Transformation of the Stäckel matrices preserving superintegrability

Abstract

If we take a superintegrable Stäckel system and make variables "faster" or "slower", that is equivalent to a trivial transformation of the Stäckel matrix and potentials, then we obtain an infinite family of superintegrable systems with explicitly defined additional integrals of motion. We present some examples of such transformations associated with angle variables expressed via logarithmic functions.

Elliptic curve arithmetic and superintegrable systems

“…In physical terms a, b and ω 1 , ω 2 are action-angle variables associated with this motion, whereas Euler's algebraic constraint (1.5) is an additional first integral [33]. Second quadrature in the differential form coincides with (1.6) and, therefore, we have superintegrable system with additional first integrals which are abscissa and ordinate of the third point P 3 on a projective plane…”

“…These functions can be obtained from (2.13) by using non-canonical transformation p ui → k −1 i p ui , see discussion in [33]. The corresponding quadratures…”

Superintegrable systems and Riemann-Roch theorem

“…we can "restore" the symmetry between points by using substitution So, all the superintegrable systems described in Section 2 remain superintegrable after the noncanonical transformations (3.22), see discussion in [20,22,23].…”

The Kepler Problem: Polynomial Algebra of Nonpolynomial First Integrals

“…Let us consider non-canonical transformations of momenta preserving symmetries of configuration space, but breaking symmetry between divisors [16,36,37,38,39,41,42]. It is easy to see, that transformation of momenta…”

“…According to [41,42] these Hamiltonians (2.11) are superintegrable Hamiltonians because this noncanonical transformation sends the original sum of elliptic integrals (2.5) to the sum…”