Classical Action for Harmonic Oscillator

[DeclanTKatt]

Homework Statement

Hello. I am attempting to evaluate the classical action of a harmonic oscillator by using the Euler-Lagrange equations.

Homework Equations

The Lagrangian for such an oscillator is

$$ L=(1/2)m(\dot{x}^2-\omega^2 x^2) $$

This is easy enough to solve for. The classical action is defined by $$ S_{cl} = \int L dt$$

The Attempt at a Solution

I know what the answer is, but I am having difficulty achieving it. So far I have used:
$$x=\sin (\omega t) $$
$$\dot{x}=\omega \cos(\omega t)$$

Substituted these into the Lagrangian and then integrated, with respect to t, for the classical action. This did not provide the proper results.

Any suggestions would be greatly appreciated. Thanks

 

[malawi_glenn]

So far I have used:
x=sin⁡(ωt)
x˙=ωcos⁡(ωt)

That is not the most general form of x(t) and the velocity.

Just try to evaulate this integral $\displaystyle \dfrac{m}{2} \int (\dot x{}^2 -\omega^2 x^2) \mathrm{d} t$

Hint: calculate this first $\displaystyle \int \dot x{}^2 \mathrm{d} t$ using integration by parts.
After that, you could figure out a way how to go further, hint number 2: what is the relation between $\ddot x$ and $x$ for an HO?