Transformations of the perturbed two-body problem to unperturbed harmonic oscillators

Transformations are given which change the perturbed planar problem of two bodies into unperturbed and undamped harmonic oscillators with constant coefficients. The orginally singular, nonlinear and Lyapunov unstable equations become in this way regular, linear, and the stable solution may be written down immediately in terms of the new variables. Transformations of the independent and dependent variables are treated separately as well as jointly. Using arbitrary and special functions for the transformations allows a systematic discussion of previously introduced and new anomalies.For the unperturbed two-body problem the theorem is proved according to which if the transformations are power-functions of the radial variable, then only the eccentric and the true anomalies with the corresponding transformations of the radial variable will result in harmonic oscillators.Important practical applications are to increase autonomous operations in space, since by replacing lengthy numerical integrations by transformations, computer requirements are significantly reduced.