Symplectic algorithms based on the principle of least action and generating functions

Abstract

SUMMARYIn this paper, the principle of least action and generating functions are used to construct symplectic numerical algorithms for finite dimensional autonomous Hamiltonian systems. The approximate action is obtained by approximating the generalized coordinates and momentums by Lagrange polynomials and performing Gaussian quadrature. Based on the principle of least action and the requirements of a canonical transformation, different types of symplectic algorithms have been constructed by choosing different types of independent variables at two ends of the time step. The symmetric property of the four types of symplectic algorithms proposed in this paper is discussed, and the exact linear stability domain for small m, n and g is discussed. The linear stability and precision of different types of symplectic algorithms are tested using numerical examples. Copyright © 2011 John Wiley & Sons, Ltd.