Brief layout of strategy:

1. Write down Lorentz force ${\bf F}$.

2. Find (velocity-dependent) potential $U$ for ${\bf F}$.

3. Write down Lagrangian $L$.

4. Perform Legendre transformation ${\bf v}\to {\bf p}$.

5. Write down Hamiltonian $H$.

Brief layout of strategy:

1. Write down Lorentz force ${\bf F}$.

2. Find (velocity-dependent) potential $U$ for ${\bf F}$.

3. Write down Lagrangian $L=T-U$.

4. Define the canonical (as opposed to kinetic/mechanical) momentum ${\bf p}:=\frac{\partial L}{\partial{\bf v}}$.

5. Perform Legendre transformation ${\bf v}\to {\bf p}$.

6. Write down Hamiltonian $H={\bf v}\cdot{\bf p}-L$.