Lie-poisson integrators: A hamiltonian, variational approach

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.