Hamiltonian Systems
by Lewis Napper (University of Surrey, UK)
In this talk, we provide an introduction to the use of geometry as a tool for studying PDEs of Monge–Ampère type. We begin by reviewing some models of physical systems featuring Monge–Ampère equations, before demonstrating that all such equations may be thought of as a pair of constraints on the phase space of the system they describe, with submanifolds satisfying these constraints representing solutions.
Taking the Poisson equation for the pressure of an incompressible fluid flow in two dimensions as a case study, we demonstrate how properties of these submanifolds correspond to properties of the solutions. In particular, we highlight the existence of a Lychagin–Rubtsov metric whose signature and curvature are determined by the accumulation of vorticity and strain. This example naturally leads to a framework for studying incom- pressible Navier–Stokes flows on Riemannian manifolds of arbitrary dimension, including an extension of the Weiss–Okubo criterion for a vortex. If time permits, we will discuss how topological information about vortices may be obtained in two and three dimensions.
This talk is based on collaboration with Martin Wolf, Ian Roulstone, and Volodya Rubtsov.
https://www.arxiv.org/abs/2302.11604v2 and references therein.
