Lie-poisson integrators: A hamiltonian, variational approach

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Abstract

In this paper we present a systematic and general method for developing variational integrators for Lie- Poisson Hamiltonian systems living in a finite-dimensional space g*, the dual of Lie algebra associated with a Lie group G. These integrators are essentially different discretized versions of the Lie-Poisson variational principle, or a modified Lie-Poisson variational principle proposed in this paper. We present three different integrators, including symplectic, variational Lie-Poisson integrators on G×g* and on g×g*, as well as an integrator on g* that is symplectic under certain conditions on the Hamiltonian. Examples of applications include simulations of free rigid body rotation and the dynamics of N point vortices on a sphere. Simulation results verify that some of these variational Lie-Poisson integrators are good candidates for geometric simulation of those two Lie-Poisson Hamiltonian systems.

Original languageEnglish (US)
Pages (from-to)1609-1644
Number of pages36
JournalInternational Journal for Numerical Methods in Engineering
Volume82
Issue number13
DOIs
StatePublished - Jun 25 2010

All Science Journal Classification (ASJC) codes

  • Numerical Analysis
  • Engineering(all)
  • Applied Mathematics

Keywords

  • Hamiltonian
  • Lie-poisson systems
  • Variational integrators

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