TY - GEN A1 - Leimkuhler, Benedict A1 - Reich, Sebastian T1 - The Numerical Solution of Constrained Hamiltonian Systems. N2 - A Hamiltonian system subject to smooth constraints can typically be viewed as a Hamiltonian system on a manifold. Numerical computations, however, must be performed in $ R^n$. In this paper, canonical transformations from ``Hamiltonian differential--algebraic equations'' to ODEs in Euclidean space are considered. In \S2, canonical parameterizations or local charts are developed and it is shown how these can be computed in a practical framework. In \S3 we consider the construction of unconstrained Hamiltonian ODE systems in the space in which the constraint manifold is embedded which preserve the constraint manifold as an integral invariant and whose flow reduces to the flow of the constrained system along the manifold. It is shown that certain of these unconstrained Hamiltonian systems force Lyapunov stability of the constraint--invariants, while others lead to an unstable invariant. In \S4, we compare various projection techniques which might be incorporated to better insure preservation of the constraint--invariants in the context of numerical discretization. Numerical experiments illustrate the degree to which the constraint and symplectic invariants are maintained under discretization of various formulations. {\bf Keywords:} differential--algebraic equations, Hamiltonian systems, canonical discretization schemes. {\bf AMS(MOS):} subject classification 65L05. T3 - ZIB-Report - SC-92-16 KW - differential algebraic equations KW - Hamiltonian system KW - canonical discretization schemes Y1 - 1992 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-868 ER - TY - GEN A1 - Holder, Thomas A1 - Leimkuhler, Benedict A1 - Reich, Sebastian T1 - Explicit Variable Step-Size and Time-Reversible Integration N2 - A variable step-size, semi-explicit variant of the explicit Störmer-Verlet method has been proposed for the time-reversible integration of Newton's equations of motion by Huang & Leimkuhler. Here we propose a fully explicit version of this approach applicable to explicit and symmetric integration methods for general time-reversible differential equations. As applications, we discuss the variable step-size, time-reversible, and fully explicit integration of rigid body motion and reversible Nos\'e-Hoover dynamics. T3 - ZIB-Report - SC-98-17 KW - reversible differential equations KW - time integration KW - adaptive methods Y1 - 1998 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-3607 ER -