Geometric integration theory can be employed when numerically solving ODEs or
PDEs with constraints. In this paper, we present several one-step algorithms of
various orders for ODEs on a collection of spheres. To demonstrate the
versatility of these algorithms, we present representative calculations for
reduced free rigid body motion (a conservative ODE) and a discretization of
micromagnetics (a dissipative PDE). We emphasize the role of isotropy in
geometric integration and link numerical integration schemes to modern
differential geometry through the use of partial connection forms; this
theoretical framework generalizes moving frames and connections on principal
bundles to manifolds with nonfree actions.Comment: This paper appeared in prin