[QUE/CM-04010]

Starting form the Lagrangian $$ L = {1\over 2}M\, \dot{\vec{r}}^2 + {e\over c}\,\vec{A}(\vec{r})\cdot \dot{\vec{r}} -e\phi $$ for a charged particle in external electric and magnetic fields, $\vec{E}$ and $\vec{B}$, show that the Hamiltonian is given by $$ H = {1\over 2M}\,(\vec{p} - e \vec{A})^2 + e \phi$$