4 − −
If we come back tobinibtial inbtegralbs of motion H1,2 then these separation relations go over
to the equation
: Φ(q,p) = 8q(c q)p2 a2(c 2q) H + H (3.14)
2 2 1 2
C − − − −
(cid:16)8q(c q)p2 a2(c 2q) H pH (cid:17) 2c (c 2q)=0.
2 2 1 2 4 2
× − − − − − − −
(cid:16) p (cid:17)
Proposition 1 The variables of separation (q ,p ) lie on the hyperelliptic algebraic curve of
i i
C
thegenustwog=2definedby(3.14)andtheequations ofmotion arelinearized onitsJacobian.
Remark 1 The separation relations (3.14) may be obtained in a framework of the St¨ackel
formalizm(3.3)byusinggeneralized St¨ackelmatrixS (3.9),whoseentriesS dependon(q ,p )
ij i i
and on the integrals of motion
s =s (p ,q ) 2H
i i i i 1
−
Remark 2 At c = 0 we reproduced the Chaplygin result so that (q ,p ) lie on a pair of the
4 i i
b
elliptic curves of genus one
: 8q c q)p2 a2(c 2q) H H =0. (3.15)
1,2 ( 2 2 1 2
C − − − − ±
Remind that every algebraic curve of genus one is isomorphic topa real torus.
6
3.2 The cubic Poisson bivector
For the cubic in momenta Poissonbivector P (2.7) the entries of control matrix F look like
′
ic
4
F = iT +√c x (J iJ ) 2(x ix )J +2c x x ,
11 − 2 3 1− 2 − 1− 2 3 2 1 2− x2
3
(cid:0) c c (cid:1)
F = i J2+J2+ 4 J2+J2+ 2 +2√c x i(J iJ ) +ic2x4
21 − 1 2 x2 1 2 x2 2 3 1− 2 2 3
(cid:18) 3(cid:19)(cid:18) 3 (cid:19)
+ 2c x2 √c x (J +iJ )+2J J ,
2 3 2 3 1 2 1 2
i (cid:16) (cid:17)
F = , F =0.
12 22
4
The separated variables λ are the roots of the characteristic polynomial
1,2
i
det(F λI)=(λ λ )(λ λ )=λ2 F λ F . (3.16)
1 2 1,1 21
− − − − − 4
In the next step we have to find the conjugated momenta to these Darboux-Nijenhuis coordi-
nates.
In[5,10]we’vefoundthe separatedcoordinatesQ andthe correspondingmomenta P
1,2 1,2
for the Kowalevski-Goryachev-Chaplygingyrostat using 2 2 Lax matrix, its Baker-Akhiezer
×
vector-function and the reflection equation algebra. In the framework of Sklyanin formalism
the separated coordinates are the poles of Baker-Akhiezer function with standard simplest
normalization,whereastheseparatedmomentaareexpressedthroughthevaluesoftheelements
of the Lax matrix in the poles.
Itiseasytoseethatthe Darboux-Nijenhuisvariablesλ (3.16)arerelatedwiththe poles
1,2
Q of the Baker-Akhiezer function by the following point transformation
1,2
i
λ = Q2 . (3.17)
1,2 −2 1,2
In this case the additional knowledge of the Lax matrix allows us to introduce conjugated to
λ momenta
k
iP 1
k
µ = , P = lnB(u) , (3.18)
k k
Q 2i
k (cid:12)u=Qk
(cid:12)
where B(u) is the diagonal element of the corresponding La(cid:12)x matrix [10]
(cid:12)
B(u) = (x ix )u3+ 2J i√c (x ix ) (x ix ) (J iJ )x u2
1 2 3 2 1 2 1 2 1 2 3
− − − − − − −
+ 2x J (J iJ )(cid:16)+(cid:0) J2+J2+ c4 (x (cid:1) ix ) c (x +ix )x2 (cid:17)u
3 3 1− 2 1 2 x2 1− 2 − 2 1 2 3
(cid:18) (cid:18) 3(cid:19) (cid:19)
c
+ ic3/2x4+c x3(J +iJ )+i√c x2(J iJ )2+x J2+J2+ 4 (J iJ ).
2 3 2 3 1 2 2 3 1− 2 3 1 2 x2 1− 2
(cid:18) 3(cid:19)
Moreover, it allows us to prove that variables of separation (λ ,µ ) lie on the Jacobian of the
i i
other hyperelliptic curve of genus two
: Φ(z,λ)=z2+ 4λ2+4iH λ H z+(2a2λ+ic )2c2 =0, (3.19)
C 1 − 2 4 2
(cid:16) (cid:17)
where we putez = i√c2e2µ√2iλ.
−
So, we know the answer and, therefore, we could try to guess how to get this information
without using our knowledge of the Lax matrix. Namely, the normalized left eigenvectors of F
form the St¨ackel matrix
4iq 4iq
F =S−1diag(λ1,λ2)S, S = −1 1 −1 2 .
(cid:18) (cid:19)
7
In order to get conjugated to λ momenta µ we could use the St¨ackel potentials (3.5)
1,2 1,2
U = 4iλ H +K , such that U ,U =0, U ,λ =0, i=j.
1,2 1,2 1 2 1 2 i j
− { } { } 6
As it has been stated above we could study the following recurrence chain of the Poisson
brackets
φ = λ ,U , φ = λ ,φ ,..., φ = λ ,φ . (3.20)
1 1 1 2 1 1 i 1 i 1
{ } { } { − }
This chain is a quasi-periodic chain
φ =8iλ φ .
3 1 1
It means that the St¨ackel potential U is some trigonometric function on momenta µ and,
1 1
therefore, we could define this desired momenta in the following way
µ =f(λ )ln 8iλ φ +φ
1 1 1 1 2
(cid:16)p (cid:17)
up to canonical transformations µ µ +g(λ ). Here function f(λ ) is determined from the
1 1 1 1
→
canonicity of the bracket λ ,µ =1.
1 1
{ }
Summing up, the central idea of the proposed construction is the observation of quasi-
periodicity or of the break of the recurrence chain of the Poisson brackets (3.10) or (3.20).
We have to point out again that all the appropriate tedious calculations may be done on the
personal computer in a few seconds. So, it is a real way to get the variables of separation and
the separated relations for the given integrable system.
4 The quadratures
According to the Liouville theorem the existence of n independent integrals of motion H in
i
theinvolution H ,H =0guaranteesthatequationsofmotionmaybe solvedinquadratures.
i j
{ }
Usuallywefitsomeadditionalrequirementsonthesequadratures,asforexample,thesolutions
of equations of motion have to be single valued real functions on the real time variable t. It is
necessaryforthequalitativeanalysisofmotion,fortopologicalanalysis,forperturbationtheory
etc.
Ofcourse,weprefertohaverealsolutionsrepresentedbyanexplicitclosed-formexpression
right away. However, in nine times out of ten in order to get such single-valued solutions we
have to start with the solution of the Jacobi inversion problem. For example, for the geodesic
motiononanellipsoidandfortheKowalevskitopwehavetofindvariablesofseparations (t)
1,2
from the equations
s1 ds s2 ds
t+β = + ,
1
P(s) P(s)
Z∞ Z∞
s1 sds s2 sds
β = p + p ,
2
P(s) P(s)
Z∞ Z∞
where P is a polynomial of degree 5 or 6, anpd β are twopconstants of integration. Secondly,
1,2
we have to express the initial real variables via variables of separation s (t).
1,2
In the previous section we get the real variables of separation (q,p) and the complex vari-
ablesofseparation(λ,µ)or(Q,P). Thesevariablesarerelatedbythecanonicaltransformation
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 obtaineddirectly from the definitions (3.16,3.18) and the mapping (3.13). Using
the separation relations (3.14) we can rewrite this transformation as a quasi-point canonical
transformation [6]
λ =λ (q ,q ,H ,H )
1,2 1,2 1 2 1 2
which relatesthe Jacobianofthe hyperelliptic curve (3.14) with the Jacobianof the hyperel-
C
liptic curve (3.19). However, we can not rewrite it as a rational mapping ψ : (p,q) (λ,µ),
C →
e
8
i.e. it does not be well-studied cover of the hyperelliptic curve of genus two even at c = 0,
4
see [4] and references within. We suppose that these Jacobians are non-isogeneous in Richelot
sense [9] as well. Any further inquiry of this relation goes beyond the scope of this paper.
We have no right to escape complex variables and prefer the real variables of separation
only. It is how these variables are employed that determines the good or evil. As often as
not the equations of motion are linearized on the abelian variety, which is roughly spiking
the complexified of the corresponding Liouville real torus, see more detailed discussion in [2].
Moreover,weremindthatLyapunovcouldimprovetheKowalevskiresultandsolvetheproblem
pointed out by Painlev´e by using the complex time variable t [7]. The Ziglin method [19],
the theory of algebraically integrable systems [16] and some other modern theories deal with
the complex analytical Hamiltonian systems only. Quantum mechanics is formulated over the
complex field as well.
4.1 The real variables of separation
In this case equations of motion reads as
dq c (q q )
1 4 1 2
= 8q (c q )p 1 − ,
dt 1 2− 1 1 − (4q (c q )p2 4q (c q )p2 (q q )a2)
(cid:18) 1 2− 1 1− 2 2− 2 2− 1− 2 (cid:19)
(4.21)
dq c (q q )
2 4 1 2
= 8q (c q )p 1+ − .
dt 2 2− 2 2 (4q (c q )p2 4q (c q )p2 (q q )a2)
(cid:18) 1 2− 1 1− 2 2− 2 2− 1− 2 (cid:19)
Atc =0wehaveanaccidentaldegeneracyofthegenustwoalgebraiccurve(3.14)toaproduct
4
of two elliptic curves (3.15). Namely, if c =0 then
4
dq dq
1 2
=2dt
8q (c q )p − 8q (c q )p
1 2 1 1 2 2 2 2
− −
and we have to find functions q (t,α ,β ) from the two independent equations
1,2 1,2 1,2
q1 dq q2 dq
=β +t, and =β t.
1 2
P (q) P (q) −
Z∞ 1 Z∞ 2
Here we fix the values pof the integrals of motion H =α p, and
1,2 1,2
P (q)=8q (c q ) a2(c 2q )+α √α
k 1,2 2 1,2 2 1,2 1 2
− − ∓
(cid:16) (cid:17)
arethecubicpolynomialsdefiningtwoellipticcurves(3.15). Accordingto[1]inthiscasewecan
get solutions of x (t) and J (t) in an explicit and closedform using the Weierstrass ℘ function
k k
and its derivative.
At c =0 we have to solve the standard system of the Abel-Jacobi equations
4
6
q1 dq q2 dq
+ =β +t,
1
16q(c q)P(q) 16q(c q)P(q)
Z∞ 2− Z∞ 2−
and
q1 dq
32q(c q) a2(c 2q)+α 8q(c q)P2(q) P(q)
Z∞ 2− 2− 1− 2−
q2 (cid:16) dq (cid:17)
+ =β .
2
Z∞ 32q(c2−q) a2(c2−2q)+α1−8q(c2−q)P2(q) P(q)
(cid:16) (cid:17)
whereP(q)meansthesolutionoftheequationΦ(q,p)=0(3.14)withrespecttop. Forexample,
we can solve this equations numerically by using the Richelot approach [9].
9
The second part of the Jacobi separation of variables method consists of the construction
of the integrable system starting with some known separated variables and some arbitrary
separated relations [17]. As an example, if we consider new separation relations
Φ(p,q)=Φ(p,q) c q2 =0,
5
−
where Φ(p,q) is given by (3.14), then we get the following generalization of the Chaplygin
b
system
c
H =H 5 (J2+J2+c x2).
1 1− 4x4 1 2 2 3
3
We couldnot getany physicallybinteresting systems by using these realvariablesof separation.
4.2 The complex variables of separation
In this section we will work with variables (P,Q) instead of (µ,λ), i.e. we will consider a
ramified two-sheeted covering given by point transformation (3.17).
The poles of the Baker-Akhiezer function lie on the Jacobian of the 2 2 Lax matrix
×
spectral curve [10]. This curve is defined by the equation with the real coefficients
Φ(y,u)=y2 (u4 2H u2+H )y+c (a2u2 c )2 =0. (4.22)
1 2 2 4
− − −
However, in order to get the separation relations we have to substitute the complex functions
Q and P on the initial real variables into this equation
1,2 1,2
u=Q , y =e2iP1,2.
1,2
The Abel-Jacobi equations have the standard form
Q1 Q2 Q1 Q2
t+β = Ω + Ω , β = Ω + Ω ,
1 1 1 2 2 2
Z∞ Z∞ Z∞ Z∞
where
∂Φ(y,u)/∂H ∂Φ(y,u)/∂H
1 2
Ω = du and Ω = du.
1 2
∂Φ(y,u)/∂y ∂Φ(y,u)/∂y
After the solutionofthe Abel-Jacobiequationswehavetoexpressthe realinitialvariablesx,J
in terms of this complex solutions Q (t) and P (t).
1,2 1,2
Ontheotherhand,thesecomplexvariablesofseparationareveryusefulfortheconstruction
of other integrable systems. Namely, substituting Q and P into other separationrelations
1,2 1,2
we can get the Hamilton function for the Kowalevski-Goryachev-Chaplygingyrostat [5, 10]
c
H =J2+J2+2J2+ρJ +c x +c (x2 x2)+c x x + 4 (4.23)
1 1 2 3 3 1 1 2 1− 2 3 1 2 x2
3
after some additbional canonical transformations. Moreover, using these complex variables we
can easily get quantum counterpart of the Chaplygin system and its generalizations [5].
5 Conclusion
Starting with integrals of motion for the generalized Chaplygin system we have found the
two polynomial in momenta Poisson bivectors (2.6) and (2.7), which are compatible with the
canonical Poisson bivector on cotangent bundle T 2 of the two-dimensionalsphere.
∗
S
Anapplicationofthecorrespondingcontrolmatricesallowsustogettwofamiliesvariables
of separation and of separated relations using methods of bi-hamiltonian geometry only. The
solutions of equations of motion in these variables are briefly discussed.
The proposedapproachmaybe useful forinvestigationsofotherintegrablesystemsonthe
sphere with integrals of motion higher order in momenta [18], for instance, the search another
real variables for the Kowalevski-Goryachev-Chaplygingyrostat(4.23).
10