Geometric Hydrodynamics of Compressible Fluids

Abstract

We develop a geometric framework for Newton's equations on infinite-dimensional configuration spaces to describe numerous fluid dynamical equations. According to V. Arnold, the Euler equations of an incompressible fluid define a geodesic flow on the group of volume-preserving diffeomorphisms of a compact manifold. It turns out that a much greater variety of hydrodynamical systems can be viewed as Newton's equations (adding a potential energy to the kinetic energy Lagrangian) on the group of all diffeomorphisms and the space of smooth probability densities. This framework encompasses compressible fluid dynamics, shallow water equations, Fisher information geometry, compressible and incompressible magnetohydrodynamics and can be adapted to include relativistic fluids and the infinite-dimensional Neumann problem. Relations between these diverse systems are described using the Madelung transform and the formalism of Hamiltonian and Poisson reduction.