Complete description of the static level sets for the system of two particles under a Van der Waals potential

摘要

We study the orbits around of a two particle system under a pairwise good potential like the one of Van der Waals. We show that the levels sets are completely determined by polynomials at most four degree that can be factorized by means of standard algebraic procedures, such as the methods of Cardan and Ferrari. The distribution of real positive roots determine the level curves and provides a complete description of the map of the equipotential zones. We show that our methods can be generalized to a family of polynomials with degree multiple of 2, 3, and 4. We carry out a comparison with numerical simulations, with the true orbits, and 2-d and 3-d pictures depicting the true isopotential zones.