The symmetric representation of the rigid body equations and their discretization

This paper analyses continuous and discrete versions of the generalized rigid body equations and the role of these equations in numerical analysis, optimal control and integrable Hamiltonian systems. In particular, we present a symmetric representation of the rigid body equations on the Cartesian product SO ( n )× SO ( n ) and study its associated symplectic structure. We describe the relationship of these ideas with the Moser-Veselov theory of discrete integrable systems and with the theory of variational symplectic integrators. Preliminary work on the ideas discussed in this paper may be found in Bloch et al (Bloch A M, Crouch P, Marsden J E and Ratiu T S 1998 Proc. IEEE Conf. on Decision and Control 37 2249-54).