The Hydrodynamic Chaplygin Sleigh

TL;DR

This paper models the motion of a rigid body with nonholonomic constraints in a potential fluid, generalizing the Chaplygin sleigh to include hydrodynamic effects, and analyzes its explicit dynamics and asymptotic behavior.

Contribution

It introduces a hydrodynamic generalization of the Chaplygin sleigh using Euler--Poincaré--Suslov equations, providing explicit solutions and analyzing fluid effects on dynamics.

Findings

Explicit integration of equations of motion.

Identification of fluid effects on asymptotic behavior.

Demonstration of new features introduced by the fluid.

Abstract

We consider the motion of rigid bodies in a potential fluid subject to certain nonholonomic constraints and show that it is described by Euler--Poincar\'e--Suslov equations. In the 2-dimensional case, when the constraint is realized by a blade attached to the body, the system provides a hydrodynamic generalization of the Chaplygin sleigh, whose dynamics are studied in detail. Namely, the equations of motion are integrated explicitly and the asymptotic behavior of the system is determined. It is shown how the presence of the fluid brings new features to such a behavior.