A Note on the Kepler Problem

J. Moser proved that the flow arising in the Kepler problem and restricted to the manifold of the constant energy E < 0 is equivalent to the geodesic flow on a sphere. This was proved by means of some algebraic manipulations with the Hamilton function. In a similar way Yu. S. Osipov proved that this flow is equivalent to the geodesic flow on the Euclidean space for E = 0 and on the Lobachevskii space for E < 0. In this paper results of such kind are related to the approach to the Kepler problem suggested by Hamilton (this approach seems to be the simplest one). For the planar Kepler problem one first considers the picture arising on the hodograph plane, where the hodograph curves turn out to be circles or arcs of circles. For fixed E one obtains a net (2-parameter linear system) of circles which in the case of E < 0 can be obtained from the system of great circles on a sphere by a stereographic projection; related geometric construction exists also for other E. This leads in a geometrical way to Moser's result. Moser showed also that for E < 0 the trajectory space of the covering flow on the universal covering space (which is a three-dimensional sphere $$\mathbb{B}$$ 3) is a two-dimensional sphere $$\mathbb{B}$$ 2; the corresponding map $$\mathbb{B}$$ 3 → $$\mathbb{B}$$ 2 is the Hopf fibration. An additional remark made below is that under appropriate normalizations and modifications this is the map $${\rm Z} \mapsto \left( {\varepsilon _1 ,\varepsilon _2 ,C} \right),$$ where the right-hand side contains well-known first integrals of the Kepler problem computed for the trajectory covered by the trajectory of z : C is the area constant (i.e., rotational momentum) and ∈ i are two components of the Runge-Lenz vector. Analogous statements hold for other E and for the Kepler problem in the whole space.