A canonical reduction of order for the Kepler problem

Abstract

Using a new redudion of order technique we obtain four conserved quantities of the darsical planar Kepler problem via the symmetries of the equations of motion and an extra vector field. By passing to a suitable quotient manifold of the evolution space we deduce that negative energy orbits are closed and periodic, without having to solve the differential equation. In this paper we look at the classical Kepler problem through the application of a new reduction of order technique (l). We believe that this technique provides a systematic means of producing first integrals from group actions more general than symmetries without relying on the existence of a variational principle. Moreover, it is a powerful tool for investigating global properties such as periodicity. The Kepler problem is a good vehicle for a demonstration of these features. The planar equations of motion are defined by a second-order differential equation field on five-dimensional evolution space. There are three point symmetries which take solutions of the equation of motion into other solutions: time translation, which leaves the solution curves invariant but alters their starting times; rotation about the origin, which again leaves the curves invariant but alters the initial angular position; and the transformation associated with the Runge-Leuz vector which allows the orientation and eccentricity of the orbits to remain the same but changes the semi-latus rectum. We wish to show that conserved quantities of the system, i.e. the objects which remain invariant along solution curves, can be obtained using alternative techniques to Noether's theorem (2) or solving the differential equation (3, 41. Lie's method (2) provides us with the three point symmetries mentioned above. To effect our technique a fourth vector field is required which is a symmetry of the integrable distribution made up of the second-order differential equation field and its three symmetries. To find this field we modify the usual reduction of order via symmetry method by taking successive quotients of the extended phase space by the three actions (for example, we project down to a four-dimensional manifold by setting the time coordinate to a constant, thus identifying all the orbits which only differ by a time translation) and we reduce our differential equation field to first order on a two-dimensional quotient.