Geometric Hydrodynamics: from Euler, to Poincar\'e, to Arnold

TL;DR

This paper provides an overview of the development of geometric hydrodynamics, highlighting Arnold's 1966 discovery that links Euler's equations to geodesics on volume-preserving diffeomorphisms, foundational for the field.

Contribution

It traces the historical and mathematical progression from Euler and Poincaré to Arnold, explaining the geometric interpretation of fluid dynamics and its broad implications.

Findings

Arnold's geometric interpretation of Euler's equations

Connection of hydrodynamics to infinite-dimensional Riemannian geometry

Applications to optimal transport, shape analysis, and information theory

Abstract

These are lecture notes for a short winter course at the Department of Mathematics, University of Coimbra, Portugal, December 6--8, 2018. The course was part of the 13th International Young Researchers Workshop on Geometry, Mechanics and Control. In three lectures I trace the work of three heroes of mathematics and mechanics: Euler, Poincar\'e, and Arnold. This leads up to the aim of the lectures: to explain Arnold's discovery from 1966 that solutions to Euler's equations for the motion of an incompressible fluid correspond to geodesics on the infinite-dimensional Riemannian manifold of volume preserving diffeomorphisms. In many ways, this discovery is the foundation for the field of geometric hydrodynamics, which today encompasses much more than just Euler's equations, with deep connections to many other fields such as optimal transport, shape analysis, and information theory.