Derivation of Euler-Lagrange Equation

We vary the action:

$$ \delta S = \delta \int L \, dt = 0 $$

Expanding:

$$ \delta S = \int \left( \frac{\partial L}{\partial q} \delta q + \frac{\partial L}{\partial \dot{q}} \delta \dot{q} \right) dt $$

Using integration by parts:

$$ \int \frac{\partial L}{\partial \dot{q}} \delta \dot{q} \, dt = \left[ \frac{\partial L}{\partial \dot{q}} \delta q \right] - \int \frac{d}{dt}\left(\frac{\partial L}{\partial \dot{q}}\right)\delta q \, dt $$

Assuming $\delta q = 0$ at endpoints:

$$ \delta S = \int \left[ \frac{\partial L}{\partial q} - \frac{d}{dt}\left(\frac{\partial L}{\partial \dot{q}}\right) \right] \delta q \, dt $$

For arbitrary $\delta q$:

$$ \frac{d}{dt}\left(\frac{\partial L}{\partial \dot{q}}\right) - \frac{\partial L}{\partial q} = 0 $$