Example: Simple Pendulum

Generalized coordinate: $\theta$

Kinetic energy:

$$ T = \frac{1}{2}ml^2\dot{\theta}^2 $$

Potential energy:

$$ U = mgl(1 - \cos\theta) $$

Lagrangian:

$$ L = \frac{1}{2}ml^2\dot{\theta}^2 - mgl(1 - \cos\theta) $$

Euler-Lagrange equation:

$$ ml^2\ddot{\theta} + mgl \sin\theta = 0 $$