The Lissajous-Kustaanheimo-Stiefel transformation

Abstract

The Kustaanheimo-Stiefel transformation of the Kepler problem with a time-dependent perturbation converts it into a perturbed isotropic oscillator of 4-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 3-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 analyzed 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.

arXiv commentAccepted to Celestial Mechanics and Dynamical Astronomy

References in corpus (7)

• The Eccentric Kozai Mechanism for a Test Particle
• Interpreting the Kustaanheimo-Stiefel transform in gravitational dynamics
• The eccentric Kozai-Lidov effect as a resonance phenomenon
• Kustaanheimo-Stiefel Regularization and the Quadrupolar Conjugacy
• Kustaanheimo-Stiefel transformation with an arbitrary defining vector
• Stability and chaos in Kustaanheimo-Stiefel space induced by the Hopf fibration
• Poisson and symplectic reductions of 4-DOF isotropic oscillators. The van der Waals system as benchmark