A Novel Formulation of Point Vortex Dynamics on the Sphere: Geometrical and Numerical Aspects

详细信息    查看全文

• 作者：Joris Vankerschaver (1) (2)
  Melvin Leok (1)
  
• 关键词：Point vortices ; Hopf fibration ; Symplectic integration ; Variational methods ; 37M15 ; 76B47 ; 70H03
• 刊名：Journal of Nonlinear Science
• 出版年：2014
• 出版时间：February 2014
• 年：2014
• 卷：24
• 期：1
• 页码：1-37
• 全文大小：1,344 KB
• 作者单位：Joris Vankerschaver (1) (2)
  Melvin Leok (1)
  
  1. Department of Mathematics, University of California, San Diego, 9500 Gilman Drive, La Jolla, CA, 92093-0112, USA
  2. Department of Mathematics, Imperial College London, London, SW7 2AZ, UK
  
• ISSN：1432-1467

文摘

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.