This presentation is an extension of a previous one that visualized details of the figure-eight gravitational choreography. The mathematical details of the choreography are given in the previous presentation, as well as the distinction between direct and dual variables.

The aim here is to capture the overall behavior of dynamic variables of the choreography via Fourier analysis of the integrated data. The Fourier coefficients for each dynamic variable are evaluated and generally rounded to five decimal places to eliminate numerical noise. Tabulations of these appear above a graph of the variable for immediate comparison. Approximations of each variable are also available for visualization.

When the values of any circular function at three points equally spaced in the period are added together, the result is identically zero except when the index of the function is divisible by three. Used in conjunction with constraints on the dynamic variables, this places conditions on terms in the Fourier series with indices divisible by three. Existing constraints on center of mass and angular momentum require that such terms do not appear in the Fourier series for variables involved, as noted below.

Data are presented first for direct variables, followed by dual variables, along with comments.

Direct coordinate variable $x$ :

Cosine coefficients for indices 3, 9, 15, etc. are identically zero from the center of mass constraint.

Direct coordinate variable $y$ :

Sine coefficients for indices 6, 12, 18, etc. are identically zero from the center of mass constraint.

Direct squared radial variable $x^2+y^2$ :

All even indices are significant for cosine coefficients: not a simple function.

Sum of direct squared radial variables rounded to six digits:

All terms here have indices divisible by six: summation of Fourier series leaves terms with indices divisible by three, but the radius has only even indices.

Direct radial variable $\sqrt{x^2+y^2}$ modified into a smooth function:

The square root is somewhat simpler in overall behavior than the square.

Sum of direct radial variables modified into a smooth function:

A relatively simple function. The unmodified function would have double the frequency, hence only indices divisible by six.

Direct coordinate angular variable ${tan}^{-1}\left(\frac{y}{x}\right)$ :

Although this looks suspiciously like an elliptic cosine, it can be fit decently with only a few circular cosines.

Sum of direct coordinate angular variables:

The zeros correspond to triangular configurations and the extrema to linear configurations. Indices divisible by three arise from summation of Fourier series, but the angle has only odd indices.

Direct velocity variable $\stackrel{·}{x}$ :

Sine coefficients for indices 3, 9, 15, etc. are identically zero from the center of mass constraint.

Direct velocity variable $\stackrel{·}{y}$ :

Cosine coefficients for indices 1, 6, 12, etc. are identically zero from the center of mass constraint. Not a simple function.

Direct speed variable $\sqrt{{\stackrel{·}{x}}^2+{\stackrel{·}{y}}^2}$ :

All even indices are significant for cosine coefficients: not a simple function.

Sum of direct speed variables:

All terms here have indices divisible by six: summation of Fourier series leaves terms with indices divisible by three, but the speed has only even indices.

Direct velocity angular variable ${tan}^{-1}\left(\frac{\stackrel{·}{y}}{\stackrel{·}{x}}\right)$ modified to a smooth function:

Again, simpler than an elliptic cosine.

Sum of direct velocity angular variables modified to a smooth function:

The zeros correspond to triangular configurations and the extrema to linear configurations. Indices divisible by three arise from summation of Fourier series, but the angle has only odd indices.

Component energy in direct variables:

Cosine coefficients for indices 6, 12, 18, etc. are identically zero from the energy constraint, while the coefficient for the index zero is exactly minus one sixth.

Component angular momentum in direct variables:

Cosine coefficients for indices 3, 9, 15, etc. are identically zero from the angular momentum constraint.

Dual coordinate variable $x$ :

Sine coefficients for indices 3, 9, 15, etc. are identically zero from the center of mass constraint.

Dual coordinate variable $y$ :

Cosine coefficients for indices 6, 12, 18, etc. are identically zero from the center of mass constraint. The same applies to the coefficient for the index zero for this function with even cosine indices: the function is equally distributed above and below the x-axis.

Dual squared radial variable $x^2+y^2$ :

Somewhat simpler than the corresponding direct variable.

Sum of dual squared radial variables:

All terms here have indices divisible by six: summation of Fourier series leaves terms with indices divisible by three, but the radius has only even indices.

Dual radial variable $\sqrt{x^2+y^2}$ :

Comparable in complexity to the squared variable, since all coefficients are small.

Sum of dual radial variables:

All terms here have indices divisible by six: summation of Fourier series leaves terms with indices divisible by three, but the radius has only even indices.

Dual coordinate angular variable ${tan}^{-1}\left(\frac{y}{x}\right)$ modified to a smooth function:

All odd indices are significant for sine coefficients: not a simple function.

Sum of dual coordinate angular variables:

The zeros correspond to triangular configurations and the extrema to linear configurations. Indices divisible by three arise from summation of Fourier series, but the angle has only odd indices.

Dual velocity variable $\stackrel{·}{x}$ :

Cosine coefficients for indices 3, 9, 15, etc. are identically zero from the center of mass constraint.

Dual velocity variable $\stackrel{·}{y}$ :

Sine coefficients for indices 6, 12, 18, etc. are identically zero from the center of mass constraint.

Dual speed variable $\sqrt{{\stackrel{·}{x}}^2+{\stackrel{·}{y}}^2}$ modified to a smooth function:

All odd indices are significant for cosine coefficients: not a simple function.

Sum of dual speed variables modified to a smooth function:

Another simple function, with the coefficient for the index six oddly suppressed and so not included in the approximation. The unmodified function would have double the frequency, hence only indices divisible by six.

Dual velocity angular variable ${tan}^{-1}\left(\frac{\stackrel{·}{y}}{\stackrel{·}{x}}\right)$ :

Relatively simple function.

Sum of dual velocity angular variables:

The zeros correspond to triangular configurations and the extrema to linear configurations. Indices divisible by three arise from summation of Fourier series, but the angle has only odd indices.

Component energy in dual variables:

Cosine coefficients for indices 6, 12, 18, etc. are identically zero from the energy constraint, while the coefficient for the index zero is exactly minus one sixth.

Component angular momentum in dual variables:

Cosine coefficients for indices 3, 9, 15, etc. are identically zero from the angular momentum constraint.

Inverse cubed dual radial variable $\frac{1}{r^3}$ , rounded to four digits to remove obvious numerical noise:

The necessity for many terms in the approximation is of significance.

The function $X$ in dual variables:

The function $Y$ in dual variables:

The functions X and Y have Fourier indices divisible by three by construction.

Right-hand side of the differential equation to test for simplicity:

The second derivative of individual dual radial variables is not particularly simple, implying that neither are the dual radial variables themselves.

Uploaded 2018.12.27 — Updated 2019.01.29 analyticphysics.com