Converting a Lagrangian to a Hamiltonian

[MyoPhilosopher]

Homework Statement

Convert to Hamiltonian

Relevant Equations

$$L(\theta,\dot{\theta},\phi,\dot{\phi}) = \frac12ml^2((\dot{\theta})^2 + (sin(\theta)^2)\dot{\phi}^2) + k\theta^4$$

Given the following
$$L(\theta,\dot{\theta},\phi,\dot{\phi}) = \frac12ml^2((\dot{\theta})^2 + (sin(\theta)^2)\dot{\phi}^2) + k\theta^4$$

This is my attempt:
I am not understanding if the conserved quantities (like angular momentum about the z-axis) impacts my formulation of the Hamiltonian or is it irrelevant for the transformation. (EDIT: my last term below should be negative)

 

[wrobel]

Irrelevant

 

[TSny]

As @wrobel says, the fact that $p_{\phi}$ is conserved does not affect the formulation of $H$. However, you should express $H$ in terms of the momenta $p_{\theta}$ and $p_{\phi}$ instead of the "velocities" $\dot \theta$ and $\dot\phi$.

 

[JD_PM]

Hint: By definition the Hamiltonian , in general, has to be a function of momentum $p$ and general coordinate $q$.

In your problem you need to get $H(p_{\phi},p_{\theta}, \phi, \theta)$