In mathematical physics, the Ligon–Schaaf regularization is a method to describe the singularities of collisions in the Kepler problem using the Ligon–Schaaf (regularization) map. Both are named after Thomas Ligon and Manfred Schaaf, who introduced them in 1976. (Thomas Ligon wrote his Diploma Thesis, now Master Thesis, about them under Manfred Schaaf's supervision.) Compared to other methods of regularization, the Ligon–Schaaf regularization possesses special properties like not transforming time or being able to transform all negative energy levels together. (The related Moser regularization requires a fixed energy.)
Ligon–Schaaf map Let the Hamilton function (energy) of the Kepler problem. Let furthermore:Frauenfelder & van Koert 2018, p. 53-54
The Ligon–Schaaf map is then given by:
Properties
and are orthonormal, hence , und .Ligon 2018, Eq. (30) - (32)Frauenfelder & van Koert 2018, Lemma 4.3.1 Hence the Ligon–Schaaf map indeed maps to the subset and is therefore well-defined. The Ligon–Schaaf map is a symplectomorphism.Frauenfelder & van Koert 2018, Lemma 4.3.2 With the Delaunay Hamiltonian , one has:Ligon 2018, Eq. (40)Frauenfelder & van Koert 2018, S. 53-54
With the cross product , the Runge–Lenz vector and angular momentum :Ligon 2018, Eq. (38) - (39)Frauenfelder & van Koert 2018, S. 53-54
See also
Levi-Civita regularization, based on the real Hopf fibration Kustaanheimo–Stiefel regularization, based on the complex Hopf fibration Moser regularization
Literature
References