Hamiltonian of a charged particle in an electromagnetic field

Solution

Past exam of the mathematics course of the University of Cambridge / 2023 / ii / Paper 1 / 8D / a / Solution

For electromagnetic scalar and vector potentials $\phi$ and $A$,
$$
L=\frac12m\dot r^2-q\phi+q\dot r\cdot A.
$$
With
$$
E=-\nabla\phi-\partial_tA,
\qquad
B=\nabla\times A,
$$
the Euler--Lagrange equations give the Lorentz force $m\ddot r=q(E+\dot r\times B)$, while the canonical momentum is $p=m\dot r+qA$.


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