Action-Angle Variables In Conformal Mechanics

TL;DR

This paper develops action-angle variables for conformal mechanics, constructs new integrable systems related to black hole near-horizon dynamics, and analyzes their integrability and superintegrability properties.

Contribution

It introduces a method to transform conformal mechanics into a standard form and constructs new integrable and superintegrable models linked to black hole physics.

Findings

Action-angle variables for 2D singular oscillators

New integrable generalizations of oscillator and Coulomb systems

Superintegrability of certain black hole related models

Abstract

- We have suggested using the action-angle variables for the study of a (quasi)particle in quantum ring. We have presented the action-angle variables for three two-dimensional singular oscillator systems - We have suggested a procedure of constructing new integrable systems form the known ones, by adding a radial part to the angular Hamiltonian. Using this method we have constructed a class of integrable generalizations of oscillator and Coulomb systems on N-dimensional Euclidian space, sphere and hyperboloid. - We have developed the methods of study of conformal mechanics, based on separation of the radial and angular parts of the Hamiltonian of the system. - We have proved, that by applying a proper canonical transformation one can bring the conformal mechanics associated with near-horizon motion of massive relativistic particle in the field of extremal black holes in arbitrary dimensions to the conventional conformal mechanics form. - We have studied in details the near-horizon motion of massive relativistic particle in the field of extremal Reissner-Nordstrom black hole with magnetic momentum. We have analyzed the near-horizon motion of massive relativistic particle in the field of extremal Clement-Gal'tsov (Dilaton-Axion) black hole. - We have inspected the integrability of spherical mechanics models associated with the near horizon extremal Myers-Perry black hole in arbitrary dimension for the special case that all rotation parameters are equal. We have proved the superintegrability of the new system and demonstrated that the spherical mechanics associated with the black hole in odd dimensions is maximally superintegrable, while its even-dimensional counterpart lacks for only one constant of the motion to be maximally superintegrable.