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dc.contributor.authorKalnins, Ernie G.
dc.contributor.authorMiller, W., Jr.
dc.date.accessioned2012-07-03T03:12:02Z
dc.date.available2012-07-03T03:12:02Z
dc.date.copyright2012-06-07
dc.date.issued2012
dc.identifier.citationKalnins, E.G. & Miller, W., Jr. (2012). Structure theory for extended Kepler-Coulomb 3D classical superintegrable systems. Symmetry, Integrability and Geometry: Methods and Applications (SIGMA), 8(2012), 034.en_NZ
dc.identifier.urihttps://hdl.handle.net/10289/6437
dc.description.abstractThe classical Kepler-Coulomb system in 3 dimensions is well known to be 2nd order superintegrable, with a symmetry algebra that closes polynomially under Poisson brackets. This polynomial closure is typical for 2nd order superintegrable systems in 2D and for 2nd order systems in 3D with nondegenerate (4-parameter) potentials. However the degenerate 3-parameter potential for the 3D extended Kepler-Coulomb system (also 2nd order superintegrable) is an exception, as its quadratic symmetry algebra doesn't close polynomially. The 3D 4-parameter potential for the extended Kepler-Coulomb system is not even 2nd order superintegrable. However, Verrier and Evans (2008) showed it was 4th order superintegrable, and Tanoudis and Daskaloyannis (2011) showed that in the quantum case, if a second 4th order symmetry is added to the generators, the double commutators in the symmetry algebra close polynomially. Here, based on the Tremblay, Turbiner and Winternitz construction, we consider an infinite class of classical extended Kepler-Coulomb 3- and 4-parameter systems indexed by a pair of rational numbers (k1,k2) and reducing to the usual systems when k1=k2=1. We show these systems to be superintegrable of arbitrarily high order and work out explicitly the structure of the symmetry algebras determined by the 5 basis generators we have constructed. We demonstrate that the symmetry algebras close rationally; only for systems admitting extra discrete symmetries is polynomial closure achieved. Underlying the structure theory is the existence of raising and lowering constants of the motion, not themselves polynomials in the momenta, that can be employed to construct the polynomial symmetries and their structure relations.en_NZ
dc.format.mimetypeapplication/pdf
dc.language.isoen
dc.publisherthe Department of Applied Research, Institute of Mathematics of National Academy of Sciences of Ukraineen_NZ
dc.relation.ispartofSymmetry, Integrability and Geometry: Methods and Applications
dc.relation.urihttp://www.emis.de/journals/SIGMA/2012/034/en_NZ
dc.rightsThis article has been published in the journal: Symmetry, Integrability and Geometry: Methods and Applications (SIGMA). ©2012 copyright with the authors. It is published in SIGMA under the terms of the Creative Commons Attribution-NonCommercial-ShareAlike Licence.en_NZ
dc.subjectsuperintegrabilityen_NZ
dc.subjectKepler-Coulomb systemen_NZ
dc.titleStructure Theory for Extended Kepler-Coulomb 3D Classical Superintegrable Systemsen_NZ
dc.typeJournal Articleen_NZ
dc.identifier.doi10.3842/SIGMA.2012.034en_NZ
dc.relation.isPartOfSymmetry, Integrability and Geometry: Methods and Applicationsen_NZ
pubs.begin-page1en_NZ
pubs.elements-id37695
pubs.end-page25en_NZ
pubs.volume8 Article 34en_NZ


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