HAL : hal-00363271, version 1

arXiv : 0902.4306

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ARNOLD HYDRODYNAMICS REVISITED

Jean-François Pommaret 1

(05/2007)

The purpose of this paper is to revisit the infinite Lie group theoretical framework of hydrodynamics developped by V. Arnold in 1966. First of all, we extend this approach from the Lie pseudogroup of volume preserving transformations to an arbitrary Lie pseudogroup. Then we prove that, contrary to what could be believed from the work of Arnold which is of a purely analytical nature, the same results can be obtained from a purely formal point of view. Finally, we provide the analogue for both the so-called "body" and "space" dynamical equations. We conclude by showing that even this new approach can be superseded by dynamics on Lie groupoids, along ideas pioneered by the brothers E. and F. Cosserat or H. Weyl at the beginning of the last century, on the condition to change the underlying philosophy.

1 :	Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique (CERMICS)
	Ecole des Ponts ParisTech

Domaine

:

Mathématiques/Equations aux dérivées partielles Mathématiques/Théorie des groupes Sciences de l'ingénieur/Mécanique

Mots Clés : Lie group – Lie pseudogroup – Lie groupoid – gauge theory – duality theory – adjoint operator – Spencer operator

hal-00363271, version 1

http://hal.archives-ouvertes.fr/hal-00363271

oai:hal.archives-ouvertes.fr:hal-00363271

Contributeur : Jean-François Pommaret <>

Soumis le : Mardi 24 Février 2009, 16:32:32

Dernière modification le : Samedi 3 Octobre 2009, 12:07:40