The Bunimovich stadium is a rectangle capped by semicircles in which a point particle moves at constant speed along straight lines, reflecting off the boundary in a way that the angle of incidence equals the angle of reflection. This animation, made by Phillipe Roux, shows a collection of such particles initially moving in the same…

A little-known result in Newton’s Principia Mathematica is that by adding an inverse cube force to the usual force of gravity, you could change the angular motion of a planet while leaving its radial motion unchanged. This can have strange effects, shown in the animation here.

The Kepler problem concerns a particle moving under the influence of gravity, like a planet moving around the Sun. Newton showed the orbit of such a particle is an ellipse, assuming it doesn’t fly off to infinity. There are many ways to prove this, but the most illuminating is to reparametrize time and think of the orbit as a circle in 4 dimensions. When the circle is projected down to 3-dimensional space, it becomes an ellipse. The animation in this post, created by Greg Egan, shows how this works.