Integrable systems on the sphere with elastic interaction potentials (Q1905283)

A remarkable, simple and new, example of integrable mechanical system is given. The discussion of the authors starts from a particle constrained on the unit sphere \(\mathbb{S}^n\) and interacting with its centre by a Calogero-like energy potential, that is, a suitable homogeneous polynomial of degree \(-2\). A simple calculation shows that the Lagrangian multiplier for the constraint to move on \(\mathbb{S}^n\) is identifiable with the total energy constant; furthermore, the dynamics splits into \(n\) one-dimensional uncoupled subsystems. An analogue of a generalization of the classical integrable Jacobi system on the ellipsoid is developed for the above described system.