For any pairwise potential interaction, consider masses placed at the vertices of a regular n-gon. Curiously enough there is a frequency of rotation that always leads to a solution to the equations of motion. The gravitational three-body version was first solved by Lagrange, and the appellation “choreography” indicates that the masses follow one another equally spaced along the orbit.

The Lagrangian for a planar n-body problem is

$L=\frac{m}{2}\sum _i({\stackrel{·}{x}}_i^2+{\stackrel{·}{y}}_i^2)-\sum _{\mathrm{pairs}}f\left(r_{ij}\right)r_{ij}=\sqrt{(x_i-x_j{)}^2+(y_i-y_j{)}^2}$

with the corresponding equations of motion

$m{\stackrel{··}{x}}_i=-\sum _{j\ne i}\frac{x_i-x_j}{r_{ij}}{f}^{\prime }\left(r_{ij}\right)m{\stackrel{··}{y}}_i=-\sum _{j\ne i}\frac{y_i-y_j}{r_{ij}}{f}^{\prime }\left(r_{ij}\right)$

A rotating regular n-gon of radius a has vertices at the coordinates

$$ \begin{array}{l}{x}_i=acos(\omega t+{\phi }_i)\\ y_i=asin(\omega t+{\phi }_i)\end{array}{\phi }_i=\frac{2\pi }{n}i,i=1,\dots ,n$$

and sides of length $l=2asin\left(\frac{\pi }{n}\right)$ . The squares of the separations between vertices are easily evaluated from the coordinates of the vertices,

$r_{ij}^2=2a[1-cos({\phi }_i-{\phi }_j\left)\right]$

and are constant as expected. Inserting the coordinates of the vertices into the equations of motion and equating coefficients leads to the two apparently different conditions

$\begin{array}{l}m{\omega }^{2}cos{\phi }_i=\sum _{j\ne i}\frac{cos{\phi }_i-cos{\phi }_j}{r_{ij}}{f}^{\prime }\left(r_{ij}\right)\\ m{\omega }^{2}sin{\phi }_i=\sum _{j\ne i}\frac{sin{\phi }_i-sin{\phi }_j}{r_{ij}}{f}^{\prime }\left(r_{ij}\right)\end{array}$

which are seen to be equivalent when all angles are shifted by $\frac{\pi }{2}$ . Taking $i=n$ in the first form for convenience, the square of the frequency of rotation that produces a solution to the equations of motion is

${\omega }^2=\frac{1}{2ma}\sum _{j=1}^{n-1}{r}_{ij}{f}^{\prime }\left(r_{ij}\right)$

Given the symmetry of the vertices, this can also be written

${\omega }^2=\frac{1}{ma}\sum _{j=1}^{int\left(\frac{n-1}{2}\right)}{r}_{ij}{f}^{\prime }\left(r_{ij}\right)+{\delta }_{\left(n\mathrm{even}\right)}\frac{f^{\prime }\left(2a\right)}{m}$

And now the fun begins! Consider an attractive power potential with arbitrary exponent

$f\left(r\right)=(sgnp)k{r}^p$

where the signum function ensures the potential is attractive for negative exponents. The square of the frequency that produces a solution to the equations of motion is

${\omega }^2=(sgnp)\frac{kp}{2ma}\sum _{j=1}^{n-1}{r}_{ij}^p$

Note that the signum function keeps the frequency real. Let $k=m=1$ for convenience, since the dependence on these variables is trivial. The frequency as a function of remaining parameters behaves like this:

The behavior is strikingly similar over a wide range of powers. Interestingly, for positive exponents greater than about five, the frequency for two bodies surpasses that for three bodies. The three-body solution is in this context more stable in some sense in this region than the two-body solution.

As the exponent approaches minus two, the frequency appears to approach a straight line. For exponents less than this the arrangement of frequencies curiously changes from convex to concave. It should be noted that exponents in this regime correspond to unstable circular orbits. Presumably that instability carries over to these choreographies.

Another way to look at the behavior is as a function of the exponent:

Whether the frequency curve increases more rapidly for positive or negative exponents depends on whether the radius of the n-gon is greater or less than about one, along with significant contribution from the number of bodies.

Keep in mind that the interaction coefficient k was set equal to one merely for convenience. It is still a free parameter in the system, along with the mass, both of which can be adjusted to make the evolution of the system occur more or less rapidly.

Consideration of additional potential forms is left for a future possible addition. The existence of a general solution is clear enough, and that is the main point of this presentation.

As pointed out in a previous presentation, since the distances between bodies are constant the potential for this system is identically constant regardless of the particular potential chosen. In this system the centrifugal tendencies of the bodies balance the gravitational attractions for uniform circular motion in a constant potential.

Uploaded 2022.09.10 — Updated 2022.09.12 analyticphysics.com