@misc{LeimkuhlerReich, author = {Leimkuhler, Benedict and Reich, Sebastian}, title = {The Numerical Solution of Constrained Hamiltonian Systems.}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-868}, number = {SC-92-16}, abstract = {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.}, language = {en} } @misc{BarthLeimkuhlerReich, author = {Barth, Eric and Leimkuhler, Benedict and Reich, Sebastian}, title = {A Time-Reversible Variable-Stepsize Integrator for Constrained Dynamics}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-3220}, number = {SC-97-53}, abstract = {This article considers the design and implementation of variable-timestep methods for simulating holonomically constrained mechanical systems. Symplectic variable stepsizes are briefly discussed, we then consider time-reparameterization techniques employing a time-reversible (symmetric) integration method to solve the equations of motion. We give several numerical examples, including a simulation of an elastic (inextensible, unshearable) rod undergoing large deformations and collisions with the sides of a bounding box. Numerical experiments indicate that adaptive stepping can significantly smooth the numerical energy and improve the overall efficiency of the simulation.}, language = {en} } @misc{HolderLeimkuhlerReich, author = {Holder, Thomas and Leimkuhler, Benedict and Reich, Sebastian}, title = {Explicit Variable Step-Size and Time-Reversible Integration}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-3607}, number = {SC-98-17}, abstract = {A variable step-size, semi-explicit variant of the explicit St{\"o}rmer-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.}, language = {en} }