Spinning tops and magneto-hydrodynamics: Part 2

Abstract: In part 1, we reviewed the constrained variational problem originating from the Euler-Poincaré reduction of geodesics on Lie groups with left-invariant Riemannan metric and applied it to the case of SO(3) to derive the well-known Euler equations of a free rigid body. In this talk, we replicate those steps in the case of a semidirect product between the Lie-Fréchet group of diffeomorphisms and the space of one-forms on a domain of real space. Working at a formal level, this infinite dimensional group is equipped with a right-invariant Riemannian metric, and out come incompressible ideal magneto-hydrodynamics equations from the Euler-Poincaré reduction. Rather elegantly, Alfvén's frozen-in flux theorem is seen as a consequence of the semidirect product structure (which encodes advection), and relabelling symmetry is attributable to right-invariance.

dynamical systems

Audience: researchers in the topic

Sydney Dynamics Group Seminar

Series comments: Description: Research seminar for dynamical systems topics