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UBC CS TR-93-13 Summary

Symplectic Integration of Constrained Hamiltonian Systems by Runge-Kutta Methods, April 1993 Sebastian Reich, 24 pages

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 we investigate the symplecticity of numerical integrators for constrained Hamiltonian systems. In the first part of the paper we show that those implicit Runge--Kutta methods which result in symplectic integrators for unconstrained Hamiltonian systems can be directly applied to constrained Hamiltonian systems. The resulting discretization scheme is symplectic but does not, in general, preserve the constraints. In the second part of the paper we discuss partitioned Runge--Kutta methods. Again it turns out that those partitioned Runge--Kutta methods which are symplectic for unconstrained systems can be applied to constrained Hamiltonian systems. We show that, in contrast to implicit Runge--Kutta methods, the class of symplectic partitioned Runge--Kutta methods includes methods that also preserve the constraints. In the third part of the paper we discuss constrained Hamiltonian systems with separable Hamiltonian from a Lie algebraic point of view. This approach not only provides a different approach to the numercial integration of Hamiltonian systems but also allows for a straighforward backward error analysis.

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