On the incompleteness of the classification of quadratically integrable Hamiltonian systems in the three-dimensional Euclidean space

TL;DR

This paper provides a counterexample of an integrable Hamiltonian system in three-dimensional Euclidean space that is not separable in any orthogonal coordinate system, challenging previous classifications of such systems.

Contribution

It introduces a specific example of a quadratically integrable Hamiltonian system that defies prior classification based on separability, indicating the need for revision of existing lists.

Findings

Counterexample of a non-separable integrable system

Challenges the completeness of previous classifications

Highlights the need for revised classification criteria

Abstract

We present an example of an integrable Hamiltonian system with scalar potential in the three-dimensional Euclidean space whose integrals of motion are quadratic polynomials in the momenta, yet its Hamilton-Jacobi / Schrodinger equation cannot separate in any orthogonal coordinate system. This demonstrates a loophole in the derivation of the list of quadratically integrable Hamiltonian systems in [Makarov et al., A systematic search for nonrelativistic systems with dynamical symmetries. Nuovo Cimento A Series 10, 52:1061-1084, 1967] where only separable systems were found, and the need for its revision.