Four-Dimensional Integrable Hamiltonian Systems with Simple Singular Points

Abstract

The aim of this paper is to describe the structure of an integrable Hamiltonian vector field X H on an invariant subset V of four-dimensional C ∞-smooth symplectic manifold M, the subset V containing a singular point p of X H together with all of its orbits for which p is the limit set. Let H : M → ℝ be a C ∞-smooth function on M (Hamiltonian) and K be an additional (smooth) integral of the field X H . The pair (X H , K) is called an integrable Hamiltonian vector field (briefly, IHVF) if and only if functions H, K are independent in some open dense subset of M (or in a region under consideration). Let p be a singular point of X H . Without loss of generality we assume H(p) = K(p) = 0. Henceforth we consider eigenvalues of p to be simple.KeywordsSingular PointUnstable ManifoldSymplectic ManifoldStable ManifoldOpen Dense SubsetThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.