Nilpotent integrability, reduction of dynamical systems and a third-order Calogero-Moser system

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dc.contributor.authorIbort, A.
dc.contributor.authorMarmo, G.
dc.contributor.authorRodríguez González, Miguel Ángel
dc.contributor.authorTempesta, Piergiulio
dc.date.accessioned2023-06-17T13:38:32Z
dc.date.available2023-06-17T13:38:32Z
dc.date.issued2019-10
dc.description© 2019 Springer Heidelberg. The authors acknowledge financial support from the Spanish Ministry of Economy and Competitiveness, through the Severo Ochoa Programme for Centres of Excellence in RD (SEV-2015/0554). AI would like to thank partial support provided by the MINECO research project MTM2017-84098-P and QUITEMAD+, S2013/ICE-2801. GM would like to acknowledge the support provided by the `Catedras de Excelencia' Santander/UC3M 2016-17 program. MAR and PT would like to thank partial financial support from MINECO research project FIS2015-63966-P.
dc.description.abstractWe present an algebraic formulation of the notion of integrability of dynamical systems, based on a nilpotency property of its flow: It can be explicitly described as a polynomial on its evolution parameter. Such a property is established in a purely geometric-algebraic language, in terms both of the algebra of all higher-order constants of the motion (named the nilpotent algebra of the dynamics) and of a maximal Abelian algebra of symmetries (called a Cartan subalgebra of the dynamics). It is shown that this notion of integrability amounts to the annihilator of the nilpotent algebra being contained in a Cartan subalgebra of the dynamics. Systems exhibiting this property will be said to be nilpotent-integrable. Our notion of nilpotent integrability offers a new insight into the intrinsic dynamical properties of a system, which is independent of any auxiliary geometric structure defined on its phase space. At the same time, it extends in a natural way the classical concept of integrability for Hamiltonian systems. An algebraic reduction procedure valid for nilpotent-integrable systems, generalizing the well-known reduction procedures for symplectic and/or Poisson systems on appropriate quotient spaces, is also discussed. In particular, it is shown that a large class of nilpotent-integrable systems can be obtained by reduction of higher-order free systems. The case of the third-order free system is analyzed and a non-trivial set of third-order Calogero-Moser-like nilpotent-integrable equations is obtained.
dc.description.departmentDepto. de Física Teórica
dc.description.facultyFac. de Ciencias Físicas
dc.description.refereedTRUE
dc.description.sponsorshipMinisterio de Economía y Competitividad (MINECO)
dc.description.sponsorshipComunidad de Madrid
dc.description.sponsorshipCentros de Excelencia Severo Ochoa (MINECO)
dc.description.sponsorship'Catedras de Excelencia' Santander/UC3M 2016-17 program
dc.description.statuspub
dc.eprint.idhttps://eprints.ucm.es/id/eprint/58709
dc.identifier.doi10.1007/s10231-019-00828-x
dc.identifier.issn0373-3114
dc.identifier.officialurlhttp://dx.doi.org/10.1007/s10231-019-00828-x
dc.identifier.relatedurlhttps://link.springer.com/
dc.identifier.urihttps://hdl.handle.net/20.500.14352/13904
dc.issue.number5
dc.journal.titleAnnali di matematica pura ed applicata
dc.language.isoeng
dc.page.final1540
dc.page.initial1513
dc.publisherSpringer Heidelberg
dc.relation.projectID(MTM2017-84098-P; FIS2015-63966-P)
dc.relation.projectIDQUITEMAD+-CM (S2013-ICE2801)
dc.relation.projectIDSEV-2015/0554
dc.rights.accessRightsopen access
dc.subject.cdu51-73
dc.subject.keywordDynamical systems
dc.subject.keywordIntegrable systems
dc.subject.keywordReduction methods
dc.subject.keywordLie algebras
dc.subject.keyword37N05
dc.subject.keyword37K10.
dc.subject.ucmFísica-Modelos matemáticos
dc.subject.ucmFísica matemática
dc.titleNilpotent integrability, reduction of dynamical systems and a third-order Calogero-Moser system
dc.typejournal article
dc.volume.number198
dspace.entity.typePublication
relation.isAuthorOfPublicationd781a665-7ef6-44e0-a0da-81f722f1b8ad
relation.isAuthorOfPublication46e9a666-a5cf-44c3-8726-7cbe2c61bd1a
relation.isAuthorOfPublication.latestForDiscoveryd781a665-7ef6-44e0-a0da-81f722f1b8ad
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