Period of a Pendulum (VERY TOUGH differential equation)

[Izzhov]

I have recently taken an interest in the idea that all pendulums of the same length are isochronous, and am currently trying to figure out an equation for the period of a pendulum of a given length. I started out by trying to find an equation for the angular distance the pendulum travels as a function of time, so I drew some vectors, and this is what it boiled down to:
$$\ddot{ \theta} = g \cdot sin( \theta)$$
where theta is angular distance as a function of t (time).
I realize that if I can can solve this differential equation, solve the result for t and convert theta to arc length over radius length, I will have solved the problem, but I have no idea how to solve this differential equation in the first place. Can someone help?

 

[lalbatros]

Detailled discussions of this topic can be found in many textbooks on classical mechanics.
A lot is also available on the net.
Start with this: http://scienceworld.wolfram.com/physics/Pendulum.html.
Look also on wiki and there: http://tabitha.phas.ubc.ca/wiki/index.php/Hamilton's_Equations .
The conservation of energy is useful to look at.
Look also for a "phase-space" analysis of this system.
Of interrest: the stability of the trajectory near the "x-point", a starting point for studying chaotic motion.

But, of course your choice depends on your background and your own objectives.

 

[siddharth]

For a pendulum without air resistance, you can calculate the period exactly for an arbitary amplitude less than [itex]\pi /2[/tex] (after which, the pendulum free falls) from the energy equation

Your equation looks incorrect. Did you draw the FBD of the pendulum properly? Once you get it, to solve the pendulum equation, you might want to try to
a) Make the small angle approximation
b) Make a better approximation with a taylor series, and then solve the DE

 

[StatMechGuy]

You can reduce the problem (through conservation of energy) to what's called an elliptic integral. This leads you to a study of elliptic integrals, and their various limits, which is of some interest.

Another interesting problem is the so-called "inverted pendulum" where you drive its base with a frequency $$\omega$$. This leads to all manner of cool behavior, including eventually a stabile fixed point standing straight up!