Math · chapter 8 of 13
The ones that come back around
Some starts run a closed loop and repeat forever. Three equal weights chasing each other around a figure eight is the famous one, and it was found by looking, not by solving.
A periodic orbit is one that returns to exactly where it started, at exactly the speeds it started with, and then does the whole thing again. Two bodies always do this. Three bodies almost never do — but the almost is doing some work in that sentence.
In 1993 Cris Moore, looking for closed orbits by computer, found three equal masses that chase one another around a single figure eight. All three on the same track, evenly spaced in time, no spin in the system at all. In 2000 Alain Chenciner and Richard Montgomery proved it exists, by a route worth knowing: instead of solving the equations, they showed that the path which minimises a certain total over all possible looping paths has to be that eight. The orbit is the answer to a least-effort question.
The trick that makes such searching possible is to throw away what does not matter. Three bodies make a triangle. Ignore how big the triangle is and which way it points, and what is left is its shape — and the space of all triangle shapes is a sphere. Three points on its equator are the three two-body collisions; the poles are the two equilateral triangles; Euler's straight lines lie on the equator between the collisions. Every three-body motion, however wild, is a curve drawn on that sphere. The figure eight is a curve that crosses the equator between collisions and closes.
In 2013 Milovan Šuvakov and Veljko Dmitrašinović scanned a whole sheet of starting speeds by computer and reported thirteen new families, with names off the shapes they draw — butterfly, moth, yin-yang, goggles. Xiaoming Li and Shijun Liao and colleagues then pushed the same search much further with more computing and higher precision, into the hundreds and then the thousands of families.
The orbits on this site's Orbits page were found the same way, here, on this machine: a grid of starting speeds, run forward, looking for the ones that come back near their own start, then a solver that walks each candidate in until it closes. Each one carries the distance it closes to.
Reading it: The whole state — all the positions and all the speeds — is the same after a time T as it was at the start.
Twelve equations in the plane. The solver on this site treats them as twelve residuals in three unknowns and drives them toward zero.
Reading it: Add up the motion and the pull along the whole loop. The figure eight is the loop that makes this total as small as it can be.
The action. Chenciner and Montgomery's 2000 proof works by minimising it over loops of a fixed symmetry type, which is how a shape gets proved to exist without ever being written as a formula.
Reading it: Drop the size and the direction and all possible triangles form a sphere. Collisions sit on its equator.
The shape sphere. The reason searches of this kind are possible at all: it turns an eighteen-number problem into a curve on a two-dimensional surface.
This one runs in the browser, with JavaScript on. The words above it stand on their own.
The figure eight drawn on the shape sphere: every triangle shape is a point on this ball, size and direction thrown away. The dashed circle is every straight-line arrangement, and the three red dots on it are the three two-body collisions. The orbit threads between them and closes.
Sources
- Chenciner & Montgomery, A remarkable periodic solution of the three-body problem (Annals of Mathematics, 2000)
- Moore, Braids in classical dynamics (Physical Review Letters, 1993)
- Šuvakov & Dmitrašinović, Three classes of Newtonian three-body planar periodic orbits (PRL, 2013)
- Li & Liao, More than six hundred new families of Newtonian periodic planar collisionless three-body orbits (2017)
- Montgomery, The three-body problem and the shape sphere (American Mathematical Monthly, 2015)
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