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arXiv · 1211.4560

A novel formulation of point vortex dynamics on the sphere: geometrical and numerical aspects

Joris Vankerschaver·Melvin Leok
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Abstract

In this paper, we present a novel Lagrangian formulation of the equations of motion for point vortices on the unit 2-sphere. We show first that no linear Lagrangian formulation exists directly on the 2-sphere but that a Lagrangian may be constructed by pulling back the dynamics to the 3-sphere by means of the Hopf fibration. We then use the isomorphism of the 3-sphere with the Lie group SU(2) to derive a variational Lie group integrator for point vortices which is symplectic, second-order, and preserves the unit-length constraint. At the end of the paper, we compare our integrator with classical fourth-order Runge--Kutta, the second-order midpoint method, and a standard Lie group Munthe-Kaas method.

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Joris Vankerschaver, Melvin Leok. 2012-11-19. A novel formulation of point vortex dynamics on the sphere: geometrical and numerical aspects. https://doi.org/10.1007/s00332-013-9182-5

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