{"id":{"repo_id":"unsw","oai_identifier":"oai:unsworks.library.unsw.edu.au:1959.4/53501"},"canonical_url":"https://search.dev.ndltd.org/etd/unsw/oai:unsworks.library.unsw.edu.au:1959.4/53501","repository":{"repo_id":"unsw","name":"University of New South Wales","base_url":"https://unsworks.unsw.edu.au/oai/provider"},"display":{"title":"Classification of second-order conformally-superintegrable systems","abstract":"Over the last half century the study of superintegrable systems has established itself as an interesting subject with connections to some of the earliest known dynamical systems in mathematical-physics. Systems with constants second-order in the momenta have been particularly well studied in recent years. This thesis provides a classification of non-degenerate (maximum parameter) three-dimensional second-order superintegrable systems over conformally-flat spaces. I show that, up to Staeckel equivalence, such systems can be put into correspondence with a 6 points in the extended complex plane with an action induced by the conformal-group in three dimensions. I use this correspondence, and the tools of classical-invariant theory, to determine the inequivalent orbits under this action and show there are only 10 conformal-classes. This answers an open problem by showing that no unknown systems exist on the sphere. Additional interest in these systems comes from studying their algebra of constants. In the three-dimensional maximum-parameter case this algebra is generated by the iterated Poisson brackets of the 6 linearly independent second-order constants and is known to close at finite order. These 6 second-order constants are necessarily functionally dependent, and up to now the explicit relation for their dependence has only been known on a case-by-case basis. In this thesis I demonstrate a quartic identity which provides the functional relation for a general system.","abstract_html":"Over the last half century the study of superintegrable systems has established itself as an interesting subject with connections to some of the earliest known dynamical systems in mathematical-physics. Systems with constants second-order in the momenta have been particularly well studied in recent years. This thesis provides a classification of non-degenerate (maximum parameter) three-dimensional second-order superintegrable systems over conformally-flat spaces. I show that, up to Staeckel equivalence, such systems can be put into correspondence with a 6 points in the extended complex plane with an action induced by the conformal-group in three dimensions. I use this correspondence, and the tools of classical-invariant theory, to determine the inequivalent orbits under this action and show there are only 10 conformal-classes. This answers an open problem by showing that no unknown systems exist on the sphere. Additional interest in these systems comes from studying their algebra of constants. In the three-dimensional maximum-parameter case this algebra is generated by the iterated Poisson brackets of the 6 linearly independent second-order constants and is known to close at finite order. These 6 second-order constants are necessarily functionally dependent, and up to now the explicit relation for their dependence has only been known on a case-by-case basis. In this thesis I demonstrate a quartic identity which provides the functional relation for a general system.","abstract_has_math":false,"creators":["Capel, Joshua"],"institution":"UNSW, Sydney","degree_name":null,"degree_level":null,"degree_discipline":null,"degree_department":null,"school":null,"contributors":[],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2014,"date_issued":"2014","date_published":"2014","updated_at":"2026-07-24T05:32:14Z","subjects":["Representation Theory","Superintegrability","Mathematical Physics"],"languages":["EN"],"rights":["open access","CC BY-NC-ND 3.0","free_to_read"],"rights_urls":["https://purl.org/coar/access_right/c_abf2","https://creativecommons.org/licenses/by-nc-nd/3.0/au/"],"identifier_entries":[{"key":"dc:identifier","label":"Identifier","values":["https://doi.org/10.26190/unsworks/16817"],"render_values":[{"text":"https://doi.org/10.26190/unsworks/16817","href":"https://doi.org/10.26190/unsworks/16817","code":true}]}]},"links":{"outbound_url":"http://hdl.handle.net/1959.4/53501","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:creator","label":"Author","values":["Capel, Joshua"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2014"]},{"key":"dc:publisher","label":"Institution","values":["UNSW, Sydney"]},{"key":"dc:type","label":"Dc Type","values":["doctoral thesis","http://purl.org/coar/resource_type/c_db06"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Representation Theory","Superintegrability","Mathematical Physics"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["EN"]},{"key":"dc:rights","label":"Dc Rights","values":["open access","https://purl.org/coar/access_right/c_abf2","CC BY-NC-ND 3.0","https://creativecommons.org/licenses/by-nc-nd/3.0/au/","free_to_read"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["http://hdl.handle.net/1959.4/53501","https://unsworks.unsw.edu.au/bitstreams/5e18a438-f3a2-43a8-abe2-312ca75f737c/download","https://doi.org/10.26190/unsworks/16817"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["Over the last half century the study of superintegrable systems has established itself as an interesting subject with connections to some of the earliest known dynamical systems in mathematical-physics. Systems with constants second-order in the momenta have been particularly well studied in recent years. This thesis provides a classification of non-degenerate (maximum parameter) three-dimensional second-order superintegrable systems over conformally-flat spaces. I show that, up to Staeckel equivalence, such systems can be put into correspondence with a 6 points in the extended complex plane with an action induced by the conformal-group in three dimensions. I use this correspondence, and the tools of classical-invariant theory, to determine the inequivalent orbits under this action and show there are only 10 conformal-classes. This answers an open problem by showing that no unknown systems exist on the sphere. Additional interest in these systems comes from studying their algebra of constants. In the three-dimensional maximum-parameter case this algebra is generated by the iterated Poisson brackets of the 6 linearly independent second-order constants and is known to close at finite order. These 6 second-order constants are necessarily functionally dependent, and up to now the explicit relation for their dependence has only been known on a case-by-case basis. In this thesis I demonstrate a quartic identity which provides the functional relation for a general system."]},{"key":"dc:format","label":"Dc Format","values":["application/pdf"]},{"key":"dc:title","label":"Title","values":["Classification of second-order conformally-superintegrable systems"]}]}],"canonical_facts":{"dc:creator":["Capel, Joshua"],"dc:date":["2014"],"dc:description":["Over the last half century the study of superintegrable systems has established itself as an interesting subject with connections to some of the earliest known dynamical systems in mathematical-physics. Systems with constants second-order in the momenta have been particularly well studied in recent years. This thesis provides a classification of non-degenerate (maximum parameter) three-dimensional second-order superintegrable systems over conformally-flat spaces. I show that, up to Staeckel equivalence, such systems can be put into correspondence with a 6 points in the extended complex plane with an action induced by the conformal-group in three dimensions. I use this correspondence, and the tools of classical-invariant theory, to determine the inequivalent orbits under this action and show there are only 10 conformal-classes. This answers an open problem by showing that no unknown systems exist on the sphere. Additional interest in these systems comes from studying their algebra of constants. In the three-dimensional maximum-parameter case this algebra is generated by the iterated Poisson brackets of the 6 linearly independent second-order constants and is known to close at finite order. These 6 second-order constants are necessarily functionally dependent, and up to now the explicit relation for their dependence has only been known on a case-by-case basis. In this thesis I demonstrate a quartic identity which provides the functional relation for a general system."],"dc:format":["application/pdf"],"dc:identifier":["http://hdl.handle.net/1959.4/53501","https://unsworks.unsw.edu.au/bitstreams/5e18a438-f3a2-43a8-abe2-312ca75f737c/download","https://doi.org/10.26190/unsworks/16817"],"dc:language":["EN"],"dc:publisher":["UNSW, Sydney"],"dc:rights":["open access","https://purl.org/coar/access_right/c_abf2","CC BY-NC-ND 3.0","https://creativecommons.org/licenses/by-nc-nd/3.0/au/","free_to_read"],"dc:subject":["Representation Theory","Superintegrability","Mathematical Physics"],"dc:title":["Classification of second-order conformally-superintegrable systems"],"dc:type":["doctoral thesis","http://purl.org/coar/resource_type/c_db06"]},"updated_at":"2026-07-24T05:32:14Z"}