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.