Arnold’s Hydrodynamics Revisited

The purpose of this short 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 We conclude by showing that even this new approach can be superseded by dynamics on Lie groupoids, along ideas pioneered by the brothers E. and F. Cosserat or H. Weyl, on the condition to change the underlying philosophy.