Perturbations of Integrable Systems

Abstract

In this section we give a more precise characterization of trajectories which are solutions of an integrable system. We start from Liouville’s result. Given a Hamiltonian System

$$\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} {{{{\dot{y}}}_{k}} = + {{H}_{{{{x}_{k}}}}},} \\ {{{{\dot{x}}}_{k}} = - {{H}_{{{{y}_{k}}}}},} \\ \end{array} } & {(k = 1,2,. .,n)} \\ \end{array}$$

(3.1.1)

let H = H(y;x) be analytic in a given domain D of the phase space. If n uniform integrals F₁, F₂,...,F_n, in involution, are known, in a domain D’ ⊂ D, then in D’ the system is integrable, i.e., reducible to quadratures. Let

$$ {F_i}(y;x) = {C_i} = {\text{const}} $$

(3.1.2)

for i = 1,2,...,n. In general one verifies that the closed manifolds generated by Eqs. (3.1.2.) are tori and, on these, the motion is quasiperiodic. One can actually show that this is so, in general, for a Liouville system. More precisely, the following theorem (Arnol’d, 1963) can be stated. “Let the system (3.1.1.), with n degrees of freedom, have n first uniform integrals F1,...,Fn in involution. The equations F_i = C_i define a compact manifold M = M_C in very point of which the vectors grad F_i (i = 1,2,...,n) are linearly independent in the phase space of dimensions 2n. Then M is a torus of dimension n and the point (y(t); x(t)), solution of (3.1.1) in D’, has a quasiperiodic motion on M.” This theorem justi fies the fact that we always consider intergrable systems as given by a Hamiltonian H = H_O(x), a function of the momenta only.