Method of Canonical Transformation for Constructing Mappings

Abstract

In this chapter we present the rigorous and systematic method to construct symplectic maps (3.2), particularly, Poincaré maps for generic Hamiltonian systems affected by perturbations. The method is based on the Hamilton–Jacobi method for integrating Hamiltonian equations and Jacobi’s theorem recalled in Sect. 1.2.2. As we have seen there the idea of the Hamilton–Jacobi method consists of finding such a canonical change of variables which reduces a Hamiltonian function to a form that Hamiltonian equations are easy to integrate. The canonical transformation of variables is given by a generation function satisfying to the Hamilton–Jacobi partial differential equation. If we succeed to find a complete integral, i.e., the solutions of this equation depending N independent constants of motion, then according to Jacobi’s theorem the dynamics of system is completely determined by the generating function F(q,P,t). It means that the time evolution of system (q(t), p(t)) can be found through its initial position (q(t₀), p(t₀)) by the forward, (q(t₀), p(t₀)) → (Q(t₀), P(t₀)), and the backward, (Q(t), P(t)) → (q(t), p(t)), canonical transformations (1.20) given by the generating function F(q,P, t) taken at the time instants t₀ and t, respectively. The evolution of new variables (Q, P) between these time instants is trivial and given by (1.18).