Example: Free Particle

Kinetic energy:

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

Potential energy:

$$ U = 0 $$

Lagrangian:

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

Applying Euler-Lagrange:

$$ \frac{d}{dt}(m\dot{x}) = 0 $$

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

This gives uniform motion.