The Dynamics of a Rigid Body in Potential Flow with Circulation

TL;DR

This paper derives and interprets the equations governing a rigid body's motion in potential flow with circulation, revealing geometric structures and curvature effects underlying the dynamics.

Contribution

It provides a geometric derivation of the equations of motion, interprets the Kutta-Zhukowski force as a curvature effect, and links the dynamics to geodesic flow on a central extension of SE(2).

Findings

Recovered classical equations of motion with geometric insight

Interpreted the Kutta-Zhukowski force as a curvature-related effect

Connected rigid body motion with circulation to geodesic flow on a central extension of SE(2)

Abstract

We consider the motion of a two-dimensional body of arbitrary shape in a planar irrotational, incompressible fluid with a given amount of circulation around the body. We derive the equations of motion for this system by performing symplectic reduction with respect to the group of volume-preserving diffeomorphisms and obtain the relevant Poisson structures after a further Poisson reduction with respect to the group of translations and rotations. In this way, we recover the equations of motion given for this system by Chaplygin and Lamb, and we give a geometric interpretation for the Kutta-Zhukowski force as a curvature-related effect. In addition, we show that the motion of a rigid body with circulation can be understood as a geodesic flow on a central extension of the special Euclidian group SE(2), and we relate the cocycle in the description of this central extension to a certain curvature tensor.