Geometrical theory of two-dimensional hydrodynamics with special reference to a system of point vortices

Abstract

Two-dimensional hydrodynamics is formulated geometrically with a metric defined in terms of stream function and vorticity for flows of an inviscid incompressible fluid in unbounded space. Based on this, the vorticity equation is derived as a geodesic equation over a group of diffeomorphisms of a fluid at rest at infinity with vorticity having compact supports, which is also rewritten in the form of a Hamilton's equation with a Lie–Poisson bracket. In particular, this formulation is applied to a flow induced by a finite number ( N) of point vortices. It is shown that the geodesic equation reduces to the well-known Hamiltonian system of N point vortices. This is considered as a dynamics of an internal space of finite degrees of freedom, as against the former background dynamical system described by a group of diffeomorphisms of infinite dimensions and Riemannian geometry. It is shown that the dynamical trajectories of the system of N degrees of freedom governed by Hamiltonian system of equations are the projections of the geodesics of ( N+1)-dimensional Finslerian space associated with a fundamental function obtained from the original Lagrangian of N point vortices. The Finsler geometry provides us curvature tensors and equation of geodesic deviation. As an example, the Finslerian geometry is applied to a system of a vortex pair, in which sum of two vortex strengths vanishes and the pair makes steady translational motion with a constant velocity. The scalar curvature of the motion is found to be zero, and its relation to geodesic deviation is considered.