The movement of a solid in an incompressible perfect fluid as a geodesic flow

TL;DR

This paper demonstrates that the motion of a rigid body in an incompressible perfect fluid can be understood as geodesic flow on an infinite-dimensional manifold, extending Arnold's classical results.

Contribution

It proves that classical solutions for a rigid body in a perfect fluid are geodesics of a Riemannian manifold, generalizing Arnold's work to fluid-solid interaction.

Findings

Classical solutions are geodesics of an infinite-dimensional Riemannian manifold.

The action functional is the integral of the total kinetic energy.

Extension of Arnold's theorem to fluid-solid systems.

Abstract

The motion of a rigid body immersed in an incompressible perfect fluid which occupies a three- dimensional bounded domain have been recently studied under its PDE formulation. In particular classical solutions have been shown to exist locally in time. In this note, following the celebrated result of Arnold concerning the case of a perfect incompressible fluid alone, we prove that these classical solutions are the geodesics of a Riemannian manifold of infinite dimension, in the sense that they are the critical points of an action, which is the integral over time of the total kinetic energy of the fluid-rigid body system.