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A minimal-variable symplectic integrator on spheres

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Authors Robert McLachlan
Klas Modin
Olivier Verdier
Publication year 2015
Published at Department of Mathematical Sciences, Mathematics
Language en
Links arxiv.org/abs/1402.3334
arxiv.org/pdf/1402.3334v4
Subject categories Computational Mathematics, Geometry

Abstract

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.

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