A changing-chart symplectic algorithm for rigid bodies and other Hamiltonian systems on manifolds

We revive the elementary idea of constructing symplectic integrators for Hamiltonian flows on manifolds by covering the manifold with the charts of an atlas, implementing the algorithm in each chart ( thus using coordinates) and switching among the charts whenever a coordinate singularity is approached. We show that this program can be implemented successfully by using a splitting algorithm if the Hamiltonian is the sum H-1 + H-2 of two (or more) integrable Hamiltonians. Profiting from integrability, we compute exactly the flows of H-1 and H-2 in each chart and thus compute the splitting algorithm on the manifold by means of its representative in any chart. This produces a symplectic algorithm on the manifold which possesses an interpolating Hamiltonian, and hence it has excellent properties of conservation of energy. We exemplify the method for a point constrained to the sphere and for a symmetric rigid body under the influence of positional potential forces.