Some aspects of Hamiltonian systems and symplectic algorithms

Abstract

Symplectic algorithms have been shown to be a right formalism for numerical computation of Hamiltonian systems. They are suitable to long time computation and of good qualitative properties. These properties are ensured by the fact that a symplectic algorithm approximating to a time-independent Hamiltonian system can be regarded as a perturbed time-dependent Hamiltonian system of the original one. That is, a solution of a symplectic algorithm is a solution of a certain perturbed time dependent Hamiltonian system evaluated at (time) discrete points. Moreover, a linear symplectic algorithm approximating to a linear time-independent Hamiltonian system can be regarded as a perturbed time-independent Hamiltonian system. So it has all properties that a linear Hamiltonian system has. Based on these results, stochastic webs and chaos appeared in symplectic simulation for Hamiltonian systems are also discussed.