| Numerical integrations of Hamiltonian problems appear frequently in physical and other sciences. Recent investigations strongly suggest that they should be carried out by symplectic integration, i.e., by numerical methods that preserve the symplectic structure of the phase space, thus reproduce main features of the Hamiltonian dynamics.;In this paper, we present in detail how to derive, analyze and use the numerical symplectic integration for both finite and infinite dimensional Hamiltonian systems. For finite dimensional Hamiltonian system, we summarize the existing work about the symplectic integration, and present a systematic way to construct symplectic schemes using generating function method. Then we generalize the symplectic integration to infinite dimensional Hamiltonian system, and propose three general approaches which can be used to construct symplectic schemes for the infinite dimensional Hamiltonian system.;Through our discussion, we use various numerical examples to test the effectiveness of the symplectic schemes, and demonstrate the super performance of these schemes. |
