Levi-Civita coordinates

In order to apply one of the proposed sets of elements for describing coorbital motion we formulate the equations of motion (1) of the weakly coupled Kepler motions in terms of the Levi-Civita coordinates of Section 2. In this way the unperturbed problem will be defined by linear differential equations. Using the symbols pj = mo + nij, j = 1,2 as well as complex notation iq, r2 C and the abbreviations fi, f2 C for the right-hand sides, equation (1) reads as... 

Its primary use in our applications is for expressing various commutation relations in compact form and performing calculations involving them. The Levi-Civita symbol also arises naturally in vector analysis. Thus, if et, i = 1, 2, 3, are three mutually perpendicular unit vectors defining a right-handed coordinate system, then... 

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

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. 

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,... 

Cosserat strain tensor differing from the Cosserat deformation tensor through the fundamental metric tensor Gkl KMN the usual Levi-Civita alternating tensor. In (3) and (4), a repeated index implies summation over that index, a comma followed by an index indicates a partial derivative with respect to the material coordinate carrying that index and a colon followed by an index represents total covariant differentiation with respect to the material coordinate carrying that index. 

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