1202.0197 (E. G. Kalnins et al.)

Structure theory for extended Kepler-Coulomb 3D classical
superintegrable systems    [PDF]

E. G. Kalnins, Willard Miller Jr

The 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, if a 2nd 4th order symmetry is added to the
generators, the symmetry algebra closes 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 (k_1,k_2) and reducing to the usual systems when
k_1=k_2=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.

View original: http://arxiv.org/abs/1202.0197