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

Five independent isolating orbit labels describe a regular closed, nonstationary orbit in three-dimensional space: its phase space closure is one-dimensional. Five is the maximum at a nonzero flow vector, since every conserved differential annihilates that vector in the six-dimensional tangent space. A global superintegrable Hamiltonian system example is the Kepler orbit, where specific orbital energy, angular momentum and the Laplace-Runge-Lenz vector have two vector/scalar relations leaving five independent invariants. A circular or radial orbit requires a separate treatment of any rank-degenerate formula for these invariants. Five integrals do not force an escaping trajectory to be periodic: unbound Kepler orbits also have the conserved apsidal vector. The closure criterion concerns recurrent bound motion.

Closure of one resonant orbit need not imply five smooth independent integrals on an open neighborhood of all nearby orbits. The orbit-wise count and global superintegrability must be distinguished; the distinction matters in part (b).