Give Lagrange’s equations of motion. Applying them to the motion of a planet orbiting the Sun, give expressions for the kinetic and potential energies of the planet in polar coordinates and obtain two equations of motion for the radial and angular motion. Show that the angular equation of motion can be integrated and leads to the conservation of angular momentum. By changing the radial coordinate r to u = 1/r and eliminating time, show that the radial equation of motion has the form of a differential equation for displaced simple harmonic motion. Hence obtain a solution for the shape of the orbit.