Hamiltonian Fluid Mechanics

Abstract

This paper reviews the relatively recent application of the methods of Hamiltonian mechanics to problems in fluid dynamics. By Hamiltonian mechanics I mean all of what is often called classical mechanics-the subject of the textbooks by Lanczos ( 1970), Goldstein ( 1 980), and Arnol'd (1978). Since the advent of quantum mechanics, Hamiltonian methods have played an increasingly important role in both the classical and quan­ tum mechanics of particles and fields. By comparison, the introduction of Hamiltonian methods into fluid mechanics has been tardy. Why is this so? In general mechanical systems, the Lagrangian or Hamiltonian equa­ tions of motion are coupled equations governing the locations and veloc­ ities of massive particles or rigid bodies. These coupled equations cannot generally be solved for any subset of the dependent variables without also finding all of the other dependent variables. By contrast, the conventional Eulerian fluid equations are closed equations in the velocity, density, and entropy (regarding pressure as a prescribed function of the density and entropy) that can (in principle) be solved without also finding the trajectory of every fluid particle. Once the velocity field is known, the particle tra­ jectories can always be reconstructed by solving the equations for three independent, passively advected tracers (such as the initial Cartesian com­ ponents), but these extra computations are not required if only the Eulerian fields are sought. In the special case of constant-density flow, the Eulerian equations are dramatically simpler than the general Lagrangian or Hamil­ tonian equations for the fluid. From the Hamiltonian perspective, the extraordinary simplicity of the Eulerian description derives from a symmetry property of the fluid