On a Family of Integrable Hamiltonian Systems - Sottocornola, Nicola (Discontinuity, Nonlinearity, and Complexity, 2026)
We consider a family of Hamiltonian systems with homogeneous potentials $V_n$ of degree $n$. These systems are known to be Liouville integrable and their first integrals of motion are known. We examine first the easiest case where the potential function is a cubic polynomial and we find the separation coordinates. After we prove that all the systems in the family can be completely solved in quadratures using these new coordinates.