Canonical Coordinates Defined on a Curved Poincare Section and a Relation to Micro-Canonical Averages in Nonlinear Hamiltonian Dynamical System : General Physics

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We find in general two canonically conjugate variables describing an area-preserving 2D map-ping defined on a curved Poincare section in 2D Hamiltonian dynamical systems. In the first part we find that derivation of canonical coordinates is based upon time-invariants retaining memory of initial conditions. In the next part we discuss a linear stability of a periodic trajectory defined as a fixed point on a curved surface, comparing with a Floquet theory. In the third part under an ergodicity hypothesis we discuss a relation of canonical coordinates to statistical averages on a micro-canonical ensemble for variables defined on a curved surface. A representative statistical average is a micro-canonical of an elapsed time between two consecutive crossings, defined by a ratio of two densities of states. We consider as a concrete example a Henon-Heiles Hamiltonian.
社団法人日本物理学会の論文
2000-12-15