Three-Body Problem: A Precise Simulation

The motion of three free bodies in gravitational interaction is one of the simplest examples of chaotic behaviour in nature. Such behaviour is characterised by complex non-periodic orbits, which usually exhibit a high sensitivity on small variations in the initial conditions – a feature known as the butterfly effect.

This video shows a high-precision simulation of the special three-body problem, where three identical point masses are initially at rest at the vertices of a Pythagorean triangle of axes ratios 3:4:5. As in almost all three-body problems, one of the bodies gets eventually kicked out of the system, while the other two remain stuck together forever.

The orbits of the three masses have been computed iteratively, using the "nbody" package of the R programming language. The solution shown here used a 4th order Yoshida integrator with an adaptive block time-step, chosen small enough to ensure that the solution is exact within less than a pixel for the whole duration of the simulation.

The masses M of the three bodies and the length unit L of the triangle are irrelevant in the sense that the orbits are geometrically similar for any positive choices of M and L – this is the scale invariance of gravitation. The overall scale of the orbits is proportional to L and the time taken to move through them is proportional to sqrt(L^3/(GM)), where G is the gravitational constant.

Credits:
Software: R-language with nbody package
Music: "Dreams" from Bensound.com (royalty-free)

Related literature:
https://en.wikipedia.org/wiki/Three-body_problem
https://en.wikipedia.org/wiki/Chaos_theory