Документ взят из кэша поисковой машины. Адрес оригинального документа : http://ics.org.ru/upload/iblock/4d3/110-kovalevskaya-exponents-and-poisson-structures_ru.pdf Дата изменения: Wed Oct 28 19:30:46 2015 Дата индексирования: Sun Apr 10 00:08:34 2016 Кодировка:

A. V. BORISOV, S. L. DUDOLADOV
FacultyofMechanics and Mathematics Department of Theoretical Mechanics Moscow State University Vorobievy gory, Moscow, 119899, Russia E-mail: borisov@uni.udm.ru, dsl@online.ru

;
(1.1) (1.2)
gi xi
i

KOVALEVSKAYA EXPONENTS AND POISSON STRUCTURES
Received September 30, 1999

We consider generalizations of pairing relations for Kovalevskaya exponents in quasihomogeneous systems with quasihomogeneous tensor invariants. The case of presence of a Poisson structure in the system is investigated in more detail. Wegive some examples which illustrate general theorems.

1. Quasihomogeneous systems. Kovalevskaya exponents
A system of n di erential equations

i =1 ::: n is called quasihomogeneous with quasihomogeneity exponents g1 ::: gn if vi ( g x1 ::: g xn)= g +1 vi (x1 ::: xn ) for all values of x and > 0: Thus, the equations (1.1) are invariant under substitution x t 7! t 18].
i n

xi = vi (x1 ::: xn ) _

i

7!

Remark 1. A more general de nition of quasihomogeneity of degree m is the invariance of the system (1.1) t gi xi t under transformation xi m;1 14]. All further results hold for this case as well.

7!

!

An important example of the equations (1.1), (1.2) is a system with quasihomogeneous quadratic right-hand sides in this case g1 = ::: = gn =1. Motion equations of many important problems of dynamics (Euler { Poisson equations, Kirchho equations, Euler { Poincare equations on Lie algebras, Toda lattices, etc.) are of the quasihomogeneous form. Di erentiating (1.2) with respect to and setting =1, we obtain the Euler formula for quasihomogeneous functions:
n X k
=1

i gk xk @vk =(gi +1)v @x

i

i =1 ::: n:

(1.3) (1.4) (1.5)

The equations (1.1) possess partial solutions

xi = Ci t;g i =1 ::: n where the complex constants C1 : : : Cn should satisfy the algebraic system of equations vi (C1 ::: Cn )= ;gi Ci i =1 ::: n:
i

Mathematics Sub ject Classi cation 34G20, 34L40
c REGULAR AND CHAOTIC DYNAMICS, V. 4,

3, 1999

13


;
yi = _
k
=1

A. V. BORISOV, S. L. DUDOLADOV

Let us write variational equations for the partial solution (1.5) as
n X @vi (C t @xk 1 g

;g

1

::: Cnt;g )yk :
n

(1.6)

The linear system (1.6) possesses partial solutions of the form

y1 = '1 t ;

1

::: yn = 'n t ;

g

n

i i where is an eigenvalue and ' is an eigenvector of a matrix K = kKji k Kji = @vj (C)+ gi j , @x i j is the Kronecker symbol. The matrix K is called Kovalevskaya matrix, its eigenvalues are called Kovalevskaya exponents (see 18]). One of the Kovalevskaya exponents always equals ;1 18]. If the general solution of the system (1.1) is expressed in terms of single-valued (meromorphic) functions of complex time, then Kovalevskaya exponents, except for ;1, are integer (nonnegative integer, respectively). Relations between Kovalevskaya exponents are pointed out in 12]. They occur due to the presence of an invariant tensor eld in the system (1.1). Recall, that a tensor eld T of (p q) type is called quasihomogeneous of degree m with quasihomogeneity exponents g1 : : : gn if

Tji11

::: ip g1 ::: jq (

x1 : : :

gn xn

)= m;g 1 ; ::: ;g +g
j jq

i

1+

::: +g

ip

Tji11

::: ip ::: jq

(x1 : : : xn ) :

This tensor eld is invariant for the system (1.1), if its Lie derivative along the vector eld v equals zero.

2. Hamilton equations
Let us consider quasihomogeneous equations of the form:

xi = _

X ik @H J
k

where J = kJ ik k is a constantskew-symmetric tensor of type (2, 0), H is a quasihomogeneous function of degree m +1:

@x

k

i =1 ::: n

(2.1)

H ( g1 x1 :::
n X k
=1

gn xn

)= m+1 H (x1 ::: xn ) :
g
+1

(2.2)

Checking the ful llment of the condition (1.2) and using (2.2), we obtain

J

ik m+1;g

k

@H (x) = @xk

i

n X ik @H (x) J : @xk k
=1

Let ; = diag(g1 ::: gn ): Di erentiating the latter identity with respect to and setting = 1 we obtain the following conditions in matrix form, to which the quasihomogeneity exponents should satisfy:

J; + ;J = mJ :
gk + g
k+ n 2

(2.3)

Let us note, that the equations (2.1) are Hamilton equations with the Hamiltonian H in (possibly) noncanonical variables. If J is a symplectic matrix, the conditions (2.3) have a simple form: = m:
3, 1999

14

REGULAR AND CHAOTIC DYNAMICS, V. 4,


KOVALEVSKAYA EXPONENTS AND POISSON STRUCTURES

;

moreover, there are always ;1and m + 1 among them. The following statement extends the corresponding results 16, 12] in general case of nondiagonalizable matrix K: Let us note that one should not speak about single-valuedness and meromorphy of the general solution in the nondiagonalizable situation. heorem 1. Let J be a nondegenerate skew-symmetric matrix for the equations (2.1). Then the Kovalevskaya exponents decompose into pairs, which satisfy the relations
k

It is shown in 16, 12], that in the case of diagonalizable Kovalevskaya matrix its exponents satisfy the analogous relations k + k+ n = m
2

and the structure of Jordan cel ls, associated with the exponents and (m ; ), is the same. Proof. Let us present the Kovalevskaya matrix in the form K = JB + ;, where
2 B = @i H k ( )

+ k+ n = m 2

k =1 ::: n 2

@x @x

is a symmetric matrix. Therefore, all conclusions of Theorem 1 follow from the chain of equivalent statements: det kK
1 1 det : In the last link of the chain we used the fact, that it is possible to substitute J;1 for J in (2.3). Let us generalize the above arguments on the case when the matrix J ik is not certainly nondegenerate and constant. It corresponds to the considerations of quasihomogeneous systems which admit the Poisson structure of a more general form (Lie { Poisson structures, quadratic structures, etc.) Let us preliminary prove the main theorem. heorem 2. Let us assume that the equations (1.1) admit a quasihomogeneous tensor invariant T of degree m and type (2 0): Then natural numbers form 1 to n can begrouped in a set (k1 ::: kn ) so that 1 : : : n satisfy at least r =rank T(C) relations:

, det kK ; EkT =0 , det k(K ; E)T J; k =0 , k(;BJ + ; ; E)J; k =0 , det kJ; (JB + ; +(h ; 1 ; )E)k =0
=0
1

; Ek

i

+ k = ;m
i

i =1 : : : n :

Proof. Using the expression of Lie derivative for an invariant tensor eld, it is easy to show (see 12] for details) that the tensors T and K are connected by the following relations:

;

mT ij = Ksi T sj + T is K

j s

whichwe shall write in matrix form

i: where T = kT ij k K = kK:j k. Let the matrix A be formed of column-vectors e1 ::: en which are Jordan vectors of K:

;mT = KT + TKT
KA = AK

(2.4)

REGULAR AND CHAOTIC DYNAMICS, V. 4,

3, 1999

15


;

A. V. BORISOV, S. L. DUDOLADOV

where K for de niteness has the following form: there are the Kovalevskaya exponents ( 1 ::: n ) on the principal diagonal, and there can be units above the principal diagonal. The analogous relation holds for the transposed matrix KT :

KT (A;1 )T =(A;1)T KT :
Let us denote column-vectors, which form the matrix (A;1 )T ,by f1 ::: fn : On account of (2.4), we obtain KT(A;1 )T = ;mT(A;1)T ; TKT (A;1 )T = T(A;1)T (;mE ; (K )T ) : The matrix (;mE ; (K )T ) is also of Jordan form, but now there can be ;1 under the principal diagonal. Thus, under transformation A 7;! T(A;1 )T to Jordan vectors (e1 ::: en ) there correspond r = rank T(C) independent vectors (Tf1 ::: Tfn ) which are also Jordan vectors of K with the eigenvalues (;m ; 1 ) ::: (;m ; n ): Remark 2. It is possible that i = ki in general case. Then i = ; m . The following Corollary speci es 2
the theorem in the case of skew-symmetric tensor invariant.

Kovalevskaya exponents satisfy l relations
i
s

Corollary. Let the tensor T be skew-symmetric. Then among natural numbers from 1 to n one 1 can extract two subsets with distinct numbers (i1 ::: il ) and (k1 : : : kl ) l = 2 rank T(C) such that
+ k = ;m
i

s =1 : :: l :

Indeed, a nonzero vector Tfi can not be proportional to ei in the case of skew-symmetric T. It follows from the fact that (ei fj )= ij where ( ) is the standard scalar product in R n (AA;1 = E) and the skew-symmetric propertyof T implies (Tfi fi )= 0: Since the skew-symmetric structural tensor J ij is a tensor invariant of motion equations for general Hamiltonian systems (Section 2), then it follows from the corollary that the Kovalevskaya exponents are coupled, and the number of pairs equals 1 rank J(C): 2

3. Invariant measure
As a rule, quasihomogeneous equations of dynamics (the Euler { Poisson equations, the Kirchho equations, etc.) possess an invariant measure besides the degenerate Poisson structure (determined by algebra e(3)). The existence of invariant measure imposes an additional condition on Kovalevskaya exponents. Indeed, let us assume that the system (1.1) admits a quasihomogeneous tensor invariant of type (n 0) = (x)dx1 ^ ::: ^ dxn (C) 6=0 : Then
i=1 n P i

= gi in particular, the sum of Kovalevskaya exponents equals the system dimension n for i=1 systems with homogeneous quadratic right-hand sides. This result follows from the main theorem of 12]. As it was pointed out in 7], in the homogeneous case (gi = 1) Kovalevskaya exponents are connected with multipliers of periodic solutions, and their pairing for Hamiltonian systems follows from the Poincare{Lyapunov theorem on recurrence of roots of characteristic polynomial of variational equations.

i=1

n P

i n P

= m where m is the quasihomogeneity degree of . If

is the standard measure, then

16

REGULAR AND CHAOTIC DYNAMICS, V. 4,

3, 1999


KOVALEVSKAYA EXPONENTS AND POISSON STRUCTURES

4. Examples

;

a) Let us consider a variantof the system of Lotka { Volterra type 5, 17], which can be written as xi = xi ( i+1 xi+1 ; i;1 xi;1 ) _ i =1 : : : n (4.1) x0 = xn xn+1 = x1 where i i are constants. The equations (4.1) are generalizations of the integrable periodic Volterra system, for which i = i = const 2]. A straightforward calculation of Kovalevskaya exponents for the system (4.1) shows, that they satisfy the pairing conditions i + j = 0 it corresponds to occurrence of a quadratic tensor invariant T ij (by de nition of quasihomogeneity degree). However, the ful llment of the theorem condition n n Q = Q holds, the is not su cient for the presence of the Poisson structure. If the relation i i i=1 i=1 n and under condition i = i equations (4.1) possess an additional linear integral F =(l x) l 2 R the system (4.1) is indeed Hamiltonian with the quadratic Poisson bracket J ij = C ij xi xj and linear Hamiltonian. b) Let us consider a generalized Suslov problem as another example. It describes a rigid body motion around a xed point with the nonholonomic constraint !3 =0. If the center of mass is on the principal axis, along which !3 = 0, then motion equations of the system have the form: I1 !1 = " 2 _ I2 !2 = ;" 1 _ (4.2) _ 1 = ;!2 3 _ 2 = !1 3 _ 3 = !2 1 ; !1 2 where I1 , I2 are components of the inertia tensor, " is the distance from the xed point to the center giv of mass. Calculation of Kovalevskaya exponents r es the following values: I 1 1. 1 = ;1 2 =2 3 =4 4 5 = 2 3 1+ 8 I2 , r I1 2. 1 = ;1 2 =2 3 =4 4 5 = 1 3 1+ 8 I1 . 2 2 Similar in structure, but more complicated expressions for the Kovalevskaya exponents 4 5 can be obtained in general case, when the position of the center of mass and the nonintegrable constraintin the body are not related anyhow 15]. These Kovalevskaya exponents are coupled: 1 + 3 = 4 + 5 =3. Therefore, it is natural to expect the presence of a tensor invariant in the system (4.2) and possibilityof its representation in the Hamiltonian form (2.1) with some in general nonconstant structural tensor J ik . 2 2 2 It is valid indeed, if one chooses the geometrical integral F = 1 ( 1 + 2 + 3 ) as the Hamiltonian. 2 Let x =(x1 x2 x3 x4 x5 )=(!1 !2 1 2 3 ) and rewrite (4.2) as where

xi = J ij (x) @Fj _ @x
0 0

0 00 1 0 0 !2 ;!1 0 For (4.2) J is a tensor invariant of degree 3, which satis es the Jacobi identity, what one veri es by straightforward calculations. The Casimir function of the Poisson structure J is the energy integral (of the Suslov problem) 1 ;I1 !2 + I2 !2 + " 2 = h: (4.3) 2 21
REGULAR AND CHAOTIC DYNAMICS, V. 4, 3, 1999

00 B0 B ij k = B 0 J = kJ B" B; @I

" I2

;

0

" I2

" I1

0 0 0

1 C C ;!2 C : C A !1 C
0 0

17


;
where

A. V. BORISOV, S. L. DUDOLADOV

The reduction on a symplectic sheet can be carried out explicitly, if one notes, that the equations (4.2) can be rewritten as Lagrange equations 8] after excluding 3 from (4.3)

d @L @L dt @ !i = @! _

L=T ;V

rather unexpectable, since motion equations of nonholonomic dynamics do not preserve the invariant measure in general case 11]. c) Let us consider a restricted problem in rigid body dynamics.

22 22 _ T = 1 I1 !1 + I2 !2 : 2_ These equations are not integrable for I1 6= I2 . The Hamiltonian property for the Suslov problem is

;

!1 = !2 !3 + z _

2

!2 = ;!1 ! _

3

;

z

1

!3 = _

;

2

_=

!:

(4.4)

The equations (4.4) at z = 0 were studied in 13], where their nonintegrability was shown with the help of the separatrix splitting method pictures of stochastic behavior are presented in 3]. In general case when z 6= 0 the system (4.4) is quasihomogeneous and possesses the following 0 0 i sets of partial solutions !i = !ti i = t2 , where
0 !1 =0 0 1

=2

0 !2 =0 0 2

=2i

0 !3 =2i 0 3

=0

and

2 0 0 = 2i 2 =0 3=z: z The Kovalevskaya exponents, corresponding to the chosen partial solutions, are of the form:
0 1

0 !1 =0

0 !2 =2i

0 !3 =0

where ( 1 2 3 ) are roots of the cubic equation 3 ; 9 2 +26 ; (24+8z )= 0 the solutions of which for any z have a complicated algebraic form. Apparently, it prevents from the existence of algebraic integrals of motion for the system (4.4). The occurrence of a Poisson structure for the equations (4.4) has not been also investigated. d) Motion of a ferromagnet with the Barnett { London e ect The essence of the quantum-mechanic e ect of Barnett is that a neutral ferromagnet magnetizes along the axis of rotation. In this case the magnetic moment B is connected with its angular velocity ! by relation B = 1 ! where 1 is a symmetric linear operator. The analogous moment occurs under rotation of a superconducting rigid body under the London e ect. If a body rotates in homogeneous magnetic eld with the intensity H, then it is under magnetic forces with the moment B H: Let us denote = H then equations of motion can be written as: _ _= M = M AM + M AM (4.5) = 1A A = I;1 = diag(a1 a2 a3 ) : As it is shown in 9], the equations are Hamiltonian at = A (they are reduced to the Kirchho equations, i. e. to equations on the algebra e(3)), and also at = diag( 1 2 3 ) A = E: In the last case they are integrable and reduced to the Clebsh case on the algebra e(3) by a linear coordinate transformation 4].

=(;1 0 1

1

2

3

)

18

REGULAR AND CHAOTIC DYNAMICS, V. 4,

3, 1999


KOVALEVSKAYA EXPONENTS AND POISSON STRUCTURES

;

The equations (4.5) possess two integrals F1 = (M ) F2 = ( ) and the standard invariant measure. There is the lackoftwointegrals for their integrability in general case. These integrals are F3 =(M M), F4 = (M AM) at = 0. At = diag( 1 2 3 ) using the method of separatrix splitting, one can show that for a1 = a2 6= a3 6= a1 the existence conditions for 6 at least one of additional motion integrals, generated by F3 or F4 ,have the form

X 2;
,!

a

1

3

=0

X
,!

a;1 a 1

23

;

a

32

+ 1 (a2 ; a3 )] = 0 :

(4.6)

It is obvious from (4.6), that one more integral can actually exist at = E: It is the integral of moment F3 =(M M): The system (4.5) is completely integrable at a1 = a2 = a, = E, and its additional integral is F4 = aM3 + 3 : The question concerning the Hamiltonian property of the equations (4.5) was arisen in 10], however it has not been solved yet. As it is noted in 1], the matrix should be diagonal = diag( 1 2 3 ) for the Hamiltonian property in the case of a1 6= a2 6= a3 6= a1 . Calculation of Kovalevskaya exponents at a2 = a3 = B a1 = 1 for the solution 1 (c1 ::: c6 )= 0 B gives the set (;1 1 2 2 1+ B

s

p;
2

2

(c1 ::: c6 )= i

s

2B 1

;B ; pB ;
3 3 2 1 2

1

s

3

;

2 2

c

3

22

c

; ;

Bc

2

3

; ;

Bc

3

!

2

the set (;1 2 2 2 B 1 ; B ). The pairing condition does not hold in this case, what is typical (in general situation of nondegeneracy of the structural tensor at the point (c1 ::: c6 )) for Hamiltonian systems. However, this observation can not be considered to be a strong evidence of the absence of the Hamiltonian property for the system (4.6).

B(

;

3

)

s

2B ), and for the solution

B(

1 3

;

2

)0

Bc

2

Bc

3

!

1

1

References
1] Yu. V. Barkin, A. V. Borisov. Nonintegrability of the 6] S. L. Ziglin. On the absence of an additional rst inteKirchho equations and related problems of rigid body gral in a problem of rigid body dynamics. DAN SSSR, dynamics. 5037{89, M.: VINITI, 1989 (in Russian). V. 292, 1987, 4, P. 804{807. (in Russian). 2] O. I. Bogoyavlensky. Reversing solitons. Nonlinear in- 7] V. V. Kozlov. Symmetries, topology and resonances in tegrable equations. M.: Nauka, 1991. (in Russian). Hamiltonian dynamics. Izhevsk, Izd. UdGU, 1995. (in 3] A. V. Borisov, K. V. Emelianov. Nonintegrability and Russian). stochasticity in rigid body dynamics. Izhevsk, Izd. 8] V. V. Kozlov. To the theory of integrability of equaUdGU, 1995. (in Russian). tions of nonholonomic dynamics. Uspekhi mekh., V. 8, 4] it L. E. Veselova. On two problems of rigid body dy1985, 3, P. 85{101. (in Russian). namics. Vest. MGU, Ser. Mat. Mekh., 1986, 5, 9] V. V. Kozlov. To the problem on rotation of a rigid P. 90{91. (in Russian). body in magnetic eld. Izv. AN SSSR, Ser. Mekh. 5] V. Volterra. Mathematical theory of the struggle for Tv. T., 1985, 6, P. 28{33. (in Russian). existence. M.: Nauka, 1976. Translation from French V. Volterra. Lecons sur la theorie mathematique de la 10] V. V. Kozlov. Some aspects of dynamical systems thelutte pour la vie, Gauthier { Villars, Paris, 1931. ory. eds. V. V. Kozlov, A. T. Fomenko. Geometry, difREGULAR AND CHAOTIC DYNAMICS, V. 4, 3, 1999

19


;
11] 12] 13]

A. V. BORISOV, S. L. DUDOLADOV

14]

ferential equations and mechanics, M.: MGU, 1986, P. 4{18. (in Russian). V. V. Kozlov. On existence of integral invariant of smooth dynamical systems. Prikl. Mat. Mekh., V. 51, 1987, 4, P. 538{545. (in Russian). V. V. Kozlov. Tensor invariants of quasihomogeneous systems of di erential equations and the asymptotic method of Kovalevskaya{Lyapunov. Mat. Zametki, V. 51, 1992, 2, P. 46{52. (in Russian). V. V. Kozlov, D. V. Treshchev. Nonintegrability of the general problem on rotation of a dynamically symmetric heavy rigid body with a xed point. II. Vestnik MGU, Ser. Mat. Mekh., 1986, 1, P. 39{44. (in Russian). V. V. Kozlov, S. D. Furta. Asymptotics of solutions of strongly nonlinear systems of di erential equations. M.: MGU, 1996. (in Russian).

15] V. M. Nikiforov. On single-valued solutions of the Suslov problem in homogeneous gravitational eld. eds. V. V. Kozlov, A. T. Fomenko. Geometry, di erential equations and mechanics, M.: MGU, 1986, P.106{ 108. (in Russian). 16] P. Lochak. Pairing of the Kowalevski exponents in Hamiltonian systems. Phys. Lett., V. 108A, 1985, 4, P. 188{190. 17] M. Plank. Hamiltonian structures for the ndimensional Lotka { Volterra equations. J. Math. Phys., V. 36(7), 1995, P. 3520{3534. 18] H. Yoshida. Necessary condition for the existence of algebraic rst integrals, I{II. Cel. Mech., v. 31, 1983, 4, P. 363{399.

20

REGULAR AND CHAOTIC DYNAMICS, V. 4,

3, 1999