Abstract
The equations of motion for a system of point vortices on an oriented Riemannian surface of finite topological type are presented. The equations are obtained from a Green’s function on the surface. The uniqueness of the Green’s function is established under hydrodynamic conditions at the surface’s boundaries and ends. The hydrodynamic force on a point vortex is computed using a new weak formulation of Euler’s equation adapted to the point vortex context. An analogy between the hydrodynamic force on a massive point vortex and the electromagnetic force on a massive electric charge is presented as well as the equations of motion for massive vortices. Any noncompact Riemann surface admits a unique Riemannian metric such that a single vortex in the surface does not move (“Steady Vortex Metric”). Some examples of surfaces with steady vortex metric isometrically embedded in \(\mathbb {R}^3\) are presented.
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Notes
In an arbitrary local coordinate system \((\xi ^1,\xi ^2)\), the Riemannian Laplacian is given by \(\Delta =\frac{1}{\sqrt{|g|}}\frac{\partial }{\partial \xi ^j}g^{jk}\sqrt{|g|} \frac{\partial }{\partial \xi ^k}\) where the sum over repeated indices is assumed, the Riemannian metric is given by \(g_{jk}d\xi ^j\otimes d\xi ^k, g^{jk}\) is the inverse of matrix \(g_{jk}\), and |g| is the absolute value of the determinant of the matrix \(g_{jk}\).
In the coordinates of footnote 1, \(\mathrm{d}y=*\mathrm{d}x=-\frac{1}{\sqrt{|g|}}g_{ki}\epsilon ^{il}\frac{\partial x}{\partial \xi ^l} d\xi ^k\), where \(\epsilon ^{il}=-\epsilon ^{li}\) and \(\epsilon ^{12}=1\).
The constant may be determined by the normalization \(\int _S G(q,p)\mu (q)=0\).
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Acknowledgements
The authors thank to: Jair Koiller who presented and taught them the subject, Hildeberto Cabral who invited CGR to write a review on the subject, and Björn Gustafsson for the discussions.
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Communicated by George Haller.
CGR is partially supported by supported by FAPESP (Brazil) 2011/16265-8.
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Ragazzo, C.G., de Barros Viglioni, H.H. Hydrodynamic Vortex on Surfaces. J Nonlinear Sci 27, 1609–1640 (2017). https://doi.org/10.1007/s00332-017-9380-7
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DOI: https://doi.org/10.1007/s00332-017-9380-7