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

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-Poincaré-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.

The motion of the 2D hydrodynamic Chaplygin sleigh in the presence of circulation

“…It is shown in [12] that equations (2.9) are of Euler-Poincaré type on a central extension of SE (2) and thus are Hamiltonian.…”

“…where the multiplier λ is determined from the condition v 2 = 0. These equations have been shown to be of Euler-Poincaré-Suslov type on the dual Lie algebra of a central extension of SE(2) in [12].…”

“…The hydrodynamic Chaplygin sleigh in the presence of circulation was recently considered in [12], where it was shown that it is an LL system on a certain central extension of SE(2) by R 3 , where the cocycle encodes the effects of the circulation on the body.…”

On a Lagrangian Reduction and a Deformation of Completely Integrable Systems

“…When using a central extension of the Lie group, the variable in the centre of the Lie algebra will always be a constant and thus a standard kinetic term can be taken in the Hamiltonian. We refer to Marsden et al [2007]; García-Naranjo and Vankerschaver [2013] for more details of this construction. The theorem can now be stated; see Marsden et al [2007] for the proof.…”

Fermi-like acceleration and power-law energy growth in nonholonomic systems

“…The authors of [60] assert that the Kutta-Zhukovsky condition is equivalent to a nonholonomic constraint, which is, generally speaking, incorrect from the viewpoint of physical principles of mechanics. By the way, a nonholonomic model is also used in [23,24,27] to describe the motion of a plate in a fluid. It should be noted that, when it comes to describing the motion of a rigid body in an ideal fluid, the equations with nonintegrable constraints arise within the framework of vakonomic mechanics.…”