Finding equation of motion from lagrangian

[adjklx]

Hi, I am trying to solve this problem here:

http://img201.imageshack.us/img201/7006/springqo9.jpg

We're supposed to find the equation of motion from the lagrangian and not Newton's equations.

Attempted solution:

$$L = T - U = \frac{I\omega^2}{2} + \frac{mv^2}{2} - \frac{kr^2}{2}$$
$$I = m(r^2 + l^2)$$
$$v = \frac{dr}{dt}$$

From the euler-lagrange equation $$\frac{\partial L}{\partial r} = \frac{d}{dt}\frac{\partial L}{\partial v}$$ I get:

$$m\omega^2r - kr = m \frac{d^2r}{dt^2}$$
$$(\omega^2-\frac{k}{m})r = \frac{d^2r}{dt^2}$$

If anyone can see any mistakes i'd appreciate it if they could let me know. Thanks

 

[chrisk]

Your equation

$$(\omega^2-\frac{k}{m})r = \frac{d^2r}{dt^2}$$

appears correct. Recall that

$$\frac{k}{m} = \omega^2_{spring}$$

So, this shows the special value for

$$\omega^2$$

Particularly,

$$\omega^2=\omega^2_{spring}$$