Full solution for the simple pendulum

[tataratat]

Having recently completed a session on the simple pendulum in physics, I was curious as to what the solution to θ''+(g/l)sin(θ)=0 for θ(t) was sans the sin(θ)=θ simplification.

 

[voko]

What have you done to satisfy your curiosity?

 

[sweet springs]

Elliptic integral is the key word.

 

[tataratat]

voko, prior to posting I had essentially spent about an hour or so looking through various articles, lecture notes,et c., and hadn't run into anything involving a full solution of that ODE. I did check Wolfram Alpha which pointed me in the direction of the Jacobi elliptic functions, however that did not readily lend itself to understanding the derivation, or finding a numerical solution. I looked under functions.wolfram.com for the Jacobi Amplitude function, and wasn't able to find what I was looking for

 

[voko]

The full solution of that ODE is not expressible in elementary functions. Have a look here: http://en.wikipedia.org/wiki/Pendulum_(mathematics )

 

[sweet springs]

θ''+(g/l)sin(θ)=0
Multiply $$\theta^.$$ to make $${{\theta^.}^2}^.$$, integrate and then take square root to get $$\frac{dt}{d\theta}$$

 

[voko]

θ''+(g/l)sin(θ)=0
Multiply $$\theta^.$$ to make $${{\theta^.}^2}^.$$, integrate and then take square root to get $$\frac{dt}{d\theta}$$

Or just start with conservation of energy and get the first-order ODE directly.