Arnold Hydrodynamics Revisited

TL;DR

This paper revisits Arnold's Lie group framework for hydrodynamics, extending it to arbitrary Lie pseudogroups and demonstrating that the results can be derived from formal methods, with potential generalizations to Lie groupoids.

Contribution

It extends Arnold's hydrodynamics framework from volume-preserving transformations to arbitrary Lie pseudogroups and shows formal derivations of the same results, suggesting broader applicability.

Findings

Extended the framework to arbitrary Lie pseudogroups.

Proved results can be obtained from formal, not just analytical, methods.

Suggested further generalization using Lie groupoids.

Abstract

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 We conclude by showing that even this new approach can be superseded by dynamics on Lie groupoids, along ideas pioneered by the brothers E. and F. Cosserat or H. Weyl at the beginning of the last century, on the condition to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.