The Kepler problem: polynomial algebra of non-polynomial first integrals

Abstract

The sum of elliptic integrals simultaneously determines orbits in thr Kepler problem and the addition of divisors on elliptic curves. Periodic motion of a body in physical space is defined by symmetries, whereas periodic motion of divisors is defined by a fixed point on the curve. Algebra of the first integrals associated with symmetries is a well-known mathematical object, whereas algebra of the first integrals associated with coordinates of fixed points is unknown. In this paper, we discuss polynomial algebras of non-polynomial first integrals of superintegrable systems associated with elliptic curves.

LaTex with AMS fonts, 18 pages

References in corpus (5)

• On superintegrable systems separable in Cartesian coordinates
• Backlund transformations and divisor doubling
• Elliptic curve arithmetic and superintegrable systems
• Transformation of the Stackel matrices preserving superintegrability
• On bi-Hamiltonian formulation of the perturbed Kepler problem

Cited by in corpus (4)

• Superintegrable systems and Riemann-Roch theorem
• Reducible Abelian varieties and Lax matrices for Euler's problem of two fixed centres
• Reduction of divisors for classical superintegrable $GL(3)$ magnetic chain
• More on superintegrable models on spaces of constant curvature