Hamilton-Pontryagin Integrators on Lie Groups: Introduction and Structure-Preserving Properties

Nawaf Bou-Rabee; Jerrold E. Marsden

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arXiv · 0801.0996

Hamilton-Pontryagin Integrators on Lie Groups: Introduction and Structure-Preserving Properties

Nawaf Bou-Rabee·Jerrold E. Marsden

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Abstract

In this paper structure-preserving time-integrators for rigid body-type mechanical systems are derived from a discrete Hamilton-Pontryagin variational principle. From this principle one can derive a novel class of variational partitioned Runge-Kutta methods on Lie groups. Included among these integrators are generalizations of symplectic Euler and St\"{o}rmer-Verlet integrators from flat spaces to Lie groups. Because of their variational design, these integrators preserve a discrete momentum map (in the presence of symmetry) and a symplectic form. In a companion paper, we perform a numerical analysis of these methods and report on numerical experiments on the rigid body and chaotic dynamics of an underwater vehicle. The numerics reveal that these variational integrators possess structure-preserving properties that methods designed to preserve momentum (using the coadjoint action of the Lie group) and energy (for example, by projection) lack.

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Nawaf Bou-Rabee, Jerrold E. Marsden. 2008-01-07. Hamilton-Pontryagin Integrators on Lie Groups: Introduction and Structure-Preserving Properties. https://arxiv.org/abs/0801.0996

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