A Novel Formulation of Point Vortex Dynamics on the Sphere: Geometrical and Numerical Aspects
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.
KeywordsPoint vortices Hopf fibration Symplectic integration Variational methods
Mathematics Subject Classification37M15 76B47 70H03
We are very grateful to the referees of this paper, whose comments and observations significantly improved our exposition.
We would like to dedicate this paper to the memory of Hassan Aref, whose kind encouragement and insightful remarks at the 2010 SIAM-SEAS meeting at the University of North Carolina, Charlotte, provided the initial stimulus for this work. Furthermore, we would like to thank J.D. Brown, C. Burnett, B. Cheng, F. Gay-Balmaz, M. Gotay, D. Holm, E. Kanso, S.D. Kelly, P. Newton, T. Ohsawa, B. Shashikanth and A. Stern for stimulating discussions and helpful remarks.
M.L. and J.V. are partially supported by NSF grants DMS-1010687, CMMI-1029445, and DMS-1065972. J.V. is on leave from a Postdoctoral Fellowship of the Research Foundation–Flanders (FWO-Vlaanderen). This work is supported by the irses project geomech (nr. 246981) within the 7th European Community Framework Programme.
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