The motion of a particle of charge $q$ and mass $m$ in an electromagnetic field with scalar potential $\phi(\mathbf{r}, t)$ and vector potential $\mathbf{A}(\mathbf{r}, t)$ is characterized by the Lagrangian

$$L=\frac{m \dot{\mathbf{r}}^{2}}{2}-q(\phi-\dot{\mathbf{r}} \cdot \mathbf{A})$$

(a) Show that the Euler-Lagrange equation is invariant under the gauge transformation

$$\phi \rightarrow \phi-\frac{\partial \Lambda}{\partial t}, \quad \mathbf{A} \rightarrow \mathbf{A}+\nabla \Lambda$$

for an arbitrary function $\Lambda(\mathbf{r}, t)$.

(b) Derive the equations of motion in terms of the electric and magnetic fields $\mathbf{E}(\mathbf{r}, t)$ and $\mathbf{B}(\mathbf{r}, t)$.

[Recall that $\mathbf{B}=\nabla \times \mathbf{A}$ and $\mathbf{E}=-\nabla \phi-\frac{\partial \mathbf{A}}{\partial t}$.]