Levi-Civita transformation

We start by performing a change of coordinates, the Levi-Civita transformation, which can be written (in the present case) as... 

Denoting by i the imaginary unit, the Levi Civita transformation is obtained setting... 

It is immediate to check that the Levi-Civita transformation is characterized by the following properties ... 

Figure 5. The Levi-Civita transformation doubles the angles at the origin.

Coming back to the Levi-Civita transformation (5), we observe that... 

Remark We recall that in the planar case the Levi-Civita transformation is given by... 

Celletti, A. (2002) The Levi-Civita, KS and radial-inversion regularizing transformations, in Singularities in Gravitational Systems, D. Benest, C. Froeschle eds., Springer-Verlag, Berlin, Heidelberg, 25-48... 

We first restrict ourselves to the two-dimensional case and take advantage of the fact that Levi-Civita s regularizing transformation (Levi-Civita 1920) has the agreeable property of transforming perturbed Kepler problems into perturbed harmonic oscillators, i.e. into perturbed linear problems. For a recent account of regularization theory see the article (Celletti 2002) and other contributions in the same volume. 

The first step of Levi-Civita s regularization consists of introducing the fictitious time r by the differential relation dt = r dr (differentiation with respect to r will be denoted by primes). In view of the step to follow we write the result of transforming equation (3) in complex form, where... 

The Levi-Civita tensor is a natural generalization of the totally antisymmetric third-rank tensor 6, as defined by Eq. (2.9). For Lorentz transformations A with det A = +1 the Levi-Civita pseudo-tensor is indeed a tensor which has... 

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