The Exact Computation of the Free Rigid Body Motion and Its Use in Splitting Methods

In this article we thoroughly investigate the possibility of employing the exact computation of the free rigid body motion as a component of splitting methods for problems of rigid bodies subject to external forces. We review various matrix and quaternion representations of the solution of the free rigid body equation, which involve Jacobi ellipic functions and elliptic integrals, and are amenable to numerical computations. We consider implementations which are exact, (= computed to machine precision), and semi–exact, (= approximated via quadrature formulas). We perform a set of thorough numerical comparisons to state of the art geometrical integrators for rigid bodies, such as the modified discrete Moser–Veselov method. The numerical computations indicate that these techniques, combined with splitting methods, can be profitably applied to the numerical integration of torqued rigid bodies.