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 (7)

• Superintegrability with third order invariants in quantum and classical mechanics
• Quantum integrability of quadratic Killing tensors
• Polynomial Poisson Algebras for Classical Superintegrable Systems with a Third Order Integral of Motion
• An exactly solvable three-particle problem with three-body interaction
• Maximal superintegrability of Benenti systems
• Invariant classification of the rotationally symmetric R-separable webs for the Laplace equation in Euclidean space
• A class of superintegrable systems of Calogero type

Cited by in corpus (26)

• Classical and Quantum Superintegrability with Applications
• An infinite family of solvable and integrable quantum systems on a plane
• Periodic orbits for an infinite family of classical superintegrable systems
• A Recurrence Relation Approach to Higher Order Quantum Superintegrability
• Two-Variable Wilson Polynomials and the Generic Superintegrable System on the 3-Sphere
• Tools for Verifying Classical and Quantum Superintegrability
• A new approach to the higher-order superintegrability of the Tremblay-Turbiner-Winternitz system
• Generalizations of a method for constructing first integrals of a class of natural Hamiltonians and some remarks about quantization
• The Tremblay-Turbiner-Winternitz system as extended Hamiltonian
• First Integrals of Extended Hamiltonians in n+1 Dimensions Generated by Powers of an Operator
• Polynomial constants of motion for Calogero-type systems in three dimensions
• Extensions of Hamiltonian systems dependent on a rational parameter
• Superintegrable Extensions of Superintegrable Systems
• Structure results for higher order symmetry algebras of 2D classical superintegrable systems
• Extended Hamiltonians, Coupling-Constant Metamorphosis and the Post-Winternitz System
• On the Extended-Hamiltonian Structure of Certain Superintegrable Systems on Constant-Curvature Riemannian and Pseudo-Riemannian Surfaces
• Stationary Flows Revisited
• Extended Hamiltonians and shift, ladder functions and operators
• Integrable and Superintegrable Extensions of the Rational Calogero-Moser Model in 3 Dimensions
• Searching for integrable Hamiltonian systems with Platonic symmetries
• Examples of Complete Solvability of 2D Classical Superintegrable Systems
• Superintegrable systems with position dependent mass: master symmetry and action-angle methods
• A Geometric Study of Superintegrable Systems
• Three and four-body systems in one dimension: integrability, superintegrability and discrete symmetries
• Carter Constant and Superintegrability
• Extensions of natural Hamiltonians