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.