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.