On the Existence of Periodic Orbits for the Fixed Homogeneous Circle Problem

We prove the existence of some types of periodic orbits for a particle moving in Euclidean three-space under the influence of the gravitational force induced by a fixed homogeneous circle. These types include periodic orbits very far and very near the homogeneous circle, as well as eight and spiral periodic orbits. In this paper we use geometric arguments to demonstrate the existence of some types of periodic orbits for the movement of a particle in Euclidean three-space IR on which the only acting force is the gravitational force induced by a fixed homogeneous circle. The study presented is purely analytical. Interestingly all we could find in the literature about the fixed homogeneous circle problem were a few different ways of expressing the potential function. These expressions appear in classical potential theory books. Among these expressions are the one expressed in terms of elliptic integrals of the first kind and the one using the arithmetic-geometric mean given by Gauss. Essentially all expressions of the potential known today had already appeared in Poincare’s Théorie du Potentiel Newtonien [3], published first in 1899. Hence little has been done, at least in the past century, in the study of this problem. It is interesting to note that the results proved here use only elementary geometric constructions (but the technical details are sometimes a little involving). Before we state our main results we fix some notation. We are interested in the study of the movement in IR of a particle P under the influence of the gravitational force induced by a fixed homogeneous circle C. Denote by r = (x, y, z) ∈ IR − C the position of the particle P and by ṙ = (ẋ, ẏ, ż) its velocity. According to Newton’s Law the movement of P obeys the following second order differential equation: