Superintegrable three-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 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 quantum symmetry operators associated with the first integrals considered here but do not analyze the corresponding quantum dynamics. The conformal multiseparability is discussed and examples of conformal first integrals are given. The systems considered here in generality include the Calogero, Wolfes, and other threebody interactions widely studied in mathematical physics.

Periodic orbits for an infinite family of classical superintegrable systems

“…In this case the trajectories are never bounded (see (3.1)) and the motion cannot be periodic. A third integral of motion can still exist and the authors suggest a possible form of the additional integral [11].…”

“…The same classical system was studied in [11] for ω = 0 and α = 0. In this case the trajectories are never bounded (see (3.1)) and the motion cannot be periodic.…”

Extensions of Hamiltonian systems dependent on a rational parameter

“…for any rational λ = m/n. This Hamiltonian generalizes the Calogero three particle chain without harmonic term (obtained for λ = 3, see [2], also known as Jacobi system) and it was the starting point of our work in [3]. This is a particular case of the TTW system [11].…”

“…Example 2. (See also [6]) We consider the system of two uncoupled harmonic oscillators (2). By rescaling u = λy and dividing the Hamiltonian by the constant factor λ 2 , we get the equivalent Hamiltonian…”

“…This superintegrable system was already considered in [2], where it was remarked the existence of additional polynomial first integrals for several values of λ ∈ Q and a general form for odd positive integers was conjectured. Then, a generalization of the system was introduced by Tremblay, Turbiner and Winternitz (TTW) [11] and, subsequently, studied by many authors (see for instance [8,7,9] and references therein), proving the superintegrability of the TTW system, and of several related new systems, for any rational value of λ.…”