Superintegrable 3-body systems on the line

Abstract

We consider classical three-body interactions on a Euclidean line depending on the reciprocal distance of the particles and admitting four functionally independent quadratic in the momenta first integrals. These systems are superseparable (i.e. multiseparable), superintegrable and equivalent (up to rescalings) to a one-particle system in the three-dimensional Euclidean space. Common features of the dynamics are discussed. We show how to determine the quantum symmetry operators associated with the first integrals considered here but do not analyze the corresponding quantum dynamics. The conformal superseparability is proved and examples of conformal first integrals are given. The systems considered here in generality include the Calogero, Wolfes, and other three-body interactions widely studied in mathematical physics.

Corrected typos. Some improvements

References in corpus (4)

• Quantum integrability of quadratic Killing tensors
• Polynomial Poisson Algebras for Classical Superintegrable Systems with a Third Order Integral of Motion
• Invariant classification of the rotationally symmetric R-separable webs for the Laplace equation in Euclidean space
• A class of superintegrable systems of Calogero type