Date of Graduation
Eberly College of Arts and Sciences
This is an exposition of a new compactification of Euclidean space enabling the study of trajectories at and approaching spatial infinity, as well as results obtained for polynomial differential systems and Celestial Mechanics. The Lorenz system has an attractor for all real values of its parameters, and almost all of the complete quadratic systems share an interesting feature with the Lorenz system.;Informed by these results, the main theorem is established via a contraction mapping arising from an integral equation derived from the Celestial Mechanics equations of motion. We establish the existence of an open set of initial conditions through which pass solutions without singularities, to Newton's gravitational equations in Euclidean space on a semi-infinite interval in forward time, for which every pair of particles separates like a multiple of time, as time tends to infinity. The solutions are constructible as series with rapid uniform convergence and their asymptotic behavior to any order is prescribed. We show that this family of solutions depends on 6N parameters subject to certain constraints. This confirms the logical converse of Chazy's 100-year old result assuming solutions exist for all time, they take the form given by Bohlin: solutions of Bohlin's form do exist for all time. An easy consequence not found elsewhere is that the asymptotic directions of many configurations exiting the universe depend solely on the initial velocities and not on their initial positions.;The N-body problem is fundamental to astrodynamics, since it is an idealization to point masses of the general problem of the motion of gravitating bodies, such as spacecraft motion within the Solar System. These new trajectories model paths of real particles escaping to infinity. A particle escaping its primary on a hyperbolic trajectory in the Kepler problem is the simplest example. This work may have relevance to new interplanetary trajectories or insight into known trajectories for potential space missions.
Solomon, Daniel, "A New Compactification for Celestial Mechanics" (2017). Graduate Theses, Dissertations, and Problem Reports. 6689.