A minimal-variable symplectic integrator on spheres

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.

References in corpus (3)

• The Fascinating World of Landau-Lifshitz-Gilbert Equation: An Overview
• Symplectic Integration of Post-Newtonian Equations of Motion with Spin
• Symplectic integrators for classical spin systems

Cited by in corpus (9)

• Hamiltonians and canonical coordinates for spinning particles in curved space-time
• A Casimir preserving scheme for long-time simulation of spherical ideal hydrodynamics
• Geometric integration of classical spin dynamics via a mean-field Schrödinger equation
• Lie-Poisson methods for isospectral flows
• A minimal-variable symplectic method for isospectral flows
• Geometry of discrete-time spin systems
• Collective Heavy Top Dynamics
• Integrability of point-vortex dynamics via symplectic reduction: a survey
• B-stability of numerical integrators on Riemannian manifolds