(a) Explain how the Hamiltonian $H(\mathbf{q}, \mathbf{p}, t)$ of a system can be obtained from its Lagrangian $L(\mathbf{q}, \dot{\mathbf{q}}, t)$. Deduce that the action can be written as

$$S=\int(\mathbf{p} \cdot d \mathbf{q}-H d t)$$

Show that Hamilton's equations are obtained if the action, computed between fixed initial and final configurations $\mathbf{q}\left(t_{1}\right)$ and $\mathbf{q}\left(t_{2}\right)$, is minimized with respect to independent variations of $\mathbf{q}$ and $\mathbf{p}$.

(b) Let $(\mathbf{Q}, \mathbf{P})$ be a new set of coordinates on the same phase space. If the old and new coordinates are related by a type-2 generating function $F_{2}(\mathbf{q}, \mathbf{P}, t)$ such that

$$\mathbf{p}=\frac{\partial F_{2}}{\partial \mathbf{q}}, \quad \mathbf{Q}=\frac{\partial F_{2}}{\partial \mathbf{P}}$$

deduce that the canonical form of Hamilton's equations applies in the new coordinates, but with a new Hamiltonian given by

$$K=H+\frac{\partial F_{2}}{\partial t}$$

(c) For each of the Hamiltonians (i) $H=H(p)$, (ii) $H=\frac{1}{2}\left(q^{2}+p^{2}\right)$,

express the general solution $(q(t), p(t))$ at time $t$ in terms of the initial values given by $(Q, P)=(q(0), p(0))$ at time $t=0$. In each case, show that the transformation from $(q, p)$ to $(Q, P)$ is canonical for all values of $t$, and find the corresponding generating function $F_{2}(q, P, t)$ explicitly.