Semester

Spring

Date of Graduation

2022

Document Type

Dissertation

Degree Type

PhD

College

Eberly College of Arts and Sciences

Department

Mathematics

Committee Chair

Harry Gingold, Ph.D., Chair

Committee Member

Harvey Diamond, Ph.D.

Committee Member

Adam Halasz, Ph.D.

Committee Member

Harumi Hattori, Ph.D.

Committee Member

Wathiq Abdul-Razzaq, Ph.D.

Abstract

It is shown that for the classical system of the N body problem ( Newtonian Motion), if the motion of the N particles starts from a planar initial motion at t=t_{0}, then the motion of the N particles continues to be planar for every t\in[t_{0},t_{1}], assuming that no collisions occur between the N particles. Same argument is shown about the linear motion, namely, for the classical system of the N body problem, if the motion of the N particles starts from a linear initial motion at t=t_{0}, then the motion of the N particles continues to be linear for every t\in[t_{0},t_{1}], and again, assuming that no collisions occur between the N particles. In other words, for the classical system of the N body problem, the initial planar motion and the initial linear motion at t=t_{0}, yield planar motion and linear motion respectively in the whole interval [t_{0},t_{1}],assuming that the positions of the N particles are all analytic vector functions in [t_{0},t_{1}]. Integral equations along with the Gronwall inequality are utilized with the aid of the matrix of blocks formulation to prove these two arguments. Three types of the system of differences of the classical system of the N body problem are introduced, namely, the larger system of differences of the N bodies which has N(N-1)/2,unknowns and two smaller systems of differences of the N bodies which both have (N-1) unknowns. It is known that in the classical system of the N body problem when two or more particles collide, the right side of the classical system of the N bodies becomes invalid at the collision time and the potential energy of the whole system blows up and consequently velocities of the N particles become unbounded. In order to handle such a phenomenon, the classical model of the N body problem is regularized to another model of motion that is called “ The Regularized Model of the N Bodies with \alpha=3". In the regularized model with \alpha=3, the motion continues at and after collision occurs and accelerations of the N particles become bounded, velocities of all the N bodies become continues vector functions in some interval [t_{0}-\lambda,t_{0}+\lambda],for \lambda>0. Furthermore, it is shown that the conservation of linear momentum and conservation of angular momentum hold pre collision, at, and after collision. The potential energy of the classical system of the N bodies is regularized in such a way that it becomes bounded when collisions between particles occur . Another regularized model of motion is introduced, namely, “ The Regularized Model of Motion with \alpha=2". The model of motion with \alpha=2,works when no collisions between particles occur. Howover, the right sides of the two regularized models of motion, namely, the model of motion with \alpha=2, and the model of motion with \alpha=3, have both the same upper bound. The technique of successive approximations is developed to prove that second order vectorial nonlinear autonomous differential systems \ddot{\overrightarrow{y}}=\overrightarrow{f}(\overrightarrow{y}), possess a continuum of symmetric solutions. They are shown to possess a continuum of even solutions. If \overrightarrow{f}(\overrightarrow{y}), is an odd function of \overrightarrow{y}, then \ddot{\overrightarrow{y}}=\overrightarrow{f}(\overrightarrow{y}), is shown also to possess a continuum of odd solutions. The results apply to a significant family of second order vectorial nonlinear differential systems that are not dissipative. This family of differential equations includes the celebrated N body problem of celestial mechanics and other central force problems.

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