The Three-Body Problem

Abstract

The three-body problem studies the motion of three mutually attracting gravitational bodies, given their positions and initial velocities. The chaotic nature of the model due to its high sensitivity to initial conditions renders it impossible to use in the prediction of real world phenomena. Some of the most influential mathematicians studied the problem, including Newton, Euler, Lagrange, Jacobi, and Poincaré. This paper uses the Newtonian law of gravity to model the motion of three bodies. Using python, simulations were run using three different sets of initial conditions: The conditions for the Earth-Sun-Moon system, the Lemniscate solution, and Burrau’s solution. Case studies were done on each to analyze the results they produced, and we discuss the solver error of the models. The solutions all differ from their true, analytic solutions due to this error. The three-body problem can be expanded to the n-body problem, which has various applications in the real world, such as providing models of the orbits of the outer planets, the planets in the solar system, and even all the stars in the milky way galaxy. These models cannot be used to make predictions however, since the highly chaotic dependence on unmodelled variables can cause great variation in comparison to the model. The nonexistent general analytic solution to the problem could be used to make accurate predictions of phenomena in our solar system and universe, which would further develop the human understanding of the world.