On Euler superintegrable systems

Abstract

We discuss the Euler approach to construction and to investigation of the superintegrable systems with additional quadratic and cubic integrals of motions.

On the superintegrable Richelot systems

“…It's easy to get a lot of such superintegrable Richelot systems in framework of the Jacobi separation of variables method, see [17,30,31].…”

“…In order to use the Richelot results we have to impose some constraints on the entries of the Stäckel matrix S kj (q j ), which give rise to some restrictions on the coefficients A k [17,30].…”

“…The main problem is that we want to get Hamiltonians H j in some physical variables x instead of Hamiltonians (2.11) in terms of the abstract separated variables q. According to [17,30,31] it leads to some additional restrictions on the coefficients A j in (2.8) and substitutions (2.9).…”

“…One of the best examples of this phenomenon is provided by the harmonic oscillator and the Kepler-Coulomb problem. A large number of papers have been published on super-integrability in these last years, most of them related with second-order integrals of motion (see [3,8,10,14,17,20,26,30,31] for some recent results and an extensive list of references).…”

“…where f is an arbitrary quartic [11]. The corresponding superintegrable Stäckel systems have been classified in [17]. The Abel theorem may be regarded as a generalization of these Euler results.…”