Figure-eight solution of the three-body problem

Example

Take the planar three-body problem with three equal masses. There are a period $T > 0$ and a closed curve $x \colon \R \to \R^2$ of period $T$, shaped like the figure eight, with this property. The motion $x_1(t) = x(t)$, $x_2(t) = x(t + T/3)$, $x_3(t) = x(t + 2T/3)$ is a solution. It is a periodic orbit. So the three bodies chase each other along the same curve, each one third of a period behind the next. The centre of mass stays at the crossing point of the eight, and the total angular momentum is zero. At some instants the three bodies are collinear, with one body at the midpoint of the other two.

Check of the period: since $x$ has period $T$, each $x_i$ has period $T$, so the whole motion repeats after time $T$.