Poisson brackets for the dynamically coupled system of a free boundary and a neutrally buoyant rigid body in a body-fixed frame

TL;DR

This paper derives the Hamiltonian structure and Poisson brackets for a coupled fluid-rigid body system with a free surface, revealing its underlying geometric and dynamical properties in an inviscid, irrotational setting.

Contribution

It introduces a Hamiltonian formulation for the coupled fluid-rigid body system, combining Zakharov and Lie-Poisson brackets in a body-fixed frame.

Findings

The evolution equations form a Hamiltonian system.

Poisson brackets are the sum of Zakharov and Lie-Poisson brackets.

The formulation applies under general assumptions for the coupled system.

Abstract

The fully coupled dynamic interaction problem of the free surface of an incompressible fluid and a rigid body beneath it, in an inviscid, irrotational framework and in the absence of surface tension, is considered. Evolution equations of the global momenta of the body+fluid system are derived. It is then shown that, under fairly general assumptions, these evolution equations combined with the evolution equation of the free-surface, referred to a body-fixed frame, is a Hamiltonian system. The Poisson brackets of the system are the sum of the canonical Zakharov bracket and the non-canonical Lie-Poisson bracket. Variations are performed consistent with the mixed Dirichlet-Neumann problem governing the system.