Classification of second-order conformally-superintegrable systems
type: CreativeWork
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.
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.
author: Capel, Joshua
language: EN
keywords: Representation Theory, Superintegrability, Mathematical Physics
license: https://purl.org/coar/access_right/c_abf2
license: https://creativecommons.org/licenses/by-nc-nd/3.0/au/
date published: 2014
date modified: 2026-07-24T05:32:14Z
provider: University of New South Wales
publisher: Networked Digital Library of Theses and Dissertations
type: WebPage
source organization: UNSW, Sydney

PropertyValue
property ID: Handle
value: http://hdl.handle.net/1959.4/53501

PropertyValue
property ID: DOI
value: 10.26190/unsworks/16817