Central Force Fields

Abstract

This chapter studies the characteristics of central force fields, i.e. those vector fields which point towards a fixed point. This study is of particular relevance for Newtonian gravity, because a spherical distribution of matter determines a central force field in the space. After introducting the topic, by mean of Poisson’s equation and fundaments of potential theory, we obtain the expressions of potential (and force) for simple spherical matter distributions: the point-like case, the homogeneous sphere, the Plummer sphere, the isochrone sphere. We face the problem of the analysis of motion in such potentials both quantitatively, deriving trajectories and laws of motion when possible, and qualitatively, delimiting the allowed phase space by a proper use of constants of motion. The possible simple and multiple periodicity of oscillations in such potentials is investigated, too.