On the Drach superintegrable systems

Abstract

Cubic invariants for two-dimensional degenerate Hamiltonian systems are considered by using variables of separation of the associated Stäckel problems with quadratic integrals of motion. For the superintegrable Stäckel systems the cubic invariant is shown to admit new algebro-geometric representation that is far more elementary than the all the known representations in physical variables. A complete list of all known systems on the plane which admit a cubic invariant is discussed.

16 pages, Latex2e+Amssymb

References in corpus (7)

• Quadratic Poisson algebras for two dimensional classical superintegrable systems and quadratic associative algebras for quantum superintegrable systems
• A unified treatment of quartic invariants at fixed and arbitrary energy
• A unified treatment of cubic invariants at fixed and arbitrary energy
• Duality between integrable Stackel systems
• Canonical transformations of the extended phase space, Toda lattices and Stackel family of integrable systems
• The Lax pairs for the Holt system
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Cited by in corpus (33)

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• Superintegrable Systems with a Third Order Integrals of Motion
• Classification of quantum superintegrable systems with quadratic integrals on two dimensional manifolds
• Polynomial Poisson Algebras for Classical Superintegrable Systems with a Third Order Integral of Motion
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• Construction of classical superintegrable systems with higher order integrals of motion from ladder operators
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• General $N^{th}$-order superintegrable systems separating in polar coordinates
• Ricci-flat spacetimes admitting higher rank Killing tensors
• Higher-order superintegrability of a Holt related potential
• On reciprocal equivalence of Stäckel systems
• Addition theorems and the Drach superintegrable systems
• Fourth-order superintegrable systems separating in Polar Coordinates. II. Standard Potentials
• Superintegrable deformations of superintegrable systems : Quadratic superintegrability and higher-order superintegrability
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• New infinite families of $N$th-order superintegrable systems separating in Cartesian coordinates
• A multiple scales approach to maximal superintegrability
• The Kepler problem: polynomial algebra of non-polynomial first integrals
• Higher order first integrals of autonomous dynamical systems
• Towards a classification of natural bi-hamiltonian systems
• Cubic first integrals of autonomous dynamical systems in $E^2$ by an algorithmic approach
• An invariant classification of cubic integrals of motion
• On two-dimensional Hamiltonian systems with sixth-order integrals of motion
• Superintegrability on the 3-dimensional spaces with curvature. Oscillator-related and Kepler-related systems on the Sphere $S^3$ and on the Hyperbolic space $H^3$