Nontrivial Solutions to the Newtonian Two-fixed Centers Problem

by Dakota Blair

A thorny problem in the domain of classical physics is the famous three body problem. A general solution to this problem has yet to be found, thus there is constant research in this area. There are many simplified versions of this problem such as the two-fixed center which are still not fully understood. This problem concerns two equal masses a fixed distance apart with a light mass that is allowed to move under the force of gravity. A physical example of this system would be a cross section perpendicular to the plane of a massive torus with a satellite moving in the plane of the cross section. At present, there

are analytical solutions for a light body far away from both masses. This problem is very similar to the two body problem, but is computationally much more difficult. This presentation will be concerned with a family of stable numerical solutions which are connected via their initial conditions. A strikingly simple pattern emerges, and a question that can be addressed with more study is whether or not this pattern is the only way to generate new orbits with similar initial conditions.