Conformal regularization of the Kepler Problem

Abstract

The manifold M of null rays through the origin of R 2 ' π+1 is diffeomorphic to S 1 x S", and it is a homogeneous space of SO(2,n + 1). This group therefore acts on T*M, which we show to be the "generating manifold" of the extended phase space of the regularized Kepler Problem. A local canonical chart in T*M is found such that the restriction to the subbundle of the null nonvanishing covectors is given by p 0 + H(q, p) = 0, where H(q, p) is the Hamiltonian of the Kepler Problem. By means of this construction, we get some results that clarify and complete the previous approaches to the problem.

On the Geometry of Extended Self-Similar Solutions of the Airy Shallow Water Equations

“…Hence (51) shows that there is a blow-up of α, γ, and ζ at the finite time t s . Therefore we can see the variable change τ = τ (t) as an analogue of the classical regularizing Kustaanheimo-Stiefel transformation for the Kepler problem (see, e.g., [9] and the references quoted therein).…”

Quantization of the Kepler manifold

“…For the proof see [7]. Let N+ be the submanifold of null non-vanishing covectors in T*M pointing into the past and into the future respectively.…”

“…The reduction T*M\-^>T + S n may be interpreted as the reduction of the extended phase space of a mechanical system to the phase space by means of the pseudo-energy integral. In fact, let us consider the geodesic motion in T*M, where the metric of S 1 x S n is the usual pseudo-Riemannian: restriction to N is equivalent to fixing the pseudo-energy and dividing by S This approach permits us to handle the KP for every sign of the energy and to introduce the regularization parameter x° without postulating it (see [7] for more details).…”

“…Unfortunately this program is not easy to realize, because the natural polarization of the coadjoint orbit is not invariant under the action of the dynamical group; or, in other words, because S n does not carry an effective action of this group. To escape this difficulty and follow the analysis pursued at classical level in [7], we consider the manifold S 1 x S n , which is a homogeneous space for 5Ό (2, n+1). The cotangent bundle T^(S 1 x S n ) is a union of five homogeneous components under the action of the same group, two of which are the Kepler manifold.…”

Canonical variables and analysis on so(n, 2)

“…It serves as a "linearization" of the conformal group of Minkowski space R n−1,1 (see e.g., [11]), the symmetry group of Maxwell's equations. Also, the group SO(n,2) plays an important rôle in the n-dimensional Kepler problem, where the compactified phase space (the Moser phase space) coincides with a coadjoint orbit of the dynamical group SO(n+1,2) [8,16,19]. In another context, the group SO(4,2) serves as the spectrum-generating symmetry group of the hydrogen atom [2,17].…”