Elliptic curve arithmetic and superintegrable systems

Abstract

Harmonic oscillator and the Kepler problem are superintegrable systems which admit more integrals of motion than degrees of freedom and all these integrals are polynomials in momenta. We present superintegrable deformations of the oscillator and the Kepler problem with algebraic and rational first integrals. Also, we discuss a family of superintegrable metrics on the two-dimensional sphere, which have similar first integrals.

17 pages, 3 figures, LaTeX with Amsfonts, with corrected misprints

References in corpus (5)

• New bi-Hamiltonian systems on the plane
• Bäcklund transformations for the nonholonomic Veselova system
• Integrable discretization and deformation of the nonholonomic Chaplygin ball
• Superintegrable Stäckel Systems on the Plane: Elliptic and Parabolic Coordinates
• On exact discretization of the cubic-quintic Duffing oscillator

Cited by in corpus (7)

• Superintegrable systems and Riemann-Roch theorem
• New superintegrable models on spaces of constant curvature
• The Kepler problem: polynomial algebra of non-polynomial first integrals
• Discretization and superintegrability all rolled into one
• Reduction of divisors for classical superintegrable $GL(3)$ magnetic chain
• Reduction of divisors and Kowalevski top
• On two-dimensional Hamiltonian systems with sixth-order integrals of motion