Lie-Poisson Hamilton-Jacobi theory and Lie-Poisson integrators

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 Lie algebras. The techniques are baseda on time-stepping with the generating function obtained as an approximate solution to the Hamilton-Jacobi equation, following ideas of deVogelaére, Channel,, and Feng. To accomplish this, the Hamilton-Jacobi theory is reduced from T*G to g*, where 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.