The Lissajous-Kustaanheimo-Stiefel transformation

The Kustaanheimo–Stiefel transformation of the Kepler problem with a time-dependent perturbation converts it into a perturbed isotropic oscillator of four-and-a-half degrees of freedom with additional constraint known as bilinear invariant. Appropriate action–angle variables for the constrained oscillator are required to apply canonical perturbation techniques in the perturbed problem. The Lissajous–Kustaanheimo–Stiefel (LKS) transformation is proposed, leading to the action–angle set which is free from singularities of the LCF variables earlier proposed by Zhao. One of the actions is the bilinear invariant, which allows the reduction back to the three-and-a-half degrees of freedom. The transformation avoids any reference to the notion of the orbital plane, which allowed to obtain the angles properly defined not only for most of the circular or equatorial orbits, but also for the degenerate, rectilinear ellipses. The Lidov–Kozai problem is analysed in terms of the LKS variables, which allow a direct study of stability for all equilibria except the circular equatorial and the polar radial orbits.