Foundations of Computational Mathematics

, Volume 9, Issue 2, pp 197-219

First online:

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

  • Nawaf Bou-RabeeAffiliated withApplied and Computational Mathematics, Caltech Email author 
  • , Jerrold E. MarsdenAffiliated withControl and Dynamical Systems, Caltech

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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.


Variational integrators Hamilton–Pontryagin Lie group integrators

Mathematics Subject Classification (2000)

37M15 58E30 65P10 70EXX 70HXX