Re: [Phys-L] Jacobi elliptic functions sn cn dn ... was: pendulum

[John Denker <jsd@av8n.com>]

On 09/24/2014 07:34 AM, Bill Nettles wrote:

> I have my cal-based physics students measure some large angle 
> periods.

That's good.  Now they know why clock builders go
to so much trouble to regulate the amplitude.....



Here's an elegant, informative, highly-readable 
reference for the math:
  W. Schwalm
  "Elliptic Functions sn, cn, dn, as Trigonometry"
  http://www.und.edu/instruct/schwalm/MAA_Presentation_10-02/handout.pdf

I heartily agree when it says:
    The approach ought to be in some classic text, 
    but I have not found it.

This is how it was explained to me, back when I was
a sorcerer's apprentice.  It uses a lot of good 
physical / geometrical insight.  It uses plain old 
trigonometry as a springboard to help understand 
sn cn dn.  This reinforces and deepens the 
understanding of trig ... as well as introducing
new ideas.

This is IMHO a fine example of good pedagogy.  It 
upholds the principle:
  "Learning proceeds from the known to the unknown."

After reading this you can read the more-common
less-intuitive presentations and get more out 
of them.

One additional word about equations {9 10 11}.
Note the pattern here:  there are three functions 
such that the product of any two is the derivative 
of the third (with some scaling of the amplitudes).
If you ever see this pattern in a problem, you 
should look for a solution in terms of sn cn dn. 

It is also nice to see to the /hyperbolic/ Jacobi 
functions hinted at on the bottom of the last page.
Given the importance of sinh and cosh (in spacetime 
trigonometry among other things) and the trigonometric 
interpretation of sn cn dn you presumably suspected 
that the two ideas could be combined.....

      sin/cos            sinh/cosh

      sn/cn/dn           snh/cnh/dnh

====

There's a typo on the second page:
  <-   k = 1 should give ordinary trigonometry
  ->   k = 0 should give ordinary trigonometry

===========================

Another reference, with lots of pretty pictures and
some mathematical insight in the complex plane:
  Hans Lundmark
  "Elliptic functions"
  http://www.mai.liu.se/~halun/complex/elliptic/

===============================================

To find out about applications of sn cn dn *other*
than the simple pendulum, try something like
  https://www.google.com/search?q=jacobi+elliptic+functions+wave+-pendulum
  https://www.google.com/search?q=jacobi+elliptic+functions+soliton+-pendulum
  https://www.google.com/search?q=jacobi+elliptic+functions+orbit+-pendulum
  
https://www.google.com/search?q=jacobi+elliptic+functions+%22mass%22+motion+-pendulum