A Jacobian Elliptic Functions as Inverse Elliptic Functions

[bamajon1974]

I need help in understanding how Jacobian Elliptic Functions are interpreted as inverses of Elliptic Functions.

Please reference the wiki page on Jacobian Elliptic functions:

https://en.wikipedia.org/wiki/Jacobi_elliptic_functions

For example, if $$u=u(φ,m)$$ is defined as $$u(φ,m) = \int_0^φ \frac{1}{ \sqrt{1-m sin^2(θ)}} \, dθ$$

where $φ$ is the amplitude $$φ=am(u)$$

Then $$y'=sin(φ)$$ is defined to be the Jacobian function $$sn(u,m)$$ or $$sin(φ)=sn(u,m)$$

I understand that the Jacobian functions are defined as inverses to the elliptic integrals. Bug where does the $sin(φ)$ come from? I think I am getting getting lost with all of the different variables. Can someone explain in as clearly as possible? Any insight would be appreciated. Thanks!

 

[aheight]

The elliptic functions are inverses of the elliptic integrals. Taking the sine of the inverse of the integral is just a convenience. If you're interested, I suggest you experiment with the functions EllipticF and JacobiSN in Mathematica. . If we have the differential equation

$$y''=g/L\sin(y)$$, then we can solve it exactly using elliptic functions. The solution is:

$$y(t)=2\arcsin\left\{k\;\text{sn}\left[\sqrt{g/l} (t-t_0),m\right]\right\}$$

It's a beautiful derivation and I invite anyone interested in non-linear DEs to review the derivation. It can be found in this text which also has a nice chapter on elliptic functions and integrals:

https://books.google.com/books/about/Introduction_to_Nonlinear_Differential_a.html?id=RgbWowrjKd4C