Sub-Riemannian geometry applied to incompressible, inviscid fluids

Abstract

One field of fluid dynamics concerns the search for variational principles. So far, the Hamiltonian view and Riemannian geometry has been applied to find geodesics for hydrodynamic systems. Compared to Riemannian geometry sub-Riemannian geometry can be applied to search for geodesics of constrained systems such as vortex flows, where the vortex motion is restricted by rotations that are expressed by vortex-related conservation laws. The Nambu formulation of incompressible, inviscid fluid dynamics provides a nilpotent Lie algebra for two- and three-dimensional fluids, called Vortex-Heisenberg algebra, which makes it natural to apply sub-Riemannian geometry. As a first approach we consider discretized models. The resulting vortex geodesics for two-dimensional incompressible, inviscid discrete point vortices is a known special point vortex constellation, the equilibrium. We also outline a concept for the derivation of 3D vortex geodesics.