On the inverse variational problem in nonholonomic mechanics

Olga Rossi; Jana Musilová

Communications in Mathematics (2012)

  • Volume: 20, Issue: 1, page 41-62
  • ISSN: 1804-1388

Abstract

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The inverse problem of the calculus of variations in a nonholonomic setting is studied. The concept of constraint variationality is introduced on the basis of a recently discovered nonholonomic variational principle. Variational properties of first order mechanical systems with general nonholonomic constraints are studied. It is shown that constraint variationality is equivalent with the existence of a closed representative in the class of 2-forms determining the nonholonomic system. Together with the recently found constraint Helmholtz conditions this result completes basic geometric properties of constraint variational systems. A few examples of constraint variational systems are discussed.

How to cite

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Rossi, Olga, and Musilová, Jana. "On the inverse variational problem in nonholonomic mechanics." Communications in Mathematics 20.1 (2012): 41-62. <http://eudml.org/doc/246746>.

@article{Rossi2012,
abstract = {The inverse problem of the calculus of variations in a nonholonomic setting is studied. The concept of constraint variationality is introduced on the basis of a recently discovered nonholonomic variational principle. Variational properties of first order mechanical systems with general nonholonomic constraints are studied. It is shown that constraint variationality is equivalent with the existence of a closed representative in the class of 2-forms determining the nonholonomic system. Together with the recently found constraint Helmholtz conditions this result completes basic geometric properties of constraint variational systems. A few examples of constraint variational systems are discussed.},
author = {Rossi, Olga, Musilová, Jana},
journal = {Communications in Mathematics},
keywords = {inverse problem of the calculus of variations; Helmholtz conditions; nonholonomic constraints; the nonholonomic variational principle; constraint Euler-Lagrange equations; constraint Helmholtz conditions; constraint Lagrangian; constraint ballistic motion; relativistic particle; inverse problem of calculus of variations; Helmholtz conditions; nonholonomic constraints; nonholonomic variational principle; constrained Euler-Lagrange equations; constrained Helmholtz conditions; constrained Lagrangian; constrained ballistic motion; relativistic particle},
language = {eng},
number = {1},
pages = {41-62},
publisher = {University of Ostrava},
title = {On the inverse variational problem in nonholonomic mechanics},
url = {http://eudml.org/doc/246746},
volume = {20},
year = {2012},
}

TY - JOUR
AU - Rossi, Olga
AU - Musilová, Jana
TI - On the inverse variational problem in nonholonomic mechanics
JO - Communications in Mathematics
PY - 2012
PB - University of Ostrava
VL - 20
IS - 1
SP - 41
EP - 62
AB - The inverse problem of the calculus of variations in a nonholonomic setting is studied. The concept of constraint variationality is introduced on the basis of a recently discovered nonholonomic variational principle. Variational properties of first order mechanical systems with general nonholonomic constraints are studied. It is shown that constraint variationality is equivalent with the existence of a closed representative in the class of 2-forms determining the nonholonomic system. Together with the recently found constraint Helmholtz conditions this result completes basic geometric properties of constraint variational systems. A few examples of constraint variational systems are discussed.
LA - eng
KW - inverse problem of the calculus of variations; Helmholtz conditions; nonholonomic constraints; the nonholonomic variational principle; constraint Euler-Lagrange equations; constraint Helmholtz conditions; constraint Lagrangian; constraint ballistic motion; relativistic particle; inverse problem of calculus of variations; Helmholtz conditions; nonholonomic constraints; nonholonomic variational principle; constrained Euler-Lagrange equations; constrained Helmholtz conditions; constrained Lagrangian; constrained ballistic motion; relativistic particle
UR - http://eudml.org/doc/246746
ER -

References

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  13. Sarlet, W., A direct geometrical construction of the dynamics of non-holonomic Lagrangian systems, Extracta Mathematicae, 11, 1996, 202-212 (1996) MR1424757
  14. Swaczyna, M., Volný, P., Uniform projectile motion as a nonholonomic system with a nonlinear constraint, Int. J. of Non-Linear Mechanics. Submitted 
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