Applications and extensions of the Liouville theorem on constants of motion

Applications and extensions of the Liouville theorem on constants of motion

We give an elementary proof of the Liouville theorem, which allows us to obtain n constants of motion in addition to n given constants of motion in involution, for a mechanical system with n degrees of freedom, and we give some examples of its application. For a given set of n constants of motion that are not in involution with respect to the standard symplectic structure, there exist symplectic structures with respect to which these constants will be in involution and the Liouville theorem can then be applied. Using the fact that any second-order ordinary differential equation (not necessarily related to a mechanical problem) can be expressed in the form of the Hamilton equations, the knowledge of a first integral of the equation allows us to find its general solution.

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Main Author: Torres del Castillo,G.F.
Format: Digital revista
Language:English
Published: Sociedad Mexicana de Física 2011
Online Access:http://www.scielo.org.mx/scielo.php?script=sci_arttext&pid=S0035-001X2011000300012
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spelling oai:scielo:S0035-001X20110003000122011-10-06Applications and extensions of the Liouville theorem on constants of motionTorres del Castillo,G.F. Hamilton-Jacobi equation constants of motion symplectic structures We give an elementary proof of the Liouville theorem, which allows us to obtain n constants of motion in addition to n given constants of motion in involution, for a mechanical system with n degrees of freedom, and we give some examples of its application. For a given set of n constants of motion that are not in involution with respect to the standard symplectic structure, there exist symplectic structures with respect to which these constants will be in involution and the Liouville theorem can then be applied. Using the fact that any second-order ordinary differential equation (not necessarily related to a mechanical problem) can be expressed in the form of the Hamilton equations, the knowledge of a first integral of the equation allows us to find its general solution.info:eu-repo/semantics/openAccessSociedad Mexicana de FísicaRevista mexicana de física v.57 n.3 20112011-06-01info:eu-repo/semantics/articletext/htmlhttp://www.scielo.org.mx/scielo.php?script=sci_arttext&pid=S0035-001X2011000300012en
institution SCIELO
collection OJS
country México
countrycode MX
component Revista
access En linea
databasecode rev-scielo-mx
tag revista
region America del Norte
libraryname SciELO
language English
format Digital
author Torres del Castillo,G.F.
spellingShingle Torres del Castillo,G.F.
Applications and extensions of the Liouville theorem on constants of motion
author_facet Torres del Castillo,G.F.
author_sort Torres del Castillo,G.F.
title Applications and extensions of the Liouville theorem on constants of motion
title_short Applications and extensions of the Liouville theorem on constants of motion
title_full Applications and extensions of the Liouville theorem on constants of motion
title_fullStr Applications and extensions of the Liouville theorem on constants of motion
title_full_unstemmed Applications and extensions of the Liouville theorem on constants of motion
title_sort applications and extensions of the liouville theorem on constants of motion
description We give an elementary proof of the Liouville theorem, which allows us to obtain n constants of motion in addition to n given constants of motion in involution, for a mechanical system with n degrees of freedom, and we give some examples of its application. For a given set of n constants of motion that are not in involution with respect to the standard symplectic structure, there exist symplectic structures with respect to which these constants will be in involution and the Liouville theorem can then be applied. Using the fact that any second-order ordinary differential equation (not necessarily related to a mechanical problem) can be expressed in the form of the Hamilton equations, the knowledge of a first integral of the equation allows us to find its general solution.
publisher Sociedad Mexicana de Física
publishDate 2011
url http://www.scielo.org.mx/scielo.php?script=sci_arttext&pid=S0035-001X2011000300012
work_keys_str_mv AT torresdelcastillogf applicationsandextensionsoftheliouvilletheoremonconstantsofmotion
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