Integrable systems of quartic oscillators. II

Abstract

Several completely integrable, indeed solvable, Hamiltonian many-body problems are exhibited, characterized by Newtonian equations of motion ("acceleration equal force"), with linear and cubic forces, in N-dimensional space (N being an arbitrary positive integer, with special attention to N=2, namely motions in a plane, and N=3, namely motions in ordinary three-dimensional space). All the equations of motion are written in covariant form ("N-vector equal N-vector"), entailing their rotational invariance. The corresponding Hamiltonians are of normal type, with the kinetic energy quadratic in the canonical momenta, and the potential energy quadratic and quartic in the canonical coordinates.