On a theorem of Liouville concerning integrable problems of dynamics

Abstract

1. Liouville proved that if in a system with n degrees of freedom, p˙=−∂H∂q,q˙=∂H∂pp˙=−∂H∂q,q˙=∂H∂p \label{1} \dot{p} = -\frac{\partial H}{\partial q}, \dot{q} = \frac{\partial H}{\partial p} (1) the first n integrals in the involution H = F 1, F 2, ..., F n are known, then the system is integrable in quadratures (see [1]). Many examples are known of integrable problems. It has often been noted that in these examples the bounded invariant manifolds determined by the equations F i = f i = const (i = 1, ..., n) turn out to be tori and motions on them are conditionally periodic. We shall prove that this situation is necessary in any problem which is integrable in the indicated sense. The proof is based on simple topological considerations.