The Lagrangian for a particle of mass $m$ and charge $e$ moving in a magnetic field with position vector $\mathbf{r}=(x, y, z)$ is given by

$$L=\frac{1}{2} m \dot{\mathbf{r}}^{2}+e \frac{\dot{\mathbf{r}} \cdot \mathbf{A}}{c}$$

where the vector potential $\mathbf{A}(\mathbf{r})$, which does not depend on time explicitly, is related to the magnetic field $\mathbf{B}$ through

$$\mathbf{B}=\nabla \times \mathbf{A}$$

Write down Lagrange's equations and use them to show that the equation of motion of the particle can be written in the form

$$m \ddot{\mathbf{r}}=e \frac{\dot{\mathbf{r}} \times \mathbf{B}}{c}$$

Deduce that the kinetic energy, $T$, is constant.

When the magnetic field is of the form $\mathbf{B}=(0,0, d F / d x)$ for some specified function $F(x)$, show further that

$$\dot{x}^{2}=\frac{2 T}{m}-\frac{(e F(x)+C)^{2}}{m^{2} c^{2}}+D$$

where $C$ and $D$ are constants.