Foundations of Computational Mathematics

, Volume 9, Issue 2, pp 197–219

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

Authors

    • Applied and Computational MathematicsCaltech
  • Jerrold E. Marsden
    • Control and Dynamical SystemsCaltech
Article

DOI: 10.1007/s10208-008-9030-4

Cite this article as:
Bou-Rabee, N. & Marsden, J.E. Found Comput Math (2009) 9: 197. doi:10.1007/s10208-008-9030-4

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.

Keywords

Variational integratorsHamilton–PontryaginLie group integrators

Mathematics Subject Classification (2000)

37M1558E3065P1070EXX70HXX

Copyright information

© SFoCM 2008