Lie-Poisson integrators: a Hamiltonian, variational approach

Z. Ma and C. W. Rowley

Int. J. Numer. Methods Eng. 82(13) : 1609-1644, June 2010.


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.

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