Discretization and superintegrability all rolled into one

Abstract

Abelian integrals appear in mathematical descriptions of various physical processes. According to Abel's theorem these integrals are related to motion of a set of points along a plane curve around xed points, which are rarely used in physical applications. We propose to interpret coordinates of the xed points either as parameters of exact discretization or as additional rst integrals for equations of motion reduced to Abelian quadratures on a symmetric product of algebraic curve.

Reduction of divisors and Clebsch system

“…Such constant reduced divisors also appear for various superintegrable systems [36,37,38] and for the Kowalevski top [39].…”

“…In this note, we try to reproduce one of these reduced divisors using the Euler-Abel construction of reduced divisors, symmetries of the Clebsch system, and poles of the Baker-Akhiezer function on the spectral curve of the 3 × 3 Lax matrix. A standard reduction leads to a reduced divisor which is a constant of motion similar to superintegrable systems [36,37,38] and the Kowalevski top [39]. Nevertheless, following [16], we can break the symmetry of the Clebsch system and take arbitrary prime divisor P 4 or P 5 from the support of semi-reduced divisor D ′ = P 4 + P 5 (2.9) as the desired separated variable.…”

On two-dimensional Hamiltonian systems with sixth-order integrals of motion

“…In 1935 Drach studied two-dimensional systems in the pseudo-Euclidean plane with cubic invariants [6]. Seven of the ten Drach potentials are superintegrable with two quadratic additional integrals [17] and, therefore, cubic Drach invariants exist due to the Abel and Riemann-Roch theorems [20,21].…”