we denote it by ~k" E. G. Shuvalov showed that shifts . . . of. invariants give a complete, v• family of functions for t...

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we denote it by ~k" E. G. Shuvalov showed that shifts . . . of. invariants give a complete, v• family of functions for the Lie algebra si(2, C ) | k+1 for k = i, 2, 3, 4.

in-

~k

THEOREM 4.4.2 (see [9]).

Let L be the space dual to one of the Lie algebras L12, L 9, Then on L there exists a complete, involutive family of functions which are first integrals of the Euler equations described in parts 3.4-3.9 of Chap. i, i.e., these equations are completely integrable in the Liouville sense.

so(3)OE3, Ah, m, LmGA0,n (corresponding notation was introduced in Sec. 3, Chap. i).

CHAPTER 4 QUESTIONS OF NONINTEGRABILITY IN HAMILTONIAN MECHANICS I.

Poincarg Method of Proving Nonintegrability

I.i. Perturbation Theory and the Investigation of Systems Close to Integrable Systems. In the space of all Hamiltonians open regions are distinguished which sometimes fill out almost the entire space and consist of Hamiltonians f of "general position" for which the corresponding Hamiltonian systems v = sgrad f are not Liouville integrable (or not integrabie in some other more general sense). The picture we have described is not a rigorously proved theorem, since in the formulation presented above too many objects are in need of a correct specification which is not always possible. Nevertheless, results presently known of "negative character," i.e., results asserting the nonintegrability of many concrete types of systems, make it possible to view the principle formulated above as some experimental observation which may serve as a guide in the study of concrete systems. Thus, integrable cases fill a set of "measure zero" in the space of all systems. It is already clear from this that the search for integrable Hamiltonian systems is a very difficult problem, since it is necessary to somehow "guess" or algorithmically discover in the immense set of all possible Hamiltonians those rare cases when some additional symmetries cause the appearance of a sufficient number of integrals. In Chap. 2 regular methods were indicated for constructing functions on homogeneous symplectic manifolds which in practice, as a rule, give integrable Hamiltonian systems which are interesting from the viewpoint of mechanics (see Chap. 3). We shall now demonstrate to the reader that a Hamiltonian "taken at random" most often generates a nonintegrable system. Let (M, m) be a symplectic manifold, let v 0 = s g r a d H 0 be a completely integrable Hamiltonian system, and let T n be one of the compact, connected level surfaces of a collection of first integrals fl, f2,..-,fn which are functionally independent and pairwise in involution in a neighborhood of T n. As we know, T n is diffeomorphic to an n-dimensional torus. In an open neighborhood U of the torus T n we consider curvilinear "action-angle" coordinates sl,..., an, 91 ..... ~n, where ~ are angular coordinates on the torus and s i are coordinates normal to the torus. For brevity we introduce the vector notation s=(s1,.:., an), 9=(~i,-i., ~n). We represent the neighborhood U as a direct product U = D n • T n, where D n = D is an open domain in Rn(s), for example, homeomorphic to a sufficiently small disk. Thus, we have separated the regular coordinates in the neighborhood U into two groups: U(s, 9) =Dn(S) xTn(~) 9 The integral trajectories of the system v 0 = s g r a d H 0 are distributed on the torus T n (and on tori s • T n, where s6Dn), near to it), forming a rectilinear winding (see Fig. i0). As we know (see the Liouville theorem), in the coordinates (s, ~) the Hamiltonian H ~ depends only on the variables s, i.e., H = H(s) in the neighborhood U. We now consider a perturbation of the original Hamiltonian system by means of a perturbation of its Hamiltonian H 0. We consider a family of Hamiltonian systems v~=sgradH(s, ~, s), where H(s, 9, s) is a real analytic function defined on the direct product ! U X (--e0, so) and such that for E = 0 we obtain the original Hamiltonian, i.e., H(s, % 0)=H0(s). The Hamiltonian equations

OH~

~=~-~(s),

v0-----sgrad/40 in the coordinates

where ~(s)=(~1(s) .... ,~n(s))i

(s,~} can be written thus:

As we see, these equations are ex-

plicitly integrable. If the point s = s o is fixed, it determines a torus in the neighborhood U on which the vector ~(s ~ is constant (does n o t depend on ~), and hence the equations of motion can be integrated as follows: s(t)~s0, q(t)=~0-b~(s0)t. The equations of the perturbed Hamiltonian system v = s g r a d H can be written in the following form: s ~

O0~' H ~ _---~, 0H . . . . ~.) + .... where f-/(s,~)=/-/0(s)~-sff1(s,

Assuming that the parameter

2729

Graph

17'~/s)

i Fig. i0

Fig.

ii

is small, we arrive at the problem of integrating the perturbed system in a neighborhood of the completely integrable Hamiltonian system. In some cases Poincar~'s method makes it possible to prove nonintegrability of the perturbed system in a neighborhood of an integrable system. The term of the expansion /-[I(S, q)) plays a basic role in the investigation of the system for nonintegrability; this function is sometimes called the perturbing function. We expand the perturbing function in a multiple Fourier series: ff1(s, 9)-- ~

f-fm(S)e i(m'~) 9

Here

m~Z n

m = (m I ..... m n) is an integral vector running over all nodes of the integral lattice Z n of rank n, i.e., Z n - ~ Z X . . . X Z (n times). (m, ~ ) denotes the usual scalar product of the vectors n

m and ~, lee., (~, ~)-~-~ m ~ k .

Therefore,

the expansion of the function H I has the form

k=l

Hi(s, 9)---- ~

~

~I~+...+~:~

(m~..... ran)

We c o n s i d e r

the

domain (ball)

Dn t r a n s v e r s a l

to

the

Liouville

torus

Tn ( s e e

above).

Definition 1.1.1. T h e P o i n c a r ~ s e t P i n t h e d o m a i n Dn i s t h e s e t o f a l l p o i n t s S G Dn for which there exist n - 1 linearly independent integral vectors a~ .... , a n _ 1 ~ Z n such that the following conditions are satisfied: i) all scalar products

(a~, ~(s))are equal to zero, l~

in-

~k

THEOREM 4.4.2 (see [9]).

Let L be the space dual to one of the Lie algebras L12, L 9, Then on L there exists a complete, involutive family of functions which are first integrals of the Euler equations described in parts 3.4-3.9 of Chap. i, i.e., these equations are completely integrable in the Liouville sense.

so(3)OE3, Ah, m, LmGA0,n (corresponding notation was introduced in Sec. 3, Chap. i).

CHAPTER 4 QUESTIONS OF NONINTEGRABILITY IN HAMILTONIAN MECHANICS I.

Poincarg Method of Proving Nonintegrability

I.i. Perturbation Theory and the Investigation of Systems Close to Integrable Systems. In the space of all Hamiltonians open regions are distinguished which sometimes fill out almost the entire space and consist of Hamiltonians f of "general position" for which the corresponding Hamiltonian systems v = sgrad f are not Liouville integrable (or not integrabie in some other more general sense). The picture we have described is not a rigorously proved theorem, since in the formulation presented above too many objects are in need of a correct specification which is not always possible. Nevertheless, results presently known of "negative character," i.e., results asserting the nonintegrability of many concrete types of systems, make it possible to view the principle formulated above as some experimental observation which may serve as a guide in the study of concrete systems. Thus, integrable cases fill a set of "measure zero" in the space of all systems. It is already clear from this that the search for integrable Hamiltonian systems is a very difficult problem, since it is necessary to somehow "guess" or algorithmically discover in the immense set of all possible Hamiltonians those rare cases when some additional symmetries cause the appearance of a sufficient number of integrals. In Chap. 2 regular methods were indicated for constructing functions on homogeneous symplectic manifolds which in practice, as a rule, give integrable Hamiltonian systems which are interesting from the viewpoint of mechanics (see Chap. 3). We shall now demonstrate to the reader that a Hamiltonian "taken at random" most often generates a nonintegrable system. Let (M, m) be a symplectic manifold, let v 0 = s g r a d H 0 be a completely integrable Hamiltonian system, and let T n be one of the compact, connected level surfaces of a collection of first integrals fl, f2,..-,fn which are functionally independent and pairwise in involution in a neighborhood of T n. As we know, T n is diffeomorphic to an n-dimensional torus. In an open neighborhood U of the torus T n we consider curvilinear "action-angle" coordinates sl,..., an, 91 ..... ~n, where ~ are angular coordinates on the torus and s i are coordinates normal to the torus. For brevity we introduce the vector notation s=(s1,.:., an), 9=(~i,-i., ~n). We represent the neighborhood U as a direct product U = D n • T n, where D n = D is an open domain in Rn(s), for example, homeomorphic to a sufficiently small disk. Thus, we have separated the regular coordinates in the neighborhood U into two groups: U(s, 9) =Dn(S) xTn(~) 9 The integral trajectories of the system v 0 = s g r a d H 0 are distributed on the torus T n (and on tori s • T n, where s6Dn), near to it), forming a rectilinear winding (see Fig. i0). As we know (see the Liouville theorem), in the coordinates (s, ~) the Hamiltonian H ~ depends only on the variables s, i.e., H = H(s) in the neighborhood U. We now consider a perturbation of the original Hamiltonian system by means of a perturbation of its Hamiltonian H 0. We consider a family of Hamiltonian systems v~=sgradH(s, ~, s), where H(s, 9, s) is a real analytic function defined on the direct product ! U X (--e0, so) and such that for E = 0 we obtain the original Hamiltonian, i.e., H(s, % 0)=H0(s). The Hamiltonian equations

OH~

~=~-~(s),

v0-----sgrad/40 in the coordinates

where ~(s)=(~1(s) .... ,~n(s))i

(s,~} can be written thus:

As we see, these equations are ex-

plicitly integrable. If the point s = s o is fixed, it determines a torus in the neighborhood U on which the vector ~(s ~ is constant (does n o t depend on ~), and hence the equations of motion can be integrated as follows: s(t)~s0, q(t)=~0-b~(s0)t. The equations of the perturbed Hamiltonian system v = s g r a d H can be written in the following form: s ~

O0~' H ~ _---~, 0H . . . . ~.) + .... where f-/(s,~)=/-/0(s)~-sff1(s,

Assuming that the parameter

2729

Graph

17'~/s)

i Fig. i0

Fig.

ii

is small, we arrive at the problem of integrating the perturbed system in a neighborhood of the completely integrable Hamiltonian system. In some cases Poincar~'s method makes it possible to prove nonintegrability of the perturbed system in a neighborhood of an integrable system. The term of the expansion /-[I(S, q)) plays a basic role in the investigation of the system for nonintegrability; this function is sometimes called the perturbing function. We expand the perturbing function in a multiple Fourier series: ff1(s, 9)-- ~

f-fm(S)e i(m'~) 9

Here

m~Z n

m = (m I ..... m n) is an integral vector running over all nodes of the integral lattice Z n of rank n, i.e., Z n - ~ Z X . . . X Z (n times). (m, ~ ) denotes the usual scalar product of the vectors n

m and ~, lee., (~, ~)-~-~ m ~ k .

Therefore,

the expansion of the function H I has the form

k=l

Hi(s, 9)---- ~

~

~I~+...+~:~

(m~..... ran)

We c o n s i d e r

the

domain (ball)

Dn t r a n s v e r s a l

to

the

Liouville

torus

Tn ( s e e

above).

Definition 1.1.1. T h e P o i n c a r ~ s e t P i n t h e d o m a i n Dn i s t h e s e t o f a l l p o i n t s S G Dn for which there exist n - 1 linearly independent integral vectors a~ .... , a n _ 1 ~ Z n such that the following conditions are satisfied: i) all scalar products

(a~, ~(s))are equal to zero, l~

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