UZH-Logo

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.

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.

Citations

10 citations in Web of Science®
10 citations in Scopus®
Google Scholar™

Altmetrics

Downloads

174 downloads since deposited on 26 Feb 2010
40 downloads since 12 months
Detailed statistics

Additional indexing

Item Type:Journal Article, refereed, original work
Communities & Collections:07 Faculty of Science > Institute for Computational Science
Dewey Decimal Classification:530 Physics
Uncontrolled Keywords:stellar dynamics; celestial mechanics
Language:English
Date:November 2009
Deposited On:26 Feb 2010 15:18
Last Modified:05 Apr 2016 13:56
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
Publisher DOI:10.1111/j.1365-2966.2009.15437.x
Related URLs:http://arxiv.org/abs/0803.4441
Permanent URL: http://doi.org/10.5167/uzh-30853

Download

[img]
Preview
Content: Accepted Version
Filetype: PDF (Accepted manuscript, Version 3)
Size: 1MB
View at publisher
[img]
Preview
Content: Accepted Version
Filetype: PDF (Accepted manuscript, Version 2)
Size: 105kB
[img]
Preview
Content: Accepted Version
Filetype: PDF (Accepted manuscript, Version 1)
Size: 87kB

TrendTerms

TrendTerms displays relevant terms of the abstract of this publication and related documents on a map. The terms and their relations were extracted from ZORA using word statistics. Their timelines are taken from ZORA as well. The bubble size of a term is proportional to the number of documents where the term occurs. Red, orange, yellow and green colors are used for terms that occur in the current document; red indicates high interlinkedness of a term with other terms, orange, yellow and green decreasing interlinkedness. Blue is used for terms that have a relation with the terms in this document, but occur in other documents.
You can navigate and zoom the map. Mouse-hovering a term displays its timeline, clicking it yields the associated documents.

Author Collaborations