Discrete time Hamiltonian spin systems

Preprint, 2014

We construct generating functions for symplectic maps on products of 2-spheres and use them to construct symplectic integrators for classical spin systems. They are the minimal possible such generating function and use no Lagrange multipliers or canonical variables. In the single spin case, the resulting {\em spherical midpoint method} is given by W−w=X(W+w|W+w|), where X(w)=w×∇H(w), H being the generating function. We establish the basic properties of the method and describe its relationship to collective symplectic integrators for spin systems based on the Hopf map. We introduce a numerical integrator for Riemannian manifolds called the {\em Riemannian midpoint method} and determine its properties with respect to isometries and Riemannian submersions and the conditions under which the spherical and Riemannian midpoint methods coincide.