A comparison of vakonomic and nonholonomic dynamics with applications to non-invariant Chaplygin systems

Michał Jóźwikowski; Witold Respondek

doi:10.3934/jgm.2019005

Article Contents

2019, Volume 11, Issue 1: 77-122. Doi: 10.3934/jgm.2019005

This issue Previous Article A geometric perspective on the Piola identity in Riemannian settings Next Article

A comparison of vakonomic and nonholonomic dynamics with applications to non-invariant Chaplygin systems

Michał Jóźwikowski^1,2, , and
Witold Respondek^3,

1.
Institute of Mathematics, Polish Academy of Sciences, Śniadeckich 8, 00-656 Warsaw, Poland
2.
University of Warsaw, Faculty of Mathematics, Informatics and Mechanics, Banacha 2, 02-097 Warsaw, Poland
3.
Normandie Université, INSA de Rouen, LMI, 685 Avenue de l'Université 76800 Saint-Etienne-du-Rouvray, France

^* Corresponding author
^* Corresponding author

Received: September 2018

Revised: November 2018

Published: January 2019

This research was supported by the National Science Center under the grant DEC-2011/02/A/ST1/00208 "Solvability, chaos and control in quantum systems"

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Abstract

We study relations between vakonomically and nonholonomically constrained Lagrangian dynamics for the same set of linear constraints. The basic idea is to compare both situations at the level of generalized variational principles, not equations of motion as has been done so far. The method seems to be quite powerful and effective. In particular, it allows to derive, interpret and generalize many known results on non-Abelian Chaplygin systems. We apply it also to a class of systems on Lie groups with a left-invariant constraints distribution. Concrete examples of the unicycle in a potential field, the two-wheeled carriage and the generalized Heisenberg system are discussed.

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Mathematics Subject Classification: Primary: 49S05, 70H45; Secondary: 70H30, 70G45.

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Figure 1. A homotopy-generated variation and its generator $ \xi $. Note that the homotopy with a fixed end-point(s) corresponds to a generator vanishing at that end-point(s)

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Figure 2. A (non-invariant) Chaplygin system is a principal $G$-bundle $\pi:Q \to M$, equipped with a horizontal distribution ${\rm{H}} Q$ and a Lagrangian $L:{\rm{T}} Q\to\mathbb{R}$, none of which needs to be $G$-invariant.

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