The Breadth of Symplectic and Poisson Geometry

Abstract

This paper concerns the dynamics of measure-valued solutions of the EPDiff equations, standing for the Euler-Poincaré equations associated with the diffeomorphism group (of R n or of an n-dimensional manifold M ). The paper focuses on Lagrangians that are quadratic in the velocity fields and their first derivatives; that is, on geodesic motion on the diffeomorphism group with respect to a right invariant Sobolev H 1 metric. The corresponding Euler-Poincaré (EP) equations are the EPDiff equations, which coincide with the averaged template matching equations (ATME) from computer vision and agree with the Camassa-Holm (CH) equations for shallow water waves in one dimension. The corresponding equations for the volume preserving diffeomorphism group are the LAE (Lagrangian averaged Euler) equations for incompressible fluids.The paper shows that the EPDiff equations are generated by a smooth vector field on the diffeomorphism group for sufficiently smooth solutions. This is analogous to known results for incompressible fluids-both the Euler equations and the LAE equations-and it shows that for sufficiently smooth solutions, the equations are well-posed for short time. Numerical evidence suggests that, as time progresses, these smooth solutions break up into singular solutions which, at least in one dimension, exhibit soliton behavior.These non-smooth, or measure-valued, solutions are higher dimensional generalizations of the peakon solutions of the CH equation in one dimension. One of the main purposes of the paper is to show that many of the properties of these measure-valued solutions can be understood through the fact that their solution Ansatz is a momentum map. Some additional geometry is also pointed out, for example, that this momentum map is one part of a dual pair.

A jetlet hierarchy for ideal fluid dynamics

“…Upon integrating these equations to obtain α q t ( ) and α p t ( ), one can invoke (5) to obtain a time-dependent velocity field, u, which solves (1), see [7]. In other words, we have found a class of solutions to (1) parametrized by a finite-dimensional subspace of initial conditions.…”

“…That is, if the momentum =  m x u x ( ) ( ) is equal to δ − p x y ( )for some constant vector p, then the flow field u is in − + C s n ( 1) 2 . Hamiltonʼs principle for this Lagrangian leads to the EPDiff equation [7] governing the time evolution of the flow field…”