Answer Happy

Having established that all of the action takes place in a plane, let's set up polar coordinates (r, 0) on that plane, with the center of attraction at the origin. Let's also set up a z axis pointing directly out of the plane to complete the construction of cylindrical coordinates. (i) Compute the angular momentum of a planet about its center of attraction in cylindrical = coordinates and show that its magnitude is L mr²w mr²0. By part (b), this combination must remain constant, even as r(t) and (t) evolve in time. = (ii) In an infinitesimal time interval dt, the planet changes its coordinates from (r, 0) to (r + dr, 0 + de). As this happens, the line segment joining the planet and the origin sweeps out a small triangle: show that its area is approximately dA = r²d0.