A minimal-variable symplectic integrator on spheres

McLachlan, Robert; Modin, Klas; Verdier, Olivier

doi:10.1090/mcom/3153

A minimal-variable symplectic integrator on spheres
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by Robert McLachlan, Klas Modin and Olivier Verdier PDF

Math. Comp. 86 (2017), 2325-2344 Request permission

Abstract:

We construct a symplectic, globally defined, minimal-variable, 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–Lifshitz 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.

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