Fully Algebraic Description of the Static Level Sets for the System of Two Particles under a Van der Waals Potential

Abstract

We study the equipotential surfaces around of a two particle system in 3-d under a pairwise good potential as the one of Van der Waals. The level sets are completely determined by the solutions of polynomials of at most fourth degree that can be solved by standard algebraic methods. The distribution of real positive roots determines the level sets and provides a complete description of the map for the equipotential zones. Our methods can be generalized to a family of polynomials with degree multiple of 2, 3, or 4. Numerical simulations of 2-d and 3-d pictures depicting the true orbits and equipotential zones are provided.