Example: Simple Harmonic Oscillator

Kinetic energy:

$$ T = \frac{1}{2}m\dot{x}^2 $$

Potential energy:

$$ U = \frac{1}{2}kx^2 $$

Lagrangian:

$$ L = \frac{1}{2}m\dot{x}^2 - \frac{1}{2}kx^2 $$

Applying Euler-Lagrange:

$$ m\ddot{x} + kx = 0 $$