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Vortex motion and geometric function theory: the role of connections
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).ORCID iD: 0000-0002-3125-3030
2019 (English)In: Philosophical Transactions. Series A: Mathematical, physical, and engineering science, ISSN 1364-503X, E-ISSN 1471-2962, Vol. 377, no 2158, article id 20180341Article in journal (Refereed) Published
Abstract [en]

We formulate the equations for point vortex dynamics on a closed two-dimensional Riemannian manifold in the language of affine and other kinds of connections. This can be viewed as a relaxation of standard approaches, using the Riemannian metric directly, to an approach based more on local coordinates provided with a minimal amount of extra structure. The speed of a vortex is then expressed in terms of the difference between an affine connection derived from the coordinate Robin function and the Levi-Civita connection associated with the Riemannian metric. A Hamiltonian formulation of the same dynamics is also given. The relevant Hamiltonian function consists of two main terms. One of the terms is the well-known quadratic form based on a matrix whose entries are Green and Robin functions, while the other term describes the energy contribution from those circulating flows which are not implicit in the Green functions. One main issue of the paper is a detailed analysis of the somewhat intricate exchanges of energy between these two terms of the Hamiltonian. This analysis confirms the mentioned dynamical equations formulated in terms of connections. This article is part of the theme issue 'Topological and geometrical aspects ofmass and vortex dynamics'.

Place, publisher, year, edition, pages
ROYAL SOC , 2019. Vol. 377, no 2158, article id 20180341
Keywords [en]
point vortex motion, affine connection, projective connection, Robin function, renormalization, Hamiltonian
National Category
Mathematics
Identifiers
URN: urn:nbn:se:kth:diva-262761DOI: 10.1098/rsta.2018.0341ISI: 000488279300001Scopus ID: 2-s2.0-85073260708OAI: oai:DiVA.org:kth-262761DiVA, id: diva2:1363197
Note

QC 20191022

Available from: 2019-10-22 Created: 2019-10-22 Last updated: 2019-10-22Bibliographically approved

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Gustafsson, Björn

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