Abstract
A perihelion (unit) vector-the generalisation of the Runge-Lenz vector obtained by Fradkin (1967) is investigated. Its evolution, viewed as its dependence on the position and momentum vectors of a particle moving along its trajectory, demonstrates that, in general, it depends on time. For trajectories having the form of closed orbits with one pair of turning points, the perihelion vector turns out to be time independent and therefore a true integral of the motion. When a closed orbit possesses n pairs of turning points, an n-arm star of n different perihelion vectors is an invariant of the motion. In this case integrals of motion in the form of an n-rank tensor can be constructed from the perihelion vectors. Conditions for the existence of closed orbits are derived and illustrated by examples of the motion in the Kepler potential perturbed by a centrifugal-like term and in the isotropic harmonic oscillator potential similarly perturbed.