Nonholonomic LL systems on central extensions and the hydrodynamic Chaplygin sleigh with circulation

TL;DR

This paper explores nonholonomic systems on central extensions of Lie groups, analyzing their invariant measures and applying findings to the hydrodynamic Chaplygin sleigh with circulation, revealing new geometric and dynamical insights.

Contribution

It establishes a correspondence between invariant measures on original and extended groups and applies this to the hydrodynamic Chaplygin sleigh with circulation.

Findings

Invariant measures correspond between original and extended systems.

Application to the hydrodynamic Chaplygin sleigh with circulation.

Enhanced understanding of nonholonomic systems on central extensions.

Abstract

We consider nonholonomic systems whose configuration space is the central extension of a Lie group and have left invariant kinetic energy and constraints. We study the structure of the associated Euler-Poincare-Suslov equations and show that there is a one-to-one correspondence between invariant measures on the original group and on the extended group. Our results are applied to the hydrodynamic Chaplygin sleigh, that is, a planar rigid body that moves in a potential flow subject to a nonholonomic constraint modeling a fin or keel attached to the body, in the case where there is circulation around the body.