The symplectic methods for the computation of hamiltonian equations

Abstract

The present paper gives a brief survey of results from a systematic study, undertaken by the authors and their colleagues, on the symplectic approach to the numerical computation of Hamiltonian dynamical systems in finite and infinite dimensions. Both theoretical and practical aspects of the symplectic methods are considered. Almost all the real conservative physical processes can be cast in suitable Hamiltonian formulation in phase spaces with symplectic structure, which has the advantages to make the intrinsic properties and symmetries of the underlying processes more explicit than in other mathematically equivalent formulations, so we choose the Hamiltonian formalism as the basis, together with the mathematical and physical motivations of our symplectic approach for the purpose of numerical simulation of dynamical evolutions. We give some symplectic difference schemes and related general concepts for linear and nonlinear canonical systems in finite dimensions. The analysis confirms the expectation for them to behave more satisfactorily, especially in the desirable conservation properties, than the conventional schemes. We outline a general and constructive theory of generating functions and a general method of construction of symplectic difference schemes based on all possible generating functions. This is crucial for the developments of the symplectic methods. A generalization of the above theory and method to the canonical Hamiltonian eqs. in infinite dimensions is also given. The multi-level schemes, including the leapfrog one, are studied from the symplectic point of view. We given an application of symplectic schemes, with some indications of their potential usefulness, to the computation of chaos.