A comet of mass \(m\) moves in two dimensions in response to the central gravitational potential \(U=-k / r\), where \(k\) is a constant and \(r\) is the distance from the Sun. Using the Jacobi action and polar coordinates \((r, \theta)\), find the possible shapes of the comet's orbit. Show that these are (a) a parabola, if the energy of the comet is \(E=0\); (b) a hyperbola if \(E>0\); (c) an ellipse or a circle if \(E<0\), where in each case \(r=0\) at one of the foci.