Uniform function constants of motion and their first-order perturbation

Abstract

The main purpose of this work is to present some uniform functions constant of the motion, either than the well known quantities arising from spacetime symmetries. These constants are usually associated with intrinsic characteristic of the trajectories of a particle in a central potential field. We treat two cases. The first one is the Lenz vector which sometimes is found in the literature[1, 2]; the other case is associated with the isotropic harmonic oscillator, of relative importance in some simple models of classical molecular interaction. The first example is applied to describe the perturbation of the trajectories in the Rutherford scattering and the precession of the Keplerian orbit of a planet. In the other case the conserved quantity is a symmetric tensor. We find the eigenvectors and eigenvalues of that tensor while at the same time we obtain the solution to the problem of calculating the rotation rate of the orbits in first order of a perturbation parameter in the potential energy, by performing a simple coordinate transformation in the Cartesian plane. We think that the present work addresses to many aspects of Mechanics with a didactical interest in other Physics or Mathematics courses.