Past exam of the mathematics course of the University of Cambridge / 2019 / ii / Paper 2 / 14E / b / Solution

The canonical momentum is $\mathbf{p}=m\dot{\mathbf{r}}+q\mathbf{A}$, and the Legendre transform givesIts Hamilton's equations areand, componentwise,Combining the second equation with $\mathbf{p}=m\dot{\mathbf{r}}+q\mathbf{A}$ recovers the Lorentz force law.