Question

Find the Hamiltonian and write the canonical equations of a harmonic oscillator, if its Lagrangian is L=(x ̇ ² – ω²x²)

Answer 1

Hamiltonian in terms of Lagrangian is $H = p \cdot x - L$, where $p = \frac{\partial L}{\partial{\dot x}} = 2 \dot{x}$. $H$ is a function of $p, x$.

Hence, $H = 2 \dot{x} \cdot \dot{x} - \dot{x}^2 + \omega^2 x^2 = \dot{x}^2 + \omega^2 x^2 = \frac{p^2}{4} + \omega^2 x^2$.

Hamiltonian equations are $\dot{x} = \frac{\partial H}{\partial p}$, $\dot{p} = - \frac{\partial H}{\partial x}$.

Using Hamilton above: $\dot{x} = \frac{p}{2}$, $\dot{p} = -2 \omega^2 x$.

Substituting $p = 2 \dot{x}$ from the first equation into the second, obtain $2 \ddot{x} = - 2 \omega^2 x$, or $\ddot{x} + \omega^2 x = 0$, which is the equation of motion of harmonic oscillator.