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A minimal-variable symplectic integrator on spheres


Authors: Robert McLachlan, Klas Modin and Olivier Verdier
Journal: Math. Comp. 86 (2017), 2325-2344
MSC (2010): Primary 37J15, 37M15, 65P10
DOI: https://doi.org/10.1090/mcom/3153
Published electronically: November 16, 2016
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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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Additional Information

Robert McLachlan
Affiliation: Institute of Fundamental Sciences, Massey University, New Zealand
Email: r.mclachlan@massey.ac.nz

Klas Modin
Affiliation: Mathematical Sciences, Chalmers University of Technology and University of Gothenburg, Sweden
Email: klas.modin@chalmers.se

Olivier Verdier
Affiliation: Department of Computing, Mathematics and Physics, Bergen University College, Norway
Email: olivier.verdier@hib.no

DOI: https://doi.org/10.1090/mcom/3153
Received by editor(s): February 12, 2015
Received by editor(s) in revised form: October 11, 2015, and February 22, 2016
Published electronically: November 16, 2016
Article copyright: © Copyright 2016 American Mathematical Society