Poisson brackets for the dynamically interacting system of a 2D rigid cylinder and $N$ point vortices: the case of arbitrary smooth cylinder shapes

This paper basically extends the work of Shashikanth, Marsden, Burdick and Kelly [17] by showing that the Hamiltonian (Poisson bracket) structure of the dynamically interacting system of a 2-D rigid circular cylinder and $N$ point vortices, when the vortex strengths sum to zero and the circulation around the cylinder is zero, also holds when the cylinder has arbitrary (smooth) shape. This extension is a consequence of a reciprocity relation, obtainable by an application of a classical Green's formula, that holds for this problem. Moreover, even when the vortex strengths do not sum to zero but with the circulation around the cylinder still zero, it is shown that there is a Poisson bracket for the system which differs from the previous bracket by the inclusion of a 2-cocycle term. Finally, comparisons are made to the works of Borisov, Mamaev and Ramodanov [15], [16], [5], [4].