Arnold's geometric formulation of the Euler equations

Summary

Arnold's geometric formulation views the motion of an ideal incompressible fluid as a geodesic on the infinite-dimensional group of volume-preserving diffeomorphisms of the fluid domain. The metric is given by the kinetic energy of the fluid, and the Euler equations are the geodesic equations of this metric. Vladimir Arnold proposed it in 1966, extending the description of a rigid body as a geodesic on the rotation group. It brings the tools of Riemannian geometry and Lie group theory, such as curvature and conservation laws from symmetries, to hydrodynamics.