Elliptic Functions: Physics & Engineering Applications

[Winzer]

I did know where in the math forums to post this, hope this is right.
Do elliptic functions along with elliptic integrals have lots of applications in physics and engineering? Besides the pendulum? I guess if I wanted to solve an elliptic integral for a problem it would be help but is there any benefit from studying elliptic functions for application?

 

[coomast]

I did know where in the math forums to post this, hope this is right.
Do elliptic functions along with elliptic integrals have lots of applications in physics and engineering? Besides the pendulum? I guess if I wanted to solve an elliptic integral for a problem it would be help but is there any benefit from studying elliptic functions for application?

Hello Winzer,

Some time ago I found a book in a second hand bookstore (and bought it for 6.2€ ) on this subject. I haven't had the time to read it yet (first I want to get Lie transformations for DE's in my head), but I will give you the info in case you would like to buy it:

"The applications of Elliptic Functions" by Alfred George Greenhill, 1959

It contains various applications, ranging from the pendulum, evolving chains, central orbits, swinging body (ships), certain lines on Mercator charts, geodesics, etc. Beware that this is a very advanced textbook, the math seems extremely complicated if you are not into it.

If you do not have any knowledge on this, I would suggest to start with some simpler functions. A good book to start with could be:

"Fourier Analysis with applications to boundary value problems" by Murray R. Spiegel, 1995
This is not a book only for Fourier series and transformations, but includes a starting study for advanced functions as Bessel, Legendre, etc. I read it from cover to cover on a -practical just do it and see- basis. I learned these functions therefore by making a lot of exercises.

best regards,

Coomast

 

[Winzer]

Thanks coomast.

That book sounds really interesting. When you say complicated, just how complicated? The highest math I have right now is PDE's; I will be taking Application of Complex Variables. Just from what I have heard elliptic functions can be difficult. What is the math background required? I guess this would be difficult for me seeing that I have not yet taken a proof class like analysis, they have all been applied.

I will take a look at that other book though. Thanks

 

[Carl_Weggel]

Two of the important applications of elliptic integrals are:
1. The magnetic fields, forces, and inductances generated by current loops (circles) and solenoids (circular tubes).
2. The electric fields, forces, and capacitances generated by current loops (circles) and solenoids (circular tubes).
Carl_Weggel@Juno.com

 

[Mute]

Elliptic functions are heavily used in Baxter's "Exactly solved models in Statistical Mechanics" to solve many of the 2d lattice models. For example, the (zero-field) 2d Ising model.

 

[jasonRF]

Elliptic functions also arise in conformal mapping problems with polygons (Schwartz-Christoffel).

jason