# Stability and chaos in Kustaanheimo-Stiefel space induced by the Hopf fibration

@article{Roa2016StabilityAC,
title={Stability and chaos in Kustaanheimo-Stiefel space induced by the Hopf fibration},
author={Javier Roa and Hodei Urrutxua and Jes'us Pel'aez},
journal={Monthly Notices of the Royal Astronomical Society},
year={2016},
volume={459},
pages={2444-2454}
}
• Published 14 April 2016
• Mathematics
• Monthly Notices of the Royal Astronomical Society
The need for the extra dimension in Kustaanheimo–Stiefel (KS) regularization is explained by the topology of the Hopf fibration, which defines the geometry and structure of KS space. A trajectory in Cartesian space is represented by a four-dimensional manifold called the fundamental manifold. Based on geometric and topological aspects classical concepts of stability are translated to KS language. The separation between manifolds of solutions generalizes the concept of Lyapunov stability. The…

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## References

SHOWING 1-10 OF 78 REFERENCES
The KS-transformation in hypercomplex form and the quantization of the negative-energy orbit manifold of the Kepler problem
In a previous note we have shown that the KS-transformation, introduced by Kustaanheimo and Stiefel into Celestial Mechanics for the regularization of the Kepler problem, may be formulated in terms
Linearization: Laplace vs. Stiefel
• Mathematics
• 1994
The method for processing perturbed Keplerian systems known today as the linearization was already known in the XVIIIth century; Laplace seems to be the first to have codified it. We reorganize the
Interpreting the Kustaanheimo–Stiefel transform in gravitational dynamics
The Kustaanheimo–Stiefel (KS) transform turns a gravitational two-body problem into a harmonic oscillator, by going to four dimensions. In addition to the mathematical-physics interest, the KS
Chaotic Worlds: from Order to Disorder in Gravitational N-Body Dynamical Systems
• Physics
• 2006
Preface.- Lecturers.- Part I Tools of Order.- Canonical perturbation theory for nearly integrable systems A. Giorgilli, U. Locatelli.- Periodic orbits in gravitational systems J.D. Hadjedemetriou.-
The KS-transformation in hypercomplex form
In this note the KS-transformation introduced by Kustaanheimo and Stiefel into Celestial Mechanics is formulated in terms of hypercomplex numbers as the product of a quaternion and its antiinvolute.
Time-Symmetrized Kustaanheimo-Stiefel Regularization
• Physics
• 1996
In this paper we describe a new algorithm for the long-term numerical integration of the two-body problem, in which two particles interact under a Newtonian gravitational potential. Although
Perturbation theory of Kepler motion based on spinor regularization.
• Mathematics
• 1965
A regularization of Kepler motion in the three-dimensional space Ä is developed using a simple mapping of a four-dimensional space R* onto Ä. In Ä* the equations of any undisturbed Kepler motion are
Orbit propagation in Minkowskian geometry
• Physics
• 2015
The geometry of hyperbolic orbits suggests that Minkowskian geometry, and not Euclidean, may provide the most adequate description of the motion. This idea is explored in order to derive a new
Time transformations in the extended phase-space
• Mathematics
• 1975
Time transformations involving momenta in addition to the coordinates are studied from the points of view of stabilization and regularization of the equations of motion. The generalization of