Levi-Civita’s regularization

Abstract In order to describe the motion of two weakly interacting satellites of a central body we suggest to use orbital elements based on the the linear theory of Kepler motion in Levi-Civita s regularizing coordinates. The basic model is the planar three-body problem with two small masses, a model in which both regular (e.g. quasi-periodic) as well as chaotic motion can occur. 

This paper discusses the basics of this approach and illustrates it with a typical example. First, we will revisit Levi-Civita s regularization of the two-dimensional Kepler motion and introduce sets of orbital elements based on the differential equations of the harmonic oscillator. Then, the corresponding theory for the three-dimensional motion will be developed using a quaternion representation of Kustaanheimo-Stiefel (KS) regularization we present it by means of an elegant new notation. 

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 second step of Levi-Civita s regularization consists of representing the complex physical coordinate x as the square u2 of a complex variable u = m + i v,2 G C,... 

The third step of Levi-Civita s regularization process produces linear differential equations for the unperturbed problem f = 0 by combining equation (10) with the energy relation. By using x = -2 uu equation(4) becomes... 

The key observation is that Levi-Civita s conformal map (7), u i—> x = u2, not only regularizes collisions at x = 0 but also analogous singularities at x = oo. This is seen by closing the complex planes to become Riemann spheres (by adding the point at infinity) and using inversions x = 1/x, u = 1/u. 

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