Past exam of the theoretical physics I course of the University of Cambridge / 2015 / Paper 1 / 1 / a / Solution

Take positive $\theta$ toward positive $x$ from the downward vertical, as in the source diagram. The mass has position and velocityThus $T=m|\dot{\mathbf{r}}{|}^2/2$ and $V=mg(a_y-l\cos \theta )$. Expanding the squared speed givesThe prescribed support coordinates are functions of time, rather than additional dynamical variables. Terms depending only on time do not affect the Euler-Lagrange equation for $\theta$, but have been retained here to obtain the inertial-frame kinetic and potential energies directly.