[UPSC IFoS 2017] Central Force Motion and Gravitation (Q87, 8+7=15M)

(i) State and prove Hamilton's principle and use it to prove that the shortest distance between two points in space is a straight line joining them. (ii) Use Hamiltonian mechanics to find the differential equation for planetary motion, moving under force f(r) = -\frac{k}{r^2} and prove that the areal velocity is constant.