I have to prove the following equations using the Lagrangian equations, The figure shows the image, I know how to do lagrangian. I just don't know how to solve the kinetic and potential energies of this pendulum. Can someone show me what the value are for the kinetic and potential energy? I have tried but I don't know how to solve them
This is the question https://i.stack.imgur.com/UJvfm.jpg
The horizontal speed of the mass $m$ is zero, and its vertical speed is $-\dot d$. Its kinetic energy is therefore $\frac12 m \dot d^2$. Its potential energy is obviously $\text{constant}-mgd$.
The horizontal speed of the mass $M$ is $\dot{(l\sin(\theta))} = l\cos(\theta)\dot\theta$. The vertical speed of the mass $M$ is $\dot{(-d - l\cos(\theta))} = -\dot d +l\sin(\theta)\dot\theta$. Its kinetic energy is therefore $\frac12M(\dot d^2 - 2\dot d l \sin(\theta) + l^2\dot\theta^2$). Its potential energy is $-Mg(d + l\cos(\theta))$.