Carter Constant and Superintegrability

TL;DR

This paper extends Carter's theorem to systems with more degrees of freedom, providing new insights into superintegrability and Carter separability, with applications to classical superintegrable potentials.

Contribution

It generalizes Carter's theorem to N-degree systems and offers a novel approach to analyze superintegrability through Carter separability.

Findings

Extended Carter's theorem to three and N degrees of freedom.

Connected Carter constant to superintegrability.

Constructed examples of 2D superintegrable systems.

Abstract

Carter constant is a non-trivial conserved quantity of motion of a particle moving in stationary axisymmetric spacetime. In the version of the theorem originally given by Carter, due to the presence of two Killing vectors, the system effectively has two degrees of freedom. We propose an extension to the first version of Carter's theorem to a system having three degrees of freedom to find two functionally independent Carter-like integrals of motion. We further generalize the theorem to a dynamical system with $N$ degrees of freedom. We further study the implications of Carter Constant to Superintegrability and present a different approach to probe a Superintegrable system. Our formalism gives another viewpoint to a Superintegrable system using the simple observation of separable Hamiltonian according to Carter's criteria. We then give some examples by constructing some 2-Dimensional superintegrable systems based on this idea and also show that all 3-D simple classical Superintegrable potentials are also Carter separable.