Visualization of Euler’s Equations of Motion

This demo represents 100 simulated paths of Euler’s equations of motion with random initial conditions.

A simplified explanation: the way an object rotates in space can be determined by knowing its angular momentum vector. Euler’s equations of motion then provide a differential equation that explains how the angular momentum vector changes with respect to time, and by solving that equation, we can fully describe how an object rotates in space. This visualization was created by taking 100 random points on the sphere as initial angular momentum vectors and then using Euler’s equations of motion to determine how the angular momentum vector would change over time. This gives the distinct elliptical paths seen below. Note that the cones/arrows represent the direction the angular momentum vector is moving when the angular momentum vector is located at the base of the cone. Bright cones indicate where the angular momentum is changing quickly and dark cones indicate that where the angular momentum is changing slowly.

In this second figure, I used the previous data to train a model to produce these paths. On the figure, I have drawn both the “true” path and the path produced by model, and for each path, these are both drawn in the same color. Visually, there is generally extremely good agreement between the true and predicted paths, but one extremely interesting place to look at is around some of the poles of the sphere.

Around these poles, my model received relatively little data to teach it how to generate solutions there, and the quality of the model breaks down there. In fact, one lavendar path especially close to the pole demonstrates a “bifurcation” where the true and predicted paths split off from each other and go in different directions. This is because the model was unable to correctly predict the path here, and the solution behavior near the poles is very sensitive to slight changes.