A Method For Obtaining New Conservation Quantities and a Solution Method For the N-Body Collision

Abstract

Exact, numerical, and perturbative methods are commonly used to solve dynamical systems of equations. Many systems’ solutions cannot be written in terms of elementary functions thus numerical and perturbative solutions take over and provide only approximate solutions (even though perturbative solutions are, in fact, series representations of the exact solutions, but truncating higher order terms only provide approximate solutions). Therefore, if possible, solving for unique, exact solutions should be of utmost importance when determining the dynamics of a system. In general, to solve a dynamical system, there must be a sufficient number of invariant equations about symmetries, or conservation quantities, to reduce the degrees of freedom of the system. One of the most renown dynamical systems is the n-body problem; this thesis will aim to provide a sufficient number of conservation quantities for the special case of the n-body problem that involves only simultaneous elastic collisions of free particles by analytical and experimental methods as well as present general formulations of newly theorized conservation quantities associated with any dynamical system. This thesis also presents the proof of existence of analytical solutions in the three-body collision stated above using Bezout’s theorem.