Symplectic Integration of Constrained Hamiltonian Systems by Composition Methods

Abstract

Recent work reported in the literature suggests that for the long-term integration of Hamiltonian dynamical systems one should use methods that preserve the symplectic structure of the flow. In this paper the symplecticity of numerical integrators is investigated for constrained Hamiltonian systems with holonomic constraints. The following two results will be derived. (i) It is shown that any first- or second-order symplectic integrator for unconstrained problems can be generalized to constrained systems such that the resulting scheme is symplectic and preserves the constraints. Based on this, higher-order methods can be derived by the same composition methods used for unconstrained problems.(ii) Leimkuhler and Reich [Math. Comp, 63 (1994), pp. 589–605] derived symplectic integrators based on Dirac’s reformulation of the constrained problem as an unconstrained Hamiltonian system. However, although the unconstrained reformulation can be handled by direct application of any symplectic implicit Runge–Kutta m...