An ellipsoidal billiard with a quadratic potential

There exists an in infite hierarchy of integrable generalizations of the geodesic flow on an n -di- mensional ellipsoid.hese generalizations describe the motion of a point in the force fields of certain polynomial potentials.In the limit as one of semiaxes of the ellipsoidtends to zero,one obtains i...

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Autor: Fedorov, Yuri|||0000-0002-7533-975X
Formato: artículo
Fecha de publicación:2003
País:España
Recursos:Universitat Politècnica de Catalunya (UPC)
Repositorio:UPCommons. Portal del coneixement obert de la UPC
Idioma:inglés
OAI Identifier:oai:upcommons.upc.edu:2117/1199
Acesso em linha:https://hdl.handle.net/2117/1199
Access Level:acceso abierto
Palavra-chave:Hamiltonian systems
Curves
Hamiltonian dynamical systems
Lagrangian functions
ellipsoidal billiard
Hamilton, Sistemes de
Corbes
Lagrange, Funcions de
Classificació AMS::14 Algebraic geometry::14H Curves
Classificació AMS::37 Dynamical systems and ergodic theory::37J Finite-dimensional Hamiltonian, Lagrangian, contact, and nonholonomic systems
Classificació AMS::70 Mechanics of particles and systems::70H Hamiltonian and Lagrangian mechanics
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repository_id_str
spelling An ellipsoidal billiard with a quadratic potentialFedorov, Yuri|||0000-0002-7533-975XHamiltonian systemsCurvesHamiltonian dynamical systemsLagrangian functionsellipsoidal billiardHamilton, Sistemes deCorbesLagrange, Funcions deClassificació AMS::14 Algebraic geometry::14H CurvesClassificació AMS::37 Dynamical systems and ergodic theory::37J Finite-dimensional Hamiltonian, Lagrangian, contact, and nonholonomic systemsClassificació AMS::70 Mechanics of particles and systems::70H Hamiltonian and Lagrangian mechanicsThere exists an in infite hierarchy of integrable generalizations of the geodesic flow on an n -di- mensional ellipsoid.hese generalizations describe the motion of a point in the force fields of certain polynomial potentials.In the limit as one of semiaxes of the ellipsoidtends to zero,one obtains inte- grable mappings corresponding to billiards with polynomial potentials inside an (n+1)-dimensional ellipsoid. In this paper, for the first time we give explicit expressions for the ellipsoidal billiard with a quadratic (Hooke)potential,its representation in Lax form,and a theta function solution.We also indicate the generating function of the restriction of the potential billiard map to a level set of an energy type integral. The methodwe use to obtain theta function solutions is different from those applied earlier and is based on the calculation of limit values of meromorphic functions on generalized Jacobians.20032003-01-0120072007-10-01journal articlehttp://purl.org/coar/resource_type/c_6501NAhttp://purl.org/coar/version/c_be7fb7dd8ff6fe43info:eu-repo/semantics/articleapplication/pdfhttps://hdl.handle.net/2117/1199reponame:UPCommons. Portal del coneixement obert de la UPCinstname:Universitat Politècnica de Catalunya (UPC)Inglésengopen accesshttp://purl.org/coar/access_right/c_abf2Attribution-NonCommercial-NoDerivs 2.5 Spainhttp://creativecommons.org/licenses/by-nc-nd/2.5/es/info:eu-repo/semantics/openAccessoai:upcommons.upc.edu:2117/11992026-05-27T15:37:01Z
dc.title.none.fl_str_mv An ellipsoidal billiard with a quadratic potential
title An ellipsoidal billiard with a quadratic potential
spellingShingle An ellipsoidal billiard with a quadratic potential
Fedorov, Yuri|||0000-0002-7533-975X
Hamiltonian systems
Curves
Hamiltonian dynamical systems
Lagrangian functions
ellipsoidal billiard
Hamilton, Sistemes de
Corbes
Lagrange, Funcions de
Classificació AMS::14 Algebraic geometry::14H Curves
Classificació AMS::37 Dynamical systems and ergodic theory::37J Finite-dimensional Hamiltonian, Lagrangian, contact, and nonholonomic systems
Classificació AMS::70 Mechanics of particles and systems::70H Hamiltonian and Lagrangian mechanics
title_short An ellipsoidal billiard with a quadratic potential
title_full An ellipsoidal billiard with a quadratic potential
title_fullStr An ellipsoidal billiard with a quadratic potential
title_full_unstemmed An ellipsoidal billiard with a quadratic potential
title_sort An ellipsoidal billiard with a quadratic potential
dc.creator.none.fl_str_mv Fedorov, Yuri|||0000-0002-7533-975X
author Fedorov, Yuri|||0000-0002-7533-975X
author_facet Fedorov, Yuri|||0000-0002-7533-975X
author_role author
dc.subject.none.fl_str_mv Hamiltonian systems
Curves
Hamiltonian dynamical systems
Lagrangian functions
ellipsoidal billiard
Hamilton, Sistemes de
Corbes
Lagrange, Funcions de
Classificació AMS::14 Algebraic geometry::14H Curves
Classificació AMS::37 Dynamical systems and ergodic theory::37J Finite-dimensional Hamiltonian, Lagrangian, contact, and nonholonomic systems
Classificació AMS::70 Mechanics of particles and systems::70H Hamiltonian and Lagrangian mechanics
topic Hamiltonian systems
Curves
Hamiltonian dynamical systems
Lagrangian functions
ellipsoidal billiard
Hamilton, Sistemes de
Corbes
Lagrange, Funcions de
Classificació AMS::14 Algebraic geometry::14H Curves
Classificació AMS::37 Dynamical systems and ergodic theory::37J Finite-dimensional Hamiltonian, Lagrangian, contact, and nonholonomic systems
Classificació AMS::70 Mechanics of particles and systems::70H Hamiltonian and Lagrangian mechanics
description There exists an in infite hierarchy of integrable generalizations of the geodesic flow on an n -di- mensional ellipsoid.hese generalizations describe the motion of a point in the force fields of certain polynomial potentials.In the limit as one of semiaxes of the ellipsoidtends to zero,one obtains inte- grable mappings corresponding to billiards with polynomial potentials inside an (n+1)-dimensional ellipsoid. In this paper, for the first time we give explicit expressions for the ellipsoidal billiard with a quadratic (Hooke)potential,its representation in Lax form,and a theta function solution.We also indicate the generating function of the restriction of the potential billiard map to a level set of an energy type integral. The methodwe use to obtain theta function solutions is different from those applied earlier and is based on the calculation of limit values of meromorphic functions on generalized Jacobians.
publishDate 2003
dc.date.none.fl_str_mv 2003
2003-01-01
2007
2007-10-01
dc.type.none.fl_str_mv journal article
http://purl.org/coar/resource_type/c_6501
NA
http://purl.org/coar/version/c_be7fb7dd8ff6fe43
dc.type.openaire.fl_str_mv info:eu-repo/semantics/article
format article
dc.identifier.none.fl_str_mv https://hdl.handle.net/2117/1199
url https://hdl.handle.net/2117/1199
dc.language.none.fl_str_mv Inglés
eng
language_invalid_str_mv Inglés
language eng
dc.rights.none.fl_str_mv open access
http://purl.org/coar/access_right/c_abf2
Attribution-NonCommercial-NoDerivs 2.5 Spain
http://creativecommons.org/licenses/by-nc-nd/2.5/es/
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
Attribution-NonCommercial-NoDerivs 2.5 Spain
http://creativecommons.org/licenses/by-nc-nd/2.5/es/
eu_rights_str_mv openAccess
dc.format.none.fl_str_mv application/pdf
dc.source.none.fl_str_mv reponame:UPCommons. Portal del coneixement obert de la UPC
instname:Universitat Politècnica de Catalunya (UPC)
instname_str Universitat Politècnica de Catalunya (UPC)
reponame_str UPCommons. Portal del coneixement obert de la UPC
collection UPCommons. Portal del coneixement obert de la UPC
repository.name.fl_str_mv
repository.mail.fl_str_mv
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score 15,300724

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