Three-body problem

Each body is pulled toward each of the others by a force that falls off with the square of the distance; with the gravitational constant and the masses set to 1, $\ddot{\mathbf x}_i = \sum_{j \ne i} (\mathbf x_j - \mathbf x_i)/|\mathbf x_j - \mathbf x_i|^3$. Two bodies reduce to one orbiting their common centre, and the conserved energy and angular momentum pin its path to a conic section. Three bodies conserve the same quantities, but in general these no longer reduce the motion to a formula; it is computed step by step, as this page does. Its starts lie in one slice of two numbers: bodies 1 and 2 at $x = \mp 1$, body 3 between them, with velocities $(p, q)$, $(p, q)$ and $(-2p, -2q)$. The momentum and angular momentum are then zero and the energy is $E = 3(p^2 + q^2) - 5/2$. Each point of the start map is one start, coloured by when the three break up.

A close pass can draw one pair tighter and hand the energy it releases to the third body. Given enough, the third leaves, and since the momentum is zero, it and the pair fly off in opposite directions, the pair at half its speed. The page counts a run as broken up when one body is more than 5 units from a bound pair’s centre and 10 times the pair’s separation, moving away with positive energy relative to it. That is a test, not a proof; 500 such break-ups from the map, followed 300 time units further, all kept receding. Of the 72,719 starts on the map with $E < 0$, 60% break up within the 400 time units computed. Outside the circle $p^2 + q^2 = 5/6$ the energy is not negative, and the Lagrange–Jacobi identity rules out staying together: for $I = \sum |\mathbf x_i|^2$, $\ddot I = 4E + 2|U|$, so if the distances stayed bounded, $|U|$ would stay above some $c > 0$ and $I$ would grow without bound.

A periodic orbit returns to its own start. In the figure eight, found by Cris Moore in 1993 and proved to exist by Alain Chenciner and Richard Montgomery in 2000, the three chase one another round a single curve. Milovan Šuvakov and Veljko Dmitrašinović found 13 new families in this slice in 2013; four are marked on the map, their starts stored to high precision. A small deviation grows each period by a factor the orbit sets: about 1 for the figure eight and the moth, 1.31 for the dragonfly, 1.78 for the butterfly, 21 for the goggles. The goggles need no nudge: the computation’s own tiny errors grow until, after about seven periods, the orbit is lost. The nudge key changes body 1’s velocity by 0.001, which adds a little energy and angular momentum, so a nudged eight slowly turns as a whole; the record compares shapes with that turn taken out.

Newton said the Moon, pulled by both the Earth and the Sun, was the only problem that made his head ache. Henri Poincaré’s prize essay of 1890 showed why such problems resist: near some orbits the motion folds into tangles that defeat long-range prediction, the first glimpse of what is now called chaos. The bodies here are equal point masses in a plane and nothing else; they have no size, collisions and mergers are not simulated, and this is not the three suns and one planet of Liu Cixin’s novel.