A new approach to the vakonomic mechanics

The aim of this paper is to show that the Lagrange-d'Alembert and its equivalent the Gauss and Appel principle are not the only way to deduce the equations of motion of the nonholonomic systems. Instead of them, here we consider the generalization of the Hamiltonian principle for nonholonomic s...

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Autores: Llibre, Jaume|||0000-0002-9511-5999, Ramírez, Rafael Orlando|||0000-0002-4958-0291, Sadovskaia, Natalia
Formato: artículo
Fecha de publicación:2014
País:España
Recursos:Universitat Autònoma de Barcelona
Repositorio:Dipòsit Digital de Documents de la UAB
Idioma:inglés
OAI Identifier:oai:ddd.uab.cat:150728
Acesso em linha:https://ddd.uab.cat/record/150728
https://dx.doi.org/urn:doi:10.1007/s11071-014-1554-3
Access Level:acceso abierto
Palavra-chave:Chapligyn system
Constrained Lagrangian system
D' Alembert-Lagrange principle
Equation of motion
Generalized Hamiltonian principle
Newtonian model
Transpositional relations
Vakonomic mechanic
Variational principle
Vorones system
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spelling A new approach to the vakonomic mechanicsLlibre, Jaume|||0000-0002-9511-5999Ramírez, Rafael Orlando|||0000-0002-4958-0291Sadovskaia, NataliaChapligyn systemConstrained Lagrangian systemD' Alembert-Lagrange principleEquation of motionGeneralized Hamiltonian principleNewtonian modelTranspositional relationsVakonomic mechanicVariational principleVorones systemThe aim of this paper is to show that the Lagrange-d'Alembert and its equivalent the Gauss and Appel principle are not the only way to deduce the equations of motion of the nonholonomic systems. Instead of them, here we consider the generalization of the Hamiltonian principle for nonholonomic systems with nonzero transpositional relations. By applying this variational principle which takes into the account transpositional relations different from the classical ones we deduce the equations of motion for the nonholonomic systems with constraints that in general are nonlinear in the velocity. These equations of motion coincide, except perhaps in a zero Lebesgue measure set, with the classical differential equations deduced with d'Alembert-Lagrange principle. We provide a new point of view on the transpositional relations for the constrained mechanical systems: the virtual variations can produce zero or non-zero transpositional relations. In particular the independent virtual variations can produce non-zero transpositional relations. For the unconstrained mechanical systems the virtual variations always produce zero transpositional relations. We conjecture that the existence of the nonlinear constraints in the velocity must be sought outside of the Newtonian model. All our results are illustrated with precise examples. 22014-01-0120142014-01-01Articlehttp://purl.org/coar/resource_type/c_6501AMhttp://purl.org/coar/version/c_ab4af688f83e57aainfo:eu-repo/semantics/articleapplication/pdfhttps://ddd.uab.cat/record/150728https://dx.doi.org/urn:doi:10.1007/s11071-014-1554-3reponame:Dipòsit Digital de Documents de la UABinstname:Universitat Autònoma de BarcelonaInglésengMinisterio de Ciencia e Innovación https://doi.org/10.13039/501100004837 MTM2008-03437Agència de Gestió d'Ajuts Universitaris i de Recerca https://doi.org/10.13039/501100003030 2009/SGR-410European Commission https://doi.org/10.13039/501100000780 318999European Commission https://doi.org/10.13039/501100000780 316338open accesshttp://purl.org/coar/access_right/c_abf2Aquest material està protegit per drets d'autor i/o drets afins. Podeu utilitzar aquest material en funció del que permet la legislació de drets d'autor i drets afins d'aplicació al vostre cas. Per a d'altres usos heu d'obtenir permís del(s) titular(s) de drets.https://rightsstatements.org/vocab/InC/1.0/info:eu-repo/semantics/openAccessoai:ddd.uab.cat:1507282026-06-06T12:50:31Z
dc.title.none.fl_str_mv A new approach to the vakonomic mechanics
title A new approach to the vakonomic mechanics
spellingShingle A new approach to the vakonomic mechanics
Llibre, Jaume|||0000-0002-9511-5999
Chapligyn system
Constrained Lagrangian system
D' Alembert-Lagrange principle
Equation of motion
Generalized Hamiltonian principle
Newtonian model
Transpositional relations
Vakonomic mechanic
Variational principle
Vorones system
title_short A new approach to the vakonomic mechanics
title_full A new approach to the vakonomic mechanics
title_fullStr A new approach to the vakonomic mechanics
title_full_unstemmed A new approach to the vakonomic mechanics
title_sort A new approach to the vakonomic mechanics
dc.creator.none.fl_str_mv Llibre, Jaume|||0000-0002-9511-5999
Ramírez, Rafael Orlando|||0000-0002-4958-0291
Sadovskaia, Natalia
author Llibre, Jaume|||0000-0002-9511-5999
author_facet Llibre, Jaume|||0000-0002-9511-5999
Ramírez, Rafael Orlando|||0000-0002-4958-0291
Sadovskaia, Natalia
author_role author
author2 Ramírez, Rafael Orlando|||0000-0002-4958-0291
Sadovskaia, Natalia
author2_role author
author
dc.subject.none.fl_str_mv Chapligyn system
Constrained Lagrangian system
D' Alembert-Lagrange principle
Equation of motion
Generalized Hamiltonian principle
Newtonian model
Transpositional relations
Vakonomic mechanic
Variational principle
Vorones system
topic Chapligyn system
Constrained Lagrangian system
D' Alembert-Lagrange principle
Equation of motion
Generalized Hamiltonian principle
Newtonian model
Transpositional relations
Vakonomic mechanic
Variational principle
Vorones system
description The aim of this paper is to show that the Lagrange-d'Alembert and its equivalent the Gauss and Appel principle are not the only way to deduce the equations of motion of the nonholonomic systems. Instead of them, here we consider the generalization of the Hamiltonian principle for nonholonomic systems with nonzero transpositional relations. By applying this variational principle which takes into the account transpositional relations different from the classical ones we deduce the equations of motion for the nonholonomic systems with constraints that in general are nonlinear in the velocity. These equations of motion coincide, except perhaps in a zero Lebesgue measure set, with the classical differential equations deduced with d'Alembert-Lagrange principle. We provide a new point of view on the transpositional relations for the constrained mechanical systems: the virtual variations can produce zero or non-zero transpositional relations. In particular the independent virtual variations can produce non-zero transpositional relations. For the unconstrained mechanical systems the virtual variations always produce zero transpositional relations. We conjecture that the existence of the nonlinear constraints in the velocity must be sought outside of the Newtonian model. All our results are illustrated with precise examples.
publishDate 2014
dc.date.none.fl_str_mv 2
2014-01-01
2014
2014-01-01
dc.type.none.fl_str_mv Article
http://purl.org/coar/resource_type/c_6501
AM
http://purl.org/coar/version/c_ab4af688f83e57aa
dc.type.openaire.fl_str_mv info:eu-repo/semantics/article
format article
dc.identifier.none.fl_str_mv https://ddd.uab.cat/record/150728
https://dx.doi.org/urn:doi:10.1007/s11071-014-1554-3
url https://ddd.uab.cat/record/150728
https://dx.doi.org/urn:doi:10.1007/s11071-014-1554-3
dc.language.none.fl_str_mv Inglés
eng
language_invalid_str_mv Inglés
language eng
dc.relation.none.fl_str_mv Ministerio de Ciencia e Innovación https://doi.org/10.13039/501100004837 MTM2008-03437
Agència de Gestió d'Ajuts Universitaris i de Recerca https://doi.org/10.13039/501100003030 2009/SGR-410
European Commission https://doi.org/10.13039/501100000780 318999
European Commission https://doi.org/10.13039/501100000780 316338
dc.rights.none.fl_str_mv open access
http://purl.org/coar/access_right/c_abf2
https://rightsstatements.org/vocab/InC/1.0/
dc.rights.openaire.fl_str_mv info:eu-repo/semantics/openAccess
rights_invalid_str_mv open access
http://purl.org/coar/access_right/c_abf2
https://rightsstatements.org/vocab/InC/1.0/
eu_rights_str_mv openAccess
dc.format.none.fl_str_mv application/pdf
dc.source.none.fl_str_mv reponame:Dipòsit Digital de Documents de la UAB
instname:Universitat Autònoma de Barcelona
instname_str Universitat Autònoma de Barcelona
reponame_str Dipòsit Digital de Documents de la UAB
collection Dipòsit Digital de Documents de la UAB
repository.name.fl_str_mv
repository.mail.fl_str_mv
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score 15,300719

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