Example: Simple Harmonic Oscillator

Lagrangian:

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

Momentum:

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

Hamiltonian:

$$ H = \frac{p^2}{2m} + \frac{1}{2}kx^2 $$

Hamilton’s equations:

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

$$ \dot{p} = -kx $$