Plane pendulum in terms of the x axis

[oswaler]

Homework Statement

consider the free motion of a plane pendulum whose amplitude is not small. Show that the horizontal component of the motion may be represented by the approximate expression ( components through the 3rd order are included)

$$\ddot{x}$$+w$$^{2}_{0}$$ (1+x$$^{2}_{0}$$/L$$^{2}$$)x - Ex$$^{3}$$=0

where w$$^{2}_{0}$$=g/L , E=3g/(2L$$^{3}$$) and L is the length of the suspension.

Homework Equations

we have a second order non-linear equation: $$\ddot{\Theta}$$+w$$^{2}_{0}$$sin$$^{2}$$$$\Theta$$=0

The Attempt at a Solution

I'm pretty stuck on this. It seems that I need to get a function for x in terms of $$\Theta$$, but I'm not quite sure where to start. I see here that I'm not going to be able to use the small angle approximation sin$$\Theta$$=$$\Theta$$.

Thanks for any help.

 

[Shooting Star]

x = L(sin theta) => theta = arcsin x/L.

Try to find theta'' in terms of x'', and at some point sub in the value of theta'' from the last eqn, so as to get w0^2.