Re: [Phys-L] physical significance of the arithmetic-geometric mean

[Scott Goelzer <s.goelzer@comcast.net>]

Thanks!
This is something I can actually use in my AP classes rather than handwaving about small angles. 

Scott



******************************************
Scott Goelzer
Physics, Engineering, & Chemistry 
Coe-Brown Northwood Academy
Northwood NH
sgoelzer@coebrown.org
******************************************

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> On Mar 25, 2025, at 8:03 PM, John Denker via Phys-l <phys-l@mail.phys-l.org> wrote:
>
> Hi --
>
> Here's how to make the connection between the arithmetic-geometric
> mean (agm) and the basic laws of motion (e.g. as applied to the
> nonlinear motion of a pendulum):
>
> 1) Suppose you want to integrate (1/r)dθ around an ellipse, given the
> semi-major axis a and the semi-minor axis b. That requires doing an
> elliptic integral, which is no fun.
>
> 2) Key idea: Now suppose you have a *sequence* of ellipses, all with
> the *same* average 1/r, and suppose the sequence rapidly converges to
> a circle. This is illustrated here:
>  https://av8n.com/physics/img48/agm-ellipses.png
>
> 3) The average 1/r for a circle is obvious. Game over, you win.
>
> 4) The only question is how to construct the ellipses. It turns out
> that the a(i) and b(i) that define the agm serve the purpose. Some
> guy named Gauss cooked this up in the early 1800s.
>
> For an outline of the proof, see slide 5 (pdf page 9) here, using
> definitions from slide 4 (pdf page 7).
>
> https://ctnt-summer.math.uconn.edu/wp-content/uploads/sites/1632/2016/02/coxctnt.pdf#page=9
>
> ===============
>
> In the previous message I hinted at a closed-form approximation for
> agm(1,b). Here it is explicitly:
>
>        agm(1,b) = 0.25 + 0.25*b + 0.5*sqrt(b)
>
> and for the pendulum we set
>        b = cos(θ/2)
>
> That's accurate within 20 ppm all the way out to θ=±90°. That's useful
> only if you're stuck on a desert island without a computer that can do
> iterations. Normally it's simpler and better just to call the function
> that evaluates agm() to full double-precision accuracy. Here again is
> the exact formula, which is easier to remember than any approximation:
>
>                2π √(L/g)
>        T = ------------------
>             agm(1, cos(θ/2))
>
> and evaluating agm() is remarkably efficient.
>
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