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People make the large-amplitude pendulum out to be super-complicated,
but it's really not. It cracks me up every time I read "there is no
closed-form expression" for the period.
The question arises: In terms of what?
IF the amplitude of the swing is ±θ, you can write the period as:
2π √(L/g)
T = ------------------
agm(1, cos(θ/2))
where agm() is the "arithmetic-geometric mean". Some might complain
that the typical computer language provides cos() but doesn't provide
agm(), but the latter is ridiculously easy to calculate. Forsooth, you
can calculate agm() from scratch more easily than you can calculate
cos() from scratch.
Furthermore, even if we restrict attention to spreadsheets and such,
where you can't easily write an iterative function, you can write a
simple closed-form expression that approximates agm(1,b) when b
approaches 1 from below (as when b = cos(θ/2)). Specifically: agm(1/b)
is lower-bounded by the plain old mean (1+b)*0.5 and upper-bounded by
the geometric mean (1*b)**0.5. Either of those makes a good first
approximation for small and smallish angles. Then (!) if you combine
those, i.e. split the difference, you get an approximation that is
good to 20 ppm or better, all the way out to θ=±90°, i.e. b=√½. It's
just a few terms involving cos() and sqrt().
The alert reader will have noticed that the in previous paragraph we
carried out 1.3 iterations of the procedure that defines agm().
With 8 iterations you get 15 digits of accuracy all the way out to
θ=±179.999°. The period goes to infinity at θ=±180°, but it blows up
remarkably slowly. The period is 4% "too long" at θ=±45° and 18% "too
long" at θ=±90°.
Plot of agm() and simple approximations:
https://av8n.com/physics/img48/agm-1-b.png
High-quality reference:
https://mathworld.wolfram.com/Arithmetic-GeometricMean.html
Easier-to-read reference:
https://en.wikipedia.org/wiki/Arithmetic–geometric_mean
Application to pendulum:
https://www.johndcook.com/blog/2023/11/20/nonlinear-pendulum-period/
θ/° b agm(1,b) period 1/ub 1/lb 1/combo
0 1 1 1 1 1 1
15 0.9914449 0.9957178 1.004301 1.004296 1.004305 1.004301
30 0.9659258 0.9828891 1.017409 1.017332 1.017485 1.017409
45 0.9238795 0.9615631 1.039973 1.039566 1.040381 1.039973
60 0.8660254 0.9318084 1.073182 1.071797 1.07457 1.073182
75 0.7933533 0.8936878 1.118959 1.115229 1.122708 1.118956
90 0.7071068 0.8472131 1.180341 1.171573 1.189207 1.180324
105 0.6087614 0.7922602 1.262212 1.243192 1.281671 1.262138
120 0.5 0.7283955 1.372881 1.333333 1.414214 1.372583
135 0.3826834 0.6544727 1.527948 1.446463 1.616517 1.526769
150 0.258819 0.5674713 1.762204 1.588791 1.965631 1.757234
165 0.1305262 0.457573 2.185444 1.769088 2.767905 2.158552
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