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Published April 2009 | public
Journal Article

Hamilton–Pontryagin integrators on Lie groups part I: introduction and structure-preserving properties

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 Störmer–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.

Additional Information

© SFoCM 2008. Received: 5 February 2007. Revised: 7 January 2008. Accepted: 29 February 2008. Published online: 15 May 2008. Research partially supported by the National Science Foundation through NSF grant DMS-0204474.

Additional details

Created:
August 21, 2023
Modified:
October 18, 2023