This has been bothering me for a while now, and it keeps giving me trouble - how does one explain the Jacobi amplitude, and how does one calculate it?
I've tried just about everything there is to wrap my head around it, but it still seems to confuse me.
I've seen am(u|k), and have tried to calculate u, but I end up running in circles, as calculating u requires that I find phi (aka the Jacobi Amplitude) first.
I don't have MATLAB available to me, but I do have Wolfram|Alpha and Desmos at the ready.
$\endgroup$1 Answer
$\begingroup$Some relations:
\begin{align*} u &= F(\phi| m) \\ &= \int_{0}^{\phi} \frac{d\theta}{\sqrt{1-m\sin^2 \theta}} \\ &= \int_{0}^{\sin \phi} \frac{dx}{\sqrt{1-mx^2}} \\ &= \int_{0}^{\operatorname{sn} u} \frac{dt}{\sqrt{(1-t^2)(1-mt^2)}} \\ x &= \sin \phi \\ &= \operatorname{sn} (u|m) \\ &= \sin [F^{-1}(u|m)] \\ \operatorname{sn}^{-1} (x|m) &= F(\sin^{-1} x|m) \\ \operatorname{am} (u|m) &= \phi \\ &= F^{-1}(u|m) \\ \operatorname{sn} (u|m) &= \sin \phi \\ \end{align*}
Note that Mathematica or WolframAlpha uses $m=k^2$.
$\endgroup$ 2
