Converting between Lagrangian and Hamiltonian mechanics

We can use the Legendre Transform to convert between Lagrangian Mechanics and Hamiltonian mechanics. Since position and momentum are conjugate variables, we can convert the Lagrangian $\mathcal{L}(q,\dot{q},t)$ and the Hamiltonian $\mathcal{H}(q,p,t)$.

A detailed explanation on this conversion can be found on NYU’s lecture notes for mechanics, lecture 6.