High order symplectic conservative perturbation method for time-varying Hamiltonian system

Abstract

This paper presents a high order symplectic conservative perturbation method for linear time-varying Hamiltonian system. Firstly, the dynamic equation of Hamiltonian system is gradually changed into a high order perturbation equation, which is solved approximately by resolving the Hamiltonian coefficient matrix into a “major component” and a “high order small quantity” and using perturbation transformation technique, then the solution to the original equation of Hamiltonian system is determined through a series of inverse transform. Because the transfer matrix determined by the method in this paper is the product of a series of exponential matrixes, the transfer matrix is a symplectic matrix; furthermore, the exponential matrices can be calculated accurately by the precise time integration method, so the method presented in this paper has fine accuracy, efficiency and stability. The examples show that the proposed method can also give good results even though a large time step is selected, and with the increase of the perturbation order, the perturbation solutions tend to exact solutions rapidly.