UC Santa Cruz

In this thesis we study two types of planar N-body problems: the motion of N point masses in a plane under a strong force law of attraction and under Newton's law of attraction. We aim to understand these orbits by which free homotopy classes are realized in the configuration space.

The first part looks at the strong force law of attraction: where the force is proportional to the inverse cube of the mutual distances. This problem is especially suited to the Jacobi-Maupertuis principle, which reformulates the problem as a complete geodesic flow in $\mathbb{C}P^{N-2}$ with collisions deleted. Montgomery [54] found negatively curved circumstances for such a 3-body problem, and consequently the orbits could be described effectively by symbolic dynamics and free homotopy classes are realized uniquely. Here we show that the extension to $N>3$ is not so straightforward: the sectional curvatures have mixed signs, positive in some places and negative in others however, when restricting attention to the collinear 4-body strong force problem we do find negatively curved circumstances allowing us to describe such orbits as geodesics in the hyperbolic plane.

The second part examines the force law of Newton: where the force is proportional to the inverse square of the mutual distances. This classic problem has been well studied. F ejoz [25,26] has established the existence of periodic and quasiperiodic orbits in the lunar regions using perturbation methods. Here we describe these orbits using `syzygy sequences' to represent the homotopy classes realized in these lunar regions. Our method of proof is to show that orbits of the unperturbed problem that avoid certain isolated tangencies retain their homotopy class in the full perturbed problem.