Superintegrable Systems on 3 Dimensional Conformally Flat Spaces

Abstract

We consider Hamiltonians associated with 3 dimensional conformally flat spaces, possessing 2, 3 and 4 dimensional isometry algebras. We use the conformal algebra to build additional {\em quadratic} first integrals, thus constructing a large class of superintegrable systems and the complete Poisson algebra of first integrals. We then use the isometries to reduce our systems to 2 degrees of freedom. For each isometry algebra we give a {\em universal} reduction of the corresponding general Hamiltonian. The superintegrable specialisations reduce, in this way, to systems of Darboux-Koenigs type, whose integrals are reductions of those of the 3 dimensional system.

32 pages, 7 tables

References in corpus (11)

• Superintegrable Systems in Darboux spaces
• Superintegrability in a two-dimensional space of non-constant curvature
• Two-dimensional superintegrable metrics with one linear and one cubic integral
• The gap phenomenon in parabolic geometries
• Invariant classification of second-order conformally flat superintegrable systems
• Invariant Classification and Limits of Maximally Superintegrable Systems in 3D
• Mutation-Periodic Quivers, Integrable Maps and Associated Poisson Algebras
• First Integrals from Conformal Symmetries: Darboux-Koenigs Metrics and Beyond
• Generalised Darboux-Koenigs Metrics and 3-Dimensional Superintegrable Systems
• Poisson Algebras and 3D Superintegrable Hamiltonian Systems
• A Kaluza-Klein Reduction of Super-integrable Systems

Cited by in corpus (8)

• Symmetries of the Schroedinger-Pauli equation for neutral particles
• Algebraic approach and exact solutions of superintegrable systems in 2D Darboux spaces
• Adding Potentials to Superintegrable Systems with Symmetry
• Integrable and Superintegrable Extensions of the Rational Calogero-Moser Model in 3 Dimensions
• Dynamical symmetry algebra of two superintegrable two-dimensional systems
• Superintegrability of 3-dimensional Hamiltonian systems with conformally Euclidean metrics. Oscillator-related and Kepler-related systems
• Algebraic structures and Hamiltonians from the equivalence classes of 2D conformal algebras
• Superintegrability on the 3-dimensional spaces with curvature. Oscillator-related and Kepler-related systems on the Sphere $S^3$ and on the Hyperbolic space $H^3$