The Symmetric Representation of the Rigid Body Equations and their
Discretization
Bloch, A. M., P. Crouch, J. E. Marsden and T. S. Ratiu
Nonlinearity 15, 1309-1341
Abstract:
This paper analyzes 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) x 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 the present paper may be found in
a 1998 paper of the authors.