The Hamiltonian Structure of a 2D Rigid Circular Cylinder Interacting Dynamically with N Point Vortices

Shashikanth, B. N., J. E. Marsden, J. W. Burdick and S. D. Kelly

Phys. of Fluids 14, 1214-1227

Abstract:

This paper studies the dynamical fluid plus rigid-body system consisting of a two-dimensional rigid cylinder of general cross-sectional shape interacting with N point vortices. We derive the equations of motion for this system and show that, in particular, if the vortex strengths sum to zero and the rigid-body has a circular shape, the equations are Hamiltonian with respect to a Poisson bracket structure that is the sum of the rigid body Lie­Poisson bracket on $\mathfrak{se}(2)^{\ast}$, the dual of the Lie algebra of the Euclidean group on the plane, and the canonical Poisson bracket for the dynamics of N point vortices in an unbounded plane. We then use this Hamiltonian structure to study the linear and nonlinear stability of the moving Föppl equilibrium solutions using the energy-Casimir method.