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

TL;DR

This paper derives a complete set of first integrals for central force dynamics in multiple dimensions using an explicit algorithm, introducing generalized vectors and quantities that extend classical conserved quantities.

Contribution

It provides a novel explicit algorithmic derivation of all first integrals for central force motion without relying on symmetries or Noether's theorem, including new generalized vectors.

Findings

Derived explicit formulas for energy, angular momentum, and generalized Laplace-Rugge-Lenz vectors.

Compared properties of these vectors for different types of trajectories and forces.

Extended classical conserved quantities to more general central force problems.

Abstract

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).