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<br /> <br /> 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<br /> a) Make the small angle approximation<br /> b) Make a better approximation with a taylor series, and then solve the DE[/itex]

 

[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!