Past exam of the mathematics course of the University of Cambridge / 2005 / iii / Paper 72 / 4 / a / vi / Solution

More than three occur when symmetries or frequency degeneracies reduce the orbit's invariant closure below a generic three-dimensional invariant torus. For a nonradial orbit in a spherical gravitational potential, specific orbital energy and the three components of specific angular momentum are generically four independent isolating integrals. Their values fix the orbital plane, angular momentum magnitude and radial specific orbital energy, but a nonclosed rosette still fills a two-dimensional invariant torus. Thus more than three does not by itself imply closure. Orbital resonances can further reduce the number of independent orbital phases. The count concerns independent integrals, not arbitrarily many functions of the same invariants.