(i) A particle of mass $m$ and charge $q$, at position $\mathbf{x}$, moves in an electromagnetic field with scalar potential $\phi(\mathbf{x}, t)$ and vector potential $\mathbf{A}(\mathbf{x}, t)$. Verify that the Lagrangian

$$L=\frac{1}{2} m \dot{\mathbf{x}}^{2}-q(\phi-\dot{\mathbf{x}} \cdot \mathbf{A})$$

gives the correct equations of motion.

[Note that $\mathbf{E}=-\nabla \phi-\dot{\mathbf{A}}$ and $\mathbf{B}=\boldsymbol{\nabla} \times \mathbf{A}$.]

(ii) Consider the case of a constant uniform magnetic field, with $\mathbf{E}=\mathbf{0}$, given by $\phi=0$, $\mathbf{A}=(0, x B, 0)$, where $(x, y, z)$ are Cartesian coordinates and $B$ is a constant. Find the motion of the particle, and describe it carefully.