Pendulum with larger amplitudes

[hrlaust]

Homework Statement

I really would like some help. Next month I am starting a project with the title "pendulum with larger amplitudes", where I have to come up with a solution on how to solve the equation for the pendulum with large amplitude.

Homework Equations

This is the equation I have to come up with, but I have no idea how to get this.
http://img88.imageshack.us/img88/3345/latex2png896efbxa5.png
I know how to get the equation for the pendulum with small amplitudes, its just the rest, that kills me.

The Attempt at a Solution

I have searched the internet and this forum for hours now. And the only information I am able to find is the final equation and theory about the pendulum with small amplitudes.

I would be really happy if some of you guys are able to help me or give me some links with the theory behind the equation.

Thanks, Jonas

 

[HallsofIvy]

There is no "closed form" solution to the "large angle" pendulum problem and, solutions are NOT necessarily periodic- it is possible to give the pendulum an initial speed so that it "goes over the top" and just continues around and around- so your "period" equation couldn't hold for that. One thing you can do is use "quadrature" on the non-linear pendulum problem. Let [itex]\omega= d\theta /dt[/itex]. Then [itex]d^2\theta/dt^2= d\omega /dt= d\omega/d\theta d\theta /dt= \omega d\omega /d\theta[/itex]
The equation of motion of the pendulum becomes
[tex]\frac{d^2\theta}{dt^2}=\omega \frac{d\omega}{d\theta)= \frac{g}{l} sin(\theta)[/itex]
a relatively simple separable differential equation. Integrating you get
[tex]\frac{1}{2}\omega^2= -\frac{g}{l} cos(\theta)+ C[/tex]
Solving for [itex]\omega= d\theta /dx[/itex] gives a rather complicated root involving [itex]cos(\theta)[/itex] which cannot be integrated in closed form- it is, in fact, an "elliptic integral". If, instead, you were to graph [itex]\frac{1}{2}\omega^2= -\frac{g}{l} cos(\theta)+ C[/itex] in the [itex]\theta-\omega[/itex] plane (the "phase plane") you will see that, for sufficiently low starting speeds, the graphs are ovals around the points (0,0), ([itex]\pi[/itex],0), etc. The period will be related to the distances around those ovals.