The Poincaré-Lyapunov-Liouville-Arnol'd theorem
The author presents the following theorem: Let \(M\) be a symplectic manifold of dimension \(2n\). Suppose that a Hamiltonian flow \(X_H\), \(H \in C^\infty (M)\), possesses \(k\) \((1 \leq k \leq n)\) integrals in involution \(H= F_1,F_2,\dots, F_k\) and that there exists a \(k\)-dimensional compact connected submanifold \(S \subset M\) invariant under the Hamiltonian flows of all the \(F_i\)'s with \(dF_1,\dots, dF_k \mid S\) linearly independent. Then \(S\) is a torus. If \(k < n\) assume in addition that the fundamental group \(\pi_1(S)\) has at least one element whose monodromy (linearized Poincaré map) spectrum does not contain \(1\). Then, in some neighborhood \(U\) of \(S\) there exists a \(2k\)-dimensional symplectic submanifold \(N\) which is a trivial fibration whose fibers are \(k\)-dimensional tori \(S_\beta = N \cap (F_1,\dots, F_k)^{-1} (\beta)\), \((F_1,\dots, F_k) (S_\beta) = \beta\), one of the latter tori coinciding with \(S\). Moreover \(N\) admits action-angle variables consistent with the above fibration, and on each fiber \(S_\beta\) the Hamiltonian flows of the \(F_i\)'s are quasi-periodic. This result provides an interpolation between the Poincaré-Lyapunov theorem \((k =1)\) and the Liouville-Arnol'd theorem \((k = n)\).
scientific article; zbMATH DE number 795622