Symmetries and currents in nonholonomic mechanics

Michal Čech; Jana Musilová

Communications in Mathematics (2014)

  • Volume: 22, Issue: 2, page 159-184
  • ISSN: 1804-1388

Abstract

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In this paper we derive general equations for constraint Noethertype symmetries of a first order non-holonomic mechanical system and the corresponding currents, i.e. functions constant along trajectories of the nonholonomic system. The approach is based on a consistent and effective geometrical theory of nonholonomic constrained systems on fibred manifolds and their jet prolongations, first presented and developed by Olga Rossi. As a representative example of application of the geometrical theory and the equations of symmetries and conservation laws derived within this framework we present the Chaplygin sleigh. It is a mechanical system subject to one linear nonholonomic constraint enforcing the plane motion. We describe the trajectories of the Chaplygin sleigh and show that the usual kinetic energy conservation law holds along them, the time translation generator being the corresponding constraint symmetry and simultaneously the symmetry of nonholonomic equations of motion. Moreover, the expressions for two other currents are obtained. Remarkably, the corresponding constraint symmetries are not symmetries of nonholonomic equations of motion. The physical interpretation of results is emphasized.

How to cite

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Čech, Michal, and Musilová, Jana. "Symmetries and currents in nonholonomic mechanics." Communications in Mathematics 22.2 (2014): 159-184. <http://eudml.org/doc/269854>.

@article{Čech2014,
abstract = {In this paper we derive general equations for constraint Noethertype symmetries of a first order non-holonomic mechanical system and the corresponding currents, i.e. functions constant along trajectories of the nonholonomic system. The approach is based on a consistent and effective geometrical theory of nonholonomic constrained systems on fibred manifolds and their jet prolongations, first presented and developed by Olga Rossi. As a representative example of application of the geometrical theory and the equations of symmetries and conservation laws derived within this framework we present the Chaplygin sleigh. It is a mechanical system subject to one linear nonholonomic constraint enforcing the plane motion. We describe the trajectories of the Chaplygin sleigh and show that the usual kinetic energy conservation law holds along them, the time translation generator being the corresponding constraint symmetry and simultaneously the symmetry of nonholonomic equations of motion. Moreover, the expressions for two other currents are obtained. Remarkably, the corresponding constraint symmetries are not symmetries of nonholonomic equations of motion. The physical interpretation of results is emphasized.},
author = {Čech, Michal, Musilová, Jana},
journal = {Communications in Mathematics},
keywords = {nonholonomic mechanical systems; nonholonomic constraint submanifold; canonical distribution; reduced equations of motion; symmetries of nonholonomic systems; conservation laws; Chaplygin sleigh; nonholonomic mechanical systems; nonholonomic constraint submanifold; reduced equations of motion; symmetries; conservation laws; Chaplygin sleigh},
language = {eng},
number = {2},
pages = {159-184},
publisher = {University of Ostrava},
title = {Symmetries and currents in nonholonomic mechanics},
url = {http://eudml.org/doc/269854},
volume = {22},
year = {2014},
}

TY - JOUR
AU - Čech, Michal
AU - Musilová, Jana
TI - Symmetries and currents in nonholonomic mechanics
JO - Communications in Mathematics
PY - 2014
PB - University of Ostrava
VL - 22
IS - 2
SP - 159
EP - 184
AB - In this paper we derive general equations for constraint Noethertype symmetries of a first order non-holonomic mechanical system and the corresponding currents, i.e. functions constant along trajectories of the nonholonomic system. The approach is based on a consistent and effective geometrical theory of nonholonomic constrained systems on fibred manifolds and their jet prolongations, first presented and developed by Olga Rossi. As a representative example of application of the geometrical theory and the equations of symmetries and conservation laws derived within this framework we present the Chaplygin sleigh. It is a mechanical system subject to one linear nonholonomic constraint enforcing the plane motion. We describe the trajectories of the Chaplygin sleigh and show that the usual kinetic energy conservation law holds along them, the time translation generator being the corresponding constraint symmetry and simultaneously the symmetry of nonholonomic equations of motion. Moreover, the expressions for two other currents are obtained. Remarkably, the corresponding constraint symmetries are not symmetries of nonholonomic equations of motion. The physical interpretation of results is emphasized.
LA - eng
KW - nonholonomic mechanical systems; nonholonomic constraint submanifold; canonical distribution; reduced equations of motion; symmetries of nonholonomic systems; conservation laws; Chaplygin sleigh; nonholonomic mechanical systems; nonholonomic constraint submanifold; reduced equations of motion; symmetries; conservation laws; Chaplygin sleigh
UR - http://eudml.org/doc/269854
ER -

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