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

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.

40 pages, 10 figures. v2: simplified numerical algorithm, discussion of conservation laws

References in corpus (3)

• Continuous and discrete Clebsch variational principles
• Vortex dynamics in $R^4$
• Un-reduction

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• Conservative Integrators for Many-body Problems
• Hamiltonian dynamics of several rigid bodies interacting with point vortices
• Relative periodic solutions of the n-vortex problem on the sphere
• Geometry of discrete-time spin systems
• Initializing and stabilizing variational multistep algorithms for modeling dynamical systems
• Shape Dynamics of $N$ Point Vortices on the Sphere
• Variational integrators for perturbed non-canonical Hamiltonian systems