Floating rigid bodies: a note on the conservativeness of the hydrostatic effects

TL;DR

This paper proves the conservativeness of hydrostatic forces on floating rigid bodies using Lagrangian mechanics, introduces a reduced problem framework, and discusses stability and oscillations around equilibrium.

Contribution

It explicitly derives the hydrostatic potential and applies the Routh procedure to simplify the dynamical analysis of floating bodies.

Findings

Hydrostatic forces are proven to be conservative.

A reduced dynamical problem with three variables is formulated.

The concept of pseudo-stability is introduced and analyzed.

Abstract

Within the framework of Lagrangian mechanics, the conservativeness of the hydrostatic forces acting on a floating rigid body is proved. The representation of the associated hydrostatic potential is explicitly worked out. The invariance of the resulting Lagrangian with respect surge, sway and yaw motions is used in connection with the Routh procedure in order to convert the original dynamical problem into a reduced one, in three independent variables. This allows to put on rational grounds the study of hydrostatic equilibrium, introducing the concept of pseudo--stability, meant as stability with respect to the reduced problem. The small oscillations of the system around a pseudo-stable equilibrium configuration are discussed.