(i) Explain the concept of a canonical transformation from coordinates $\left(q^{a}, p^{a}\right)$ to $\left(Q^{a}, P^{a}\right)$. Derive the transformations corresponding to generating functions $F_{1}\left(t, q^{a}, Q^{a}\right)$ and $F_{2}\left(t, q^{a}, P^{a}\right)$.

(ii) A particle moving in an electromagnetic field is described by the Lagrangian

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

where $c$ is constant

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

(b) Show that $\mathbf{E}$ and $\mathbf{B}$ are invariant under the gauge transformation

$$\mathbf{A} \rightarrow \mathbf{A}+\nabla \Lambda, \quad \phi \rightarrow \phi-\frac{1}{c} \frac{\partial \Lambda}{\partial t}$$

for $\operatorname{arbitrary} \Lambda(t, \mathbf{x})$.

(c) Construct the Hamiltonian. Find the generating function $F_{2}$ for the canonical transformation which implements the gauge transformation (1).