A new integration method of Hamiltonian systems by symplectic maps

A perturbation theory is developed for constructing stroboscopic and Poincaré maps for Hamiltonian systems with a small perturbation. It is based on a canonical transformation by which the evolution becomes unperturbed during the entire period while all perturbations are acting instantaneously during one kick per period. Matching of solutions before and after the kicks establishes a symplectic map which exactly describes the evolution. The generating function associated with this map satisfies the Hamilton-Jacobi equations. The solution of this equation is found in first order of perturbation theory. It is shown that the map reproduces correctly Poincaré sections and statistical properties of typical orbits. It is shown that the well known perturbed twist mapping and, in particular, the standard map may be obtained from the symmetric map as an approximation. The method is also applied to construct Poincaré maps at arbitrary sections of the phase space. In particularly, the maps describing a motion near the separatrix are derived.