Hamiltonian dynamics of several rigid bodies interacting with point vortices

TL;DR

This paper derives a Hamiltonian framework for the dynamics of multiple rigid bodies of arbitrary shape interacting with point vortices in a 2D fluid, enabling efficient simulation and analysis.

Contribution

It introduces a direct, first-principles derivation of the Hamiltonian dynamics for multiple rigid bodies with arbitrary shapes interacting with point vortices.

Findings

Derived a finite-dimensional Hamiltonian system for multiple bodies and vortices.

Established a Lagrangian formulation suitable for variational integrators.

Validated the approach through numerical simulations of various configurations.

Abstract

We derive the dynamics of several rigid bodies of arbitrary shape in a 2-dimensional inviscid and incompressible fluid, whose vorticity field is given by point vortices. We adopt the idea of Vankerschaver et al. (2009) to derive the Hamiltonian formulation via symplectic reduction from a canonical Hamiltonian system. The reduced system is described by a non-canonical symplectic form, which has previously been derived for a single, circular disk using heavy differential-geometric machinery in an infinite-dimensional setting. In contrast, our derivation makes use of the fact that the dynamics of the fluid, and thus the point vortex dynamics, is determined from first principles. Using this knowledge we can directly determine the dynamics on the reduced, finite-dimensional phase space, using only classical mechanics. Furthermore, our approach easily handles several bodies of arbitrary shapes. From the Hamiltonian description we derive a Lagrangian formulation, which enables the system for variational time integrators. We briefly describe how to implement such a numerical scheme and simulate different configurations for validation.