Example: Free Particle

Lagrangian:

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

Momentum:

$$ p = m\dot{x} $$

Hamiltonian:

$$ H = \frac{p^2}{2m} $$

Hamilton’s equations:

$$ \dot{x} = \frac{p}{m}, \quad \dot{p} = 0 $$