Almost Poisson Integration of Rigid Body Systems

In this paper we discuss the numerical integration of Lie-Poisson Systems using the mid-point rule. Since such systems result from the reduction of hamiltonian systems with symmetry by Lie Group actions, we also present examples of reconstruction rules for the full dynamics. A primary motivation is to preserve in the integration process, various conserved quantities of the original dynamics. A main result of this paper is a third order error estimate for the Lie-Poisson structure where h is the integration step-size. We note that Lie-Poisson systems appear naturally in many areas of physical science and engineering, including theoretical mechanics of fluids and plasmas, satellite dynamics, and polarization dynamics. In the present paper we consider a series of progressively complicated examples related to rigid body systems. We also consider a dissipative example associated to a Lie-Poisson system. The behavior of the mid-point rule and an associated reconstruction rule is numerically explored