In this talk we discuss numerical integration techniques for Lie-Poisson systems. The differential equations can be viewed as a generalization of Hamiltons equations. There are several conservations laws: energy, the symplectic structure, Casimirs and coadjoint orbits. We construct numerical methods that automatically stay on the coadjoint orbits. Moreover, they conserve the Casimirs and the energy exactly. Numerical simulations are shown to illustrate the performance of the new methods.
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