6 - Fractals in the 3-Body Problem Via Symplectic Integration

Summary

Introduction

The statement of the n-body problem is tantalizingly simple: Given the present positions and velocities of n celestial bodies, predict their motions under Newton's inverse square law of gravitation for all future time and deduce them for all past time.

This simplicity belies the fact that efforts to solve this problem, beginning particularly with the work of Henri Poincaré in the late 19th century, essentially led to the creation of the field of dynamical systems. The study of the n-body problem remains an active area of research (see, for example, [17]).

It is intriguing to consider how Poincaré's research might have benefited had he access to modern computing capabilities. That he was able to discern what are now known as fractals and chaotic behavior in the 3-body problem [19] using only paper, pencil, and mental acumen (ferocious though it was) is remarkable.

It is safe to assume, nonetheless, that even Poincaré would have gained further insight into fractal geometry and chaotic dynamics had he access to modern numerical integration algorithms. In particular, given that the equations of motion for the 3-body problem are Hamiltonian in nature, it is now clear that he would have used a symplectic integration algorithm to numerically compute solutions.