Noncommutative integrability on noncompact invariant manifolds

Abstract

The Mishchenko-Fomenko theorem on noncommutative completely integrable Hamiltonian systems on a symplectic manifold is extended to the case of noncompact invariant submanifolds.

Superintegrable systems on Poisson manifolds

“…The following two assertions clarify the structure of superintegrable systems [4,6,12,26]. Lemma 2.1: Given a symplectic manifold (Z, Ω), let π : Z → N be a fibred manifold such that, for any two functions f , f ′ constant on fibres of π, their Poisson bracket {f, f ′ } also is well.…”

“…(Theorems 2.5, 2.6 and 2.3, respectively) [6,10,12,25,26,30]. However, Definition 2.1 of a superintegrable (non-commutative completely integrable) system on a symplectic manifold is rather restrictive.…”

Time-Dependent Superintegrable Hamiltonian Systems

“…A glance at the transformation law (6) shows that the one-dimensional fiber bundle ζ (1) is a trivial affine bundle. Given its global section…”

“…Let (V * Q, H) be a time-dependent Hamiltonian system and (T * Q, H * ) be an equivalent autonomous Hamiltonian system on T * Q. An immediate consequence of the relations (12) and (22) is the following [6].…”

Global action-angle coordinates for completely integrable systems with noncompact invariant submanifolds

“…12 Theorem 1: Let the Hamiltonian vector fields ϑ i of the functions H i be complete, and let the fibers of the fibered manifold H (1) be connected and mutually diffeomorphic. Then the following hold.…”