Topology of energy surfaces and existence of transversal Poincaré sections

Details Topology of energy surfaces and existence of transversal Poincaré sections Topology of energy surfaces and existence of transversal Poincaré sections

1996-02-29

arxiv.org/abs/chao-dyn/9602023v1

J. Phys. A, 29:4977--4985, 1996

Nonlinear Sciences

Chaotic Dynamics

10 pages, LaTex

Scientific paper

10.1088/0305-4470/29/16/019

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 Poincar\'e 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 Poincar\'e section can exist.

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