Publication Date

2024

Document Type

Conference Poster

First Advisor

McCord, Chris

Second Advisor

Nicholas Karonis

Department

Department of Computer Science| Department of Mathematical Sciences

Abstract

The N-body problem is a field of study in mathematics and physics that involves predicting the motion of particles moving under their mutual gravitational attraction. It is vital in celestial mechanics, such as planning collision-free satellite orbit trajectories. When beginning to understand the N-body problem, we can start by looking at equal masses of these particles or celestial bodies. As particles move, their position and velocity change, both energy and angular momentum are conserved. Sets of constant energy and angular momentum, known as integral manifolds, are higher-dimensional figures that represent constraints of movement to a system. Integral manifolds are described using bifurcation energy levels. These energy levels indicate where there could have been a quantitative change to the solutions of the equations of motion, and thus, the appearance of the integral manifold. Bifurcation values depend on the different ways particles can form “central configurations”; arrangements of masses on a three-dimensional coordinate system that retain their shape if released simultaneously from rest. Thus far, central configurations for up to seven equal masses have all been identified and confirmed. For central configurations of eight, nine, and ten equal masses, not all central configurations are confirmed. In addition, central configurations for specialized cases of unequal masses, such as three equal masses and one unequal mass, have not yet been identified. The project takes advantage of an existing computer algorithm that can confirm all possible central configurations for up to seven equal masses. Using this, we can manipulate the program itself to identify patterns that compare or do not compare to theories on the appearance of these families of central configurations. With further understanding of these cases, we can come closer to identifying the central configurations and thus their corresponding bifurcation candidate energy levels.

Program

Conference on Undergraduate Research and Engagement: Math and Computational Sciences

Publisher

Northern Illinois University

Share

COinS