Quaternion Regularization in Celestial Mechanics

Quaternion regularisation has emerged as a powerful tool in celestial mechanics to resolve the singularities inherent in classic two-body and few-body problems. By embedding orbital motion in a four-dimensional quaternionic framework, one can transform the Newtonian equations of motion into a regular system, smoothing the singular behaviour that arises at close approaches and collisions. This approach generalises the Levi-Civita planar regularisation to spatial motion via the Kustaanheimo–Stiefel (KS) transformation, in which quaternions serve as rotation operators and link Cartesian coordinates to an isotropic oscillator formalism. The quaternionic formalism not only removes coordinate singularities but also preserves symplectic structure, offering enhanced numerical stability for long-term integrations of planetary orbits, satellite transfers and restricted three-body dynamics. Recent advances have extended the method to include perturbations from non-Keplerian potentials, relativistic corrections and time-dependent forces, broadening its scope to mission design and solar system evolution studies. By unifying geometric insights with canonical perturbation theory, quaternion regularisation stands at the intersection of analytical mechanics and computational astrodynamics, enabling precise and efficient simulations that underpin our understanding of orbital stability, resonance phenomena and chaotic transitions across a range of astrophysical and engineering applications.

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Quaternion Regularization in Celestial Mechanics publication trend

The graph below shows the total number of articles in quaternion regularization in celestial mechanics across all publications each year (not limited to Nature Index journals).

Technical terms

Quaternion: A hypercomplex number system comprising one real and three imaginary units, used to represent rotations in three dimensions.

Kustaanheimo–Stiefel (KS) transformation: A mapping from three-dimensional Cartesian coordinates to four-dimensional quaternionic variables that regularises the Kepler problem by converting it into an isotropic oscillator.

Levi-Civita transformation: A planar regularisation technique mapping two-dimensional motion onto a harmonic oscillator via a two-variable complex transformation.

Symplectic integrator: A numerical algorithm that preserves the symplectic structure of Hamiltonian systems, ensuring long-term stability in orbital integrations.

Bilinear invariant: A quadratic constraint in KS variables that maintains the dimensional reduction from the four-dimensional oscillator back to three-dimensional physical space.

References

1. Quaternion methods and models of regular celestial mechanics and astrodynamics. Applied Mathematics and Mechanics (2022).
2. The extended Lissajous–Levi-Civita transformation. Celestial Mechanics and Dynamical Astronomy (2018).
3. The Lissajous–Kustaanheimo–Stiefel transformation. Celestial Mechanics and Dynamical Astronomy (2019).
4. Alternative angle-based approach to the $\mathcal{KS}$-Map. An interpretation through symmetry and reduction. The Journal of Geometric Mechanics (2018).