Topology of energy surfaces and existence of transversal Poincaré sections

Bolsinov, Alexey; Dullin, Holger R.; Wittek, Andreas

Topology of energy surfaces and existence of transversal Poincaré sections

Authors

Bolsinov, Alexey

Dullin, Holger R.

Wittek, Andreas

Description

Two questions on the topology of compact energy surfaces of natural two degrees of freedom Hamiltonian systems in a magnetic field are discussed. We show that the topology of this 3-manifold (if it is not a unit tangent bundle) is uniquely determined by the Euler characteristic of the accessible region in configuration space. In this class of 3-manifolds for most cases there does not exist a transverse and complete Poincaré section. We show that there are topological obstacles for its existence such that only in the cases of $S^1\times S^2$ and $T^3$ such a Poincaré section can exist.
10 pages, LaTex