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Interpreting the Kustaanheimo–Stiefel transform in gravitational dynamics


Saha, P (2009). Interpreting the Kustaanheimo–Stiefel transform in gravitational dynamics. Monthly Notices of the Royal Astronomical Society, 400(1):228-231.

Abstract

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 transform has proved very useful in N-body simulations, where it helps to handle close encounters. Yet the formalism remains somewhat arcane, with the role of the extra dimension being especially mysterious. This paper shows how the basic transformation can be interpreted as a rotation in three dimensions. For example, if we slew a telescope from zenith to a chosen star in one rotation, we can think of the rotation axis and angle as the KS transform of the star. The non-uniqueness of the rotation axis encodes the extra dimension. This geometrical interpretation becomes evident on writing KS transforms in quaternion form, which also helps to derive concise expressions for regularized equations of motion.

Abstract

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 transform has proved very useful in N-body simulations, where it helps to handle close encounters. Yet the formalism remains somewhat arcane, with the role of the extra dimension being especially mysterious. This paper shows how the basic transformation can be interpreted as a rotation in three dimensions. For example, if we slew a telescope from zenith to a chosen star in one rotation, we can think of the rotation axis and angle as the KS transform of the star. The non-uniqueness of the rotation axis encodes the extra dimension. This geometrical interpretation becomes evident on writing KS transforms in quaternion form, which also helps to derive concise expressions for regularized equations of motion.

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Additional indexing

Item Type:Journal Article, refereed, original work
Communities & Collections:07 Faculty of Science > Institute for Computational Science
Dewey Decimal Classification:530 Physics
Scopus Subject Areas:Physical Sciences > Astronomy and Astrophysics
Physical Sciences > Space and Planetary Science
Uncontrolled Keywords:stellar dynamics, celestial mechanics
Language:English
Date:November 2009
Deposited On:26 Feb 2010 15:18
Last Modified:29 Jul 2020 21:35
Publisher:Wiley-Blackwell
ISSN:0035-8711
Additional Information:The attached file is a preprint (accepted version) of an article published in Monthly Notices of the Royal Astronomical Society. The definitive version is available at www3.interscience.wiley.com
OA Status:Hybrid
Publisher DOI:https://doi.org/10.1111/j.1365-2966.2009.15437.x
Related URLs:http://arxiv.org/abs/0803.4441

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Download PDF  'Interpreting the Kustaanheimo–Stiefel transform in gravitational dynamics'.
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Content: Accepted Version
Language: English
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Download PDF  'Interpreting the Kustaanheimo–Stiefel transform in gravitational dynamics'.
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Content: Accepted Version
Language: English
Filetype: PDF (Accepted manuscript, Version 1)
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Download PDF  'Interpreting the Kustaanheimo–Stiefel transform in gravitational dynamics'.
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Filetype: PDF (Nationallizenz 142-005)
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