= Solution
The <canonical momentum> is $p_i=\partial L/\partial\dot x_i=m\dot x_i+qA_i$. Its time derivative contains the total derivative along the trajectory: $\dot p_i=m\ddot x_i+q\partial_tA_i+q\dot x_j\partial_jA_i$. The <Euler-Lagrange equation> therefore gives
$$
m\ddot x_i=q\left[-\partial_i\phi-\partial_tA_i+
\dot x_j(\partial_iA_j-\partial_jA_i)\right].
$$
The <magnetic field> is $\mathbf B=\nabla\times\mathbf A$, so the last term is $(\dot{\mathbf x}\times\mathbf B)_i$. With the <electric field> $\mathbf E=-\nabla\phi-\partial_t\mathbf A$, this is precisely the <Lorentz force law>:
$$
\boxed{m\ddot{\mathbf x}=q(\mathbf E+\dot{\mathbf x}\times\mathbf B).}
$$
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