Regularization of Two-Body Collisions

Abstract

As we have seen in the discussion of the planar Kepler problem, the Kepler problem admits an obvious rotational symmetry, namely rotations of the plane, forming the group SO(2), but also admits hidden symmetries, which are generated by the Runge–Lenz vector (3.13). These hidden symmetries played an important role in the early development of quantum mechanics, in particular in Pauli’s and Fock’s discussion about the spectrum of the hydrogen atom [87, 202]. In [186] Moser explained how the Kepler flow can be embedded into the geodesic flow of the round sphere. This explains the hidden symmetries because the round metric is invariant under the group SO(3). In the case of the planar Kepler problem we obtain the geodesic flow on the two-dimensional sphere and the symmetry group becomes SO(3). In particular, the symmetry group is three-dimensional, in accordance with the fact that we have three integrals, namely angular momentum as well as the two components of the Runge–Lenz vector. We refer to the works of Hulthen [137] and Bargmann [24] for a discussion of these symmetries in terms of quantum mechanics. How the Runge–Lenz vector is related to the moment map of the Hamiltonian action of SO(3) on the cotangent bundle of the two-dimensional sphere is for example explained in [108, 150, 227].