Lie-Poisson Hamilton-Jacobi Theory and Lie-Poisson Integrators

Ge, Z. and J. E. Marsden

Phys. Lett. A, 133, 134-139

Abstract:

We present results on numerical integrators that exactly preserve momentum maps and Poisson brackets, thereby inducing integrators that preserve the natural Lie-Poisson structure on the duals of the Lie algebras. The techniques are based on time-stepping with the generating function obtained as an approximate solution to the Hamilton Jacobi equation, following ideas of de Vogelaére, Channel, and Feng. To accomplish this, the Hamilton-Jacobi theory is reduced from J^*G to $\mathfrak{g}^*$, where $\mathfrak{g}$ is the Lie algebra of a Lie group G. The algorithms exactly preserve any additional conserved quantities in the problem. An explicit algorithm is given for any semi-simple group and in particular for the Euler equation of rigid body dynamics.