Abstract
We find in general two canonically conjugate variables describing an area-preserving 2D mapping defined on a curved Poincaré 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 Hénon-Heiles Hamiltonian.
