(i) Show that Newton's equations in Cartesian coordinates, for a system of $N$ particles at positions $\mathbf{x}_{i}(t), i=1,2 \ldots N$, in a potential $V(\mathbf{x}, t)$, imply Lagrange's equations in a generalised coordinate system

$$q_{j}=q_{j}\left(\mathbf{x}_{i}, t\right) \quad, \quad j=1,2 \ldots 3 N$$

that is,

$$\frac{d}{d t}\left(\frac{\partial L}{\partial \dot{q}_{j}}\right)=\frac{\partial L}{\partial q_{j}} \quad, \quad j=1,2 \ldots 3 N$$

where $L=T-V, T(q, \dot{q}, t)$ being the total kinetic energy and $V(q, t)$ the total potential energy.

(ii) Consider a light rod of length $L$, free to rotate in a vertical plane (the $x z$ plane), but with one end $P$ forced to move in the $x$-direction. The other end of the rod is attached to a heavy mass $M$ upon which gravity acts in the negative $z$ direction.

(a) Write down the Lagrangian for the system.

(b) Show that, if $P$ is stationary, the rod has two equilibrium positions, one stable and the other unstable.

(c) The end at $P$ is now forced to move with constant acceleration, $\ddot{x}=A$. Show that, once more, there is one stable equilibrium value of the angle the rod makes with the vertical, and find it.