The Geometry and Dynamics of Interacting Rigid Bodies and Point Vortices

TL;DR

This paper derives the equations of motion for a rigid body in a 2D perfect fluid with point vortices using symplectic reduction, revealing the underlying geometric structures and Poisson brackets.

Contribution

It introduces a novel symplectic reduction framework for rigid bodies with fluid interactions, clarifying the geometric and Poisson structures involved.

Findings

Recovered known Poisson structures for the system

Provided explicit expressions for symplectic leaves

Clarified the role of curvature and cocycles in dynamics

Abstract

We derive the equations of motion for a planar rigid body of circular shape moving in a 2D perfect fluid with point vortices using symplectic reduction by stages. After formulating the theory as a mechanical system on a configuration space which is the product of a space of embeddings and the special Euclidian group in two dimensions, we divide out by the particle relabelling symmetry and then by the residual rotational and translational symmetry. The result of the first stage reduction is that the system is described by a non-standard magnetic symplectic form encoding the effects of the fluid, while at the second stage, a careful analysis of the momentum map shows the existence of two equivalent Poisson structures for this problem. For the solid-fluid system, we hence recover the ad hoc Poisson structures calculated by Shashikanth, Marsden, Burdick and Kelly on the one hand, and Borisov, Mamaev, and Ramodanov on the other hand. As a side result, we obtain a convenient expression for the symplectic leaves of the reduced system and we shed further light on the interplay between curvatures and cocycles in the description of the dynamics.