In mathematical physics, the Levi-Civita regularization is a method to transform the perturbed planar Kepler problem into a perturbed harmonic oscillator using the Levi-Civita (regularization) map. Both are named after Tullio Levi-Civita, wo introduced them in 1920.
Levi-Civita map The Levi-Civita map is an extension of the real Hopf fibration , using real and real projective space given by or the complex plane by ,Waldvogel 2010, Eq. (14) by a squared radial component. It is a map or alternatively a map given by filling with of the map:
In cartesian coordinates, the Levi-Civita map is given by:
The above squaring of the distance, hence , follows from the binomial formulas. The induced map on the cotangent bundle, better expressed in complex coordinates to shorten with complex conjugation, is given by:
Levi-Civita regularization Let the Hamilton function (energy) of the Kepler problem, then the Levi-Civita regularization sends it onto a common denominator:
See also
Kustaanheimo–Stiefel regularization, analogous using the complex Hopf fibration Moser regularization Ligon–Schaaf regularization
Literature
References