Details

The axis of the graphs for , , and is measured in units of , the complete elliptic integral of the first kind. The functions , have period , while the function has period . The term-by-term expansion of is given [4] by

.

Approximations of the Jacobian elliptic functions , , and in parametric form follow from an arbitrary-precision solution of the plane pendulum's equations of motion [1, 3]:

,

,

,

,

where are the polar coordinates of phase space. The function gives the relation between the angle and the domain of the Jacobian elliptic functions, . The approximate phase radius and time dependence are:

,

,

where the coefficients and are computable rational numbers with closed form as yet unknown (Cf. OEIS A273506, A274130 [5]).

In the complete limit, as the approximation is exact. Choosing some finite introduces error at order . One easy way to see the error is to consider the approximate period, which is directly proportional to the dependent factor:

.

In the plots, vertical lines show convergence toward the exact period as increases; however, cumulative errors always become evident if the function domain or time chosen is large enough. Convergence depends strongly on the chosen value of .

For definitions of the Seiffert spirals see the wonderful article by Erdös [3].

References

[1] A. J. Brizard, "A Primer on Elliptic Functions with Applications in Classical Mechanics." arxiv.org/abs/0711.4064.

[2] B. Klee, "Plane Pendulum and Beyond by Phase Space Geometry." arxiv.org/abs/1605.09102.

[3] P. Erdös, "Spiraling the Earth with C. G. J. Jacobi," American Journal of Physics 68, 2000 pp. 888–895. doi:10.1119/1.1285882.

[4] M. Abramowitz and I. A. Stegun. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. people.math.sfu.ca/~cbm/aands/page_591.htm.

[5] The OEIS Foundation. The Online Encyclopedia of Integer Sequences. A273506, A273507, A274130, A274131, A274076, A274078.oeis.org.