(i) The trajectory $\mathbf{x}(t)$ of a non-relativistic particle of mass $m$ and charge $q$ moving in an electromagnetic field obeys the Lorentz equation

$$m \ddot{\mathbf{x}}=q\left(\mathbf{E}+\frac{\dot{\mathbf{x}}}{c} \wedge \mathbf{B}\right) .$$

Show that this equation follows from the Lagrangian

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

where $\phi(\mathbf{x}, t)$ is the electromagnetic scalar potential and $\mathbf{A}(\mathbf{x}, t)$ the vector potential, so that

$$\mathbf{E}=-\frac{1}{c} \dot{\mathbf{A}}-\nabla \phi \text { and } \mathbf{B}=\nabla \wedge \mathbf{A}$$

(ii) Let $\mathbf{E}=0$. Consider a particle moving in a constant magnetic field which points in the $z$ direction. Show that the particle moves in a helix about an axis pointing in the $z$ direction. Evaluate the radius of the helix.