A minimal-variable symplectic integrator on spheres
Preprint, 2015

We construct a symplectic, globally defined, minimal coordinate, equivariant integrator on products of 2-spheres. Examples of corresponding Hamiltonian systems, called spin systems, include the reduced free rigid body, the motion of point vortices on a sphere, and the classical Heisenberg spin chain, a spatial discretisation of the Landau-Lifschitz equation. The existence of such an integrator is remarkable, as the sphere is neither a vector space, nor a cotangent bundle, has no global coordinate chart, and its symplectic form is not even exact. Moreover, the formulation of the integrator is very simple, and resembles the geodesic midpoint method, although the latter is not symplectic.


Robert McLachlan

Klas Modin

Chalmers, Mathematical Sciences, Mathematics

University of Gothenburg

Olivier Verdier

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Computational Mathematics



Basic sciences

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