Some new aspects of first integrals and symmetries for central force dynamics

For the general central force equations of motion in $n>1$ dimensions, a complete set of $2n$ first integrals is derived in an explicit algorithmic way without the use of dynamical symmetries or Noether's theorem. The derivation uses the polar formulation of the equations of motion and yields energy, angular momentum, a generalized Laplace-Rugge-Lenz vector, and a temporal quantity involving the time variable explicitly. A variant of the general Laplace-Rugge-Lenz vector, which generalizes Hamilton's eccentricity vector, is also obtained. The physical meaning of the general Laplace-Rugge-Lenz vector, its variant, and the temporal quantity are discussed for general central forces. Their properties are compared for precessing bounded trajectories versus non-precessing bounded trajectories, as well as unbounded trajectories, by considering an inverse-square force (Kepler problem) and a cubically perturbed inverse-square force (Newtonian revolving orbit problem).