Addition theorems and the Drach superintegrable systems

Abstract

We propose new construction of the polynomial integrals of motion related to the addition theorems. As an example we reconstruct Drach systems and get some new two-dimensional superintegrable Stäckel systems with third, fifth and seventh order integrals of motion.

On Euler superintegrable systems

“…This superintegrable system coincides with the one of the Drach systems associated with the logarithmic angle variables [15]. The same system may be obtained by using fourth substitution from (3.7).…”

“…We discuss the Euler construction of the algebraic integrals of motion for superintegrable systems at κ 1,2 = ±1 only. It will be interesting to classify the corresponding superintegrable systems for another values of κ's, because the corresponding additional integrals of motion will be higher order polynomials in momenta, see [15]. Another perspective consists in the substitution of the generic Darboux metrics into the equations (3.9) and classification of the corresponding Euler superintegrable systems on the Darboux spaces.…”

“…and 15) where Φ(H 1 , H 2 , K 2 ) is the second order polynomial such as s = K 2 − c and g 2 is quadrivariant of the quartic (2.8). Another details on the quadratic Poisson algebras of integrals of motion may be found in [4].…”

“…The remaining systems with potentials V 3 , V 4 and V 5 are defined on the Liouville surfaces. The second separated variables q 1,2 for integrals of motion H 1 , K 2 may be found by using the software proposed in [7]: It is easy to prove that the corresponding separated relations p 2 j = P ( q j ) define two different zero-genus hyperelliptic curves and lead to logarithmic angle variables [15]. Thus, for these three Euler superintegrable systems there are two different addition theorems: addition theorem for elliptic functions (2.10) and addition theorem for logarithms ln x + ln y = ln xy.…”

Superintegrable systems and Riemann-Roch theorem

“…At n = 2 equation (1.8) coincides with the Euler differential equation on an elliptic curve. The corresponding superintegrable systems are discussed in [15,17,18,19].…”