The gravitational three-body problem has some very special particular solutions in its choreographies. These are periodic solutions with the masses evenly spaced on a single orbit. Their existence arises from the highly nonlinear nature of the three-body system. In a certain sense they represent a kind of ‘quantization’ of this classical system. Understanding how they are possible could give insight into how quantum mechanics itself is possible in the context of a fully analytic theory.

The equations of motion for the three-body problem are simplest and most symmetric in coordinates that represent the Cartesian differences between the locations of the masses. Defining the difference variables

$\begin{array}{l}{x}_{12}={x}_{1}-{x}_{2}\\ {x}_{23}={x}_{2}-{x}_{3}\\ {x}_{31}={x}_{3}-{x}_{1}\end{array}\phantom{\rule{4em}{0ex}}\begin{array}{l}{y}_{12}={y}_{1}-{y}_{2}\\ {y}_{23}={y}_{2}-{y}_{3}\\ {y}_{31}={y}_{3}-{y}_{1}\end{array}\phantom{\rule{4em}{0ex}}\begin{array}{l}{r}_{12}^{2}={x}_{12}^{2}+{y}_{12}^{2}\\ {r}_{23}^{2}={x}_{23}^{2}+{y}_{23}^{2}\\ {r}_{31}^{2}={x}_{31}^{2}+{y}_{31}^{2}\end{array}$

the two-dimensional equations of motion in differences are

$\begin{array}{l}m{\stackrel{\xb7\xb7}{x}}_{12}=-3k\frac{{x}_{12}}{{r}_{12}^{3}}+X\\ m{\stackrel{\xb7\xb7}{x}}_{23}=-3k\frac{{x}_{23}}{{r}_{23}^{3}}+X\\ m{\stackrel{\xb7\xb7}{x}}_{31}=-3k\frac{{x}_{31}}{{r}_{31}^{3}}+X\end{array}\phantom{\rule{5em}{0ex}}\begin{array}{l}m{\stackrel{\xb7\xb7}{y}}_{12}=-3k\frac{{y}_{12}}{{r}_{12}^{3}}+Y\\ m{\stackrel{\xb7\xb7}{y}}_{23}=-3k\frac{{y}_{23}}{{r}_{23}^{3}}+Y\\ m{\stackrel{\xb7\xb7}{y}}_{31}=-3k\frac{{y}_{31}}{{r}_{31}^{3}}+Y\end{array}$

where the additional terms, identical for triples of equations, are

$X=k(\frac{{x}_{12}}{{r}_{12}^{3}}+\frac{{x}_{23}}{{r}_{23}^{3}}+\frac{{x}_{31}}{{r}_{31}^{3}})\phantom{\rule{7em}{0ex}}Y=k(\frac{{y}_{12}}{{r}_{12}^{3}}+\frac{{y}_{23}}{{r}_{23}^{3}}+\frac{{y}_{31}}{{r}_{31}^{3}})$

For a choreography these functions are periodic by construction with a period equal to one-third of that of the choreography. The equations of motion can be extended to higher dimensions with the only change being in making the definitions of the radial variables consistent with the spatial dimension used.

It is also worth noting that this simple symmetric form of the equations does not hold for the four-body problem or higher numbers of masses. For the three-body system each equation includes all possible terms for interactions among the masses. For the four-body problem there would be terms missing because all interactions do not enter into each equation. For example, an equation for ${x}_{12}$ in the four-body problem would not naturally have a term proportional to ${x}_{34}$ and it would have to be included manually, leading to much more complicated expressions.

For the purposes of this presentation, the focus will be restricted to three-body planar choreographies, although some extensions to three dimensions will be indicated when appropriate.

Adding triples of equations gives the constraints

${\stackrel{\xb7\xb7}{x}}_{12}+{\stackrel{\xb7\xb7}{x}}_{23}+{\stackrel{\xb7\xb7}{x}}_{31}=0\phantom{\rule{7em}{0ex}}{\stackrel{\xb7\xb7}{y}}_{12}+{\stackrel{\xb7\xb7}{y}}_{23}+{\stackrel{\xb7\xb7}{y}}_{31}=0$

which can be translated into constraints on the center of mass

$\begin{array}{r}{x}_{12}+{x}_{23}+{x}_{31}=0\\ {\stackrel{\xb7}{x}}_{12}+{\stackrel{\xb7}{x}}_{23}+{\stackrel{\xb7}{x}}_{31}=0\end{array}\phantom{\rule{7em}{0ex}}\begin{array}{r}{y}_{12}+{y}_{23}+{y}_{31}=0\\ {\stackrel{\xb7}{y}}_{12}+{\stackrel{\xb7}{y}}_{23}+{\stackrel{\xb7}{y}}_{31}=0\end{array}$

by suitable choices of the initial conditions. These constraints translate directly into a general constraint on the terms that can appear in a Fourier series for each variable.

Since the equations of motion are nonlinear, the period of the resulting solution is far from obvious. If however one scales the spatial and temporal variables simultaneously, one can set the period of the system to any desired value. Apply this to a schematic equation for the *x*-coordinate:

$\begin{array}{l}\phantom{\rule{1.3em}{0ex}}m\frac{{d}^{2}sx}{d(wt{)}^{2}}=-3k\frac{sx}{{s}^{3}{r}^{3}}+\frac{X}{{s}^{2}}\\ m\frac{{d}^{2}x}{d({\displaystyle \frac{w}{{s}^{3/2}}}t{)}^{2}}=-3k\frac{x}{{r}^{3}}+X\end{array}$

For scaling to the period of circular functions, the ratio *w* will be 2π divided by the original system period. To keep equations of motion unchanged, spatial variables must be multiplied by

$s={w}^{2/3}=(\frac{2\pi}{T}{)}^{2/3}$

Velocity variables must be multiplied by this factor divided by the ratio *w* for a multiplicative factor of
${w}^{-1/3}$ .
The cumulative effect of scaling on the energy is to divide it by the scaling factor for spatial variables, or a multiplicative factor of
${w}^{-2/3}$ .

General expansions for each Cartesian spatial variable with period scaled to that of a standard circular are

$\begin{array}{l}x\left(t\right)=\sum _{n\ge 0}{a}_{n}cosnt+\sum _{n\ge 1}{b}_{n}sinnt\\ y\left(t\right)=\sum _{n\ge 0}{c}_{n}cosnt+\sum _{n\ge 1}{d}_{n}sinnt\end{array}$

Adding three component circular functions at equally spaced times gives

$\begin{array}{r}cos\left[nt\right]+cos\left[n\right(t+\frac{2\pi}{3}\left)\right]+cos\left[n\right(t-\frac{2\pi}{3}\left)\right]=\{\phantom{\rule{.5em}{0ex}}\begin{array}{cc}0& ,\phantom{\rule{.5em}{0ex}}3\nmid n\\ 3cosnt& ,\phantom{\rule{.5em}{0ex}}3\mid n\end{array}\\ sin\left[nt\right]+sin\left[n\right(t+\frac{2\pi}{3}\left)\right]+sin\left[n\right(t-\frac{2\pi}{3}\left)\right]=\{\phantom{\rule{.5em}{0ex}}\begin{array}{cc}0& ,\phantom{\rule{.5em}{0ex}}3\nmid n\\ 3sinnt& ,\phantom{\rule{.5em}{0ex}}3\mid n\end{array}\end{array}$

For the constraints on the center of mass to hold, this implies that all Fourier coefficients with indices divisible by three must be identically zero. That immediately determines a third of the expansion coefficients for all three-body choreographies. Since this constraint arises from linear relations, it will continue to apply for higher-dimensional systems such as a three-dimensional choreography.

Knowing that Cartesian variables have no coefficients with indices divisible by three, such terms cannot occur on the left-hand sides of equations of motion. Dividing each Cartesian variable by the cube of the radial combination will reintroduce such terms on the right-hand sides of equations of motion. For the equations to hold for Fourier expansions, these terms will be cancelled by the additional functions *X* and *Y* on the right-hand sides. That is essentially the purpose of these additional functions in the equations, since they only contain Fourier expansion terms with indices divisible by three.

That means that when one wants to determine Fourier coefficients directly from the equations of motion, one can ignore these additional functions and focus on the nontrivial terms arising from the ratio of the Cartesian variables and the radial combination. Unfortunately there does not appear to be a simple way to write down the Fourier series for this ratio given the underlying expansions of Cartesian variables. One can though assume values for the expansion coefficients, numerically evaluate the expansion coefficients for the ratio and compare them to the second derivatives on the left-hand side.

With this presumably clear notation for extracting Fourier coefficients

${C}_{n}\left[x\right(t\left)\right]={a}_{n}\phantom{\rule{5em}{0ex}}{S}_{n}\left[x\right(t\left)\right]={b}_{n}$

the nontrivial expansion coefficients are thus determined by this infinite set of interlinked equations:

$\begin{array}{r}m{n}^{2}{a}_{n}-3k{C}_{n}\left[{\displaystyle \frac{x}{{r}^{3}}}\right]=0\\ m{n}^{2}{b}_{n}-3k{S}_{n}\left[{\displaystyle \frac{x}{{r}^{3}}}\right]=0\end{array}\phantom{\rule{4em}{0ex}}\begin{array}{r}m{n}^{2}{c}_{n}-3k{C}_{n}\left[{\displaystyle \frac{y}{{r}^{3}}}\right]=0\\ m{n}^{2}{d}_{n}-3k{S}_{n}\left[{\displaystyle \frac{y}{{r}^{3}}}\right]=0\end{array}$

A three-body choreography is only possible when this set of equations has a consistent solution that produces a simultaneous zero for all of them. There is a trivial solution with all coefficients zero for either Cartesian variable that should be avoided.

For given values of the expansion coefficients, the left-hand sides of these equations can be plotted graphically side by side for a range in coefficient values around the initial point. The possibility of a choreography will be indicated when all of the resulting graphs display a zero simultaneously. From each individual graph it will be clear in which direction the coefficient needs to be altered to move towards a choreography.

As a example, consider the figure eight. It is known from Fourier analysis that the *x*-coordinate can be described using sine functions with odd indices and the *y*-coordinate using cosine functions with even indices. Consider the first three nontrivial terms of each expansion:

$\begin{array}{l}x\left(t\right)={b}_{1}sint+{b}_{5}sin5t+{b}_{7}sin7t\\ y\left(t\right)={c}_{2}cos2t+{c}_{4}cos4t+{c}_{8}cos8t\end{array}$

With a mass of one third and a coupling constant of one ninth, and taking all coefficients to be positive rather, the side-by-side plots of the interlinked equations around an initial point look like this:

Adjusting the initial point, beginning with the largest coefficients first, one can literally see the approach to the choreography. A rough initial point can then be used for a numerical by clicking the button. This takes some time to evaluate and results will appear below:

These results are only good to a few significant digits, but this is to be expected for such a simple approximate expansion.

One could repeat these interactive graphics for other known choreographies but unfortunately they quickly become much more complicated, requiring six or more terms in each expansion for decently accurate results. Presenting an approach to these special solutions with twelve or more simultaneous plots becomes cumbersome and thus less clear.

Another way to see the approach to a choreography is by assuming values for the expansion coefficients and plotting the total energy and angular momentum of the system over one complete orbit. With energy in green, the same level of approximation for the figure eight gives

Both values become more constant as the choreography is approached. It is quite interesting to note that the higher-order expansion terms are critical for producing constant lines in this graphic. This makes this visualization less useful in the attempt to search for choreographies, but instructive all the same.

Given the difficulty in explicitly expanding the algebraic inverse of an expansion representing the radial variable or a power of it, one might think to try an analytic solution using the angular momentum alone since it is merely the product of two individual expansions for Cartesian coordinates and their derivatives. This unfortunately fails because of the property of choreographies noted above: adding three component circular functions at equally spaced times contains only terms with the index divisible by three. This does not provide enough equations to determine the free coefficient values. Schade!

*Uploaded 2019.03.10 — Updated 2019.03.12*
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