This short presentation has a single purpose: to evaluate the Fourier expansion coefficients for the dual and direct Cartesian coordinate variables for all of Simó’s planar three-body choreographies. The description of these variables is given in this previous presentation. The purpose of these evaluations is to allow a quick overview of the complexity of each choreography in Fourier space.

Dual coordinate variable $x$ :

Dual coordinate variable $y$ :

Direct coordinate variable $x$ :

Direct coordinate variable $y$ :

Integration of the orbits in the browser is known to have problems for a fraction of the choreographies for which the coefficients may not be as accurate as desired. Having two options for the calculation acts as a check of the values, since the scaled values must necessarily be equal to the unscaled values times the scaling factor.

Choreographies of the form

$x\left(t\right)=\sum _{3\nmid n}{b}_{n}sinnwty\left(t\right)=\sum _{3\nmid n}{c}_{n}cosnwt$

will satisfy the simultaneous equations with numerically extracted Fourier coefficients

$m{n}^2{w}^2{b}_n-3k{S}_n\left[\frac{x}{r^3}\right]=0m{n}^2{w}^2{c}_n-3k{C}_n\left[\frac{y}{r^3}\right]=0$

The graphic below can be updated with the current choreography in dual coordinates to show the behavior of the first four equations for each expansion around the initial point as described here.

The purpose of this graphic is to investigate whether or not the existence of the choreography can be inferred from the behavior of the first few coefficients.

Uploaded 2019.03.10 analyticphysics.com