# Superintegrable systems on 3-dimensional curved spaces: Eisenhart formalism and separability

## TL;DR

This paper explores superintegrable systems on 3D curved spaces using Eisenhart formalism, analyzing separability and superintegrability of Hamiltonians derived from Euclidean systems, including modifications with potentials and position-dependent masses.

## Contribution

It extends the Eisenhart formalism to analyze superintegrability and separability of four families of systems on curved three-dimensional manifolds, including potential and mass modifications.

## Key findings

- Four families of superintegrable Hamiltonians on curved spaces are identified.
- Separable and superintegrable properties are preserved under potential and mass modifications.
- The systems exhibit broken spherical symmetry on non-Euclidean manifolds.

## Abstract

The Eisenhart geometric formalism, which transforms an Euclidean natural Hamiltonian $H=T+V$ into a geodesic Hamiltonian ${\cal T}$ with one additional degree of freedom, is applied to the four families of quadratically superintegrable systems with multiple separability in the Euclidean plane. Firstly, the separability and superintegrability of such four geodesic Hamiltonians ${\cal T}_r$ ($r=a,b,c,d$) in a three-dimensional curved space are studied and then these four systems are modified with the addition of a potential ${\cal U}_r$ leading to ${\cal H}_r={\cal T}_r +{\cal U}_r$. Secondly, we study the superintegrability of the four Hamiltonians $\widetilde{{\cal H}}_r= {\cal H}_r/ \mu_r$, where $\mu_r$ is a certain position-dependent mass, that enjoys the same separability as the original system ${\cal H}_r$. All the Hamiltonians here studied describe superintegrable systems on non-Euclidean three-dimensional manifolds with a broken spherically symmetry.