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Comptes Rendus Mathématique
Volume 352, n° 3
pages 205-211 (mars 2014)
Doi : 10.1016/j.crma.2014.01.006
Received : 5 December 2013 ;  accepted : 16 January 2014
On the Hamiltonian structure of the planar steady water-wave problem with vorticity
Sur la structure hamiltonienne du problème des ondes de surface planes stationnaires avec vorticité
 

Mark D. Groves a, b , Athanasios Stylianou a
a FR 6.1 - Mathematik, Universität des Saarlandes, Postfach 151150, 66041 Saarbrücken, Germany 
b Department of Mathematical Sciences, Loughborough University, Loughborough, Leics, LE11 3TU, UK 

Abstract

We consider the stream-function formulation of the hydrodynamic problem for steady rotational water waves both with and without surface tension. A natural Lagrangian formulation is presented from which (different) Hamiltonian formulations for the two cases are derived by duality theory in the spirit of the Legendre–Fenchel transform. The treatment is systematic and clarifies a recent ad hoc approach by Kozlov and Kuznetsov [[7]].

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Résumé

Nous considérons la formulation de la fonction du courant dans le problème hydrodynamique décrivant les ondes de surface rotationnelles stationnaires, avec ou sans tension superficielle. Dans les deux cas, nous présentons une formulation lagrangienne naturelle, à partir de laquelle (différentes) formulations hamiltoniennes sont dérivées à l'aide de la théorie de la dualité, dans l'esprit de la transformée de Legendre–Fenchel. La démarche est systématique et clarifie une approche ad hoc récente de Kozlov et Kuznetsov [[7]].

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