On the differential geometry of infinite-dimensional Lie groups and its application to the hydrodynamics of perfect fluids

Abstract

In the year 1765, L. Euler [8] published the equations of rigid body motion which bear his name. It does not seem useless to mark the 200th anniversary of Euler’s equations by a modern exposition of the question.

The eulerian motions of a rigid body are the geodesics on the group of rotations of three dimensional euclidean space endowed with a left invariant metric. Basically, Euler’s theory makes use of nothing but this circumstance; hence Euler’s equations still hold for an arbitrary group. For the other groups, one obtains the “Euler equations” of rigid body motion in the n-dimensional space, the equations of the hydrodynamics of ideal fluids, etc.

Euler’s theorem on the stability of the rotations around the longest and shortest axes of the inertia ellipsoid also has analogues in the case of an arbitrary group. In the case of hydrodynamics, this analogy is an extension of Rayleigh’s theorem on the stability of flows whose velocity profile is inflexion free (see §10).

As another application of Euler’s theory, we prove in §8, the explicit formula of the riemannian curvature of a group endowed with a left invariant metric. In §11, this formula is used in the study of the curvature of the group of diffeomorphisms, whose geodesics are ideal fluid flows.

In what follows, I tried, following the call of Bourbaki [6], to always substitute blind computations for Euler’s lucid ideas.