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Gauge freedom in the N-body problem of celestial mechanics

Efroimsky, M. and Goldreich, P. (2004) Gauge freedom in the N-body problem of celestial mechanics. Astronomy and Astrophysics, 415 (3). pp. 1187-1199. ISSN 0004-6361. doi:10.1051/0004-6361:20034058.

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The goal of this paper is to demonstrate how the internal symmetry of the N-body celestial-mechanics problem can be exploited in orbit calculation. We start with summarising research reported in (Efroimsky [CITE], [CITE]; Newman & Efroimsky [CITE]; Efroimsky & Goldreich [CITE]) and develop its application to planetary equations in non-inertial frames. This class of problems is treated by the variation-of-constants method. As explained in the previous publications, whenever a standard system of six planetary equations (in the Lagrange, Delaunay, or other form) is employed for N objects, the trajectory resides on a 9(N-1)-dimensional submanifold of the 12(N-1)-dimensional space spanned by the orbital elements and their time derivatives. The freedom in choosing this submanifold reveals an internal symmetry inherent in the description of the trajectory by orbital elements. This freedom is analogous to the gauge invariance of electrodynamics. In traditional derivations of the planetary equations this freedom is removed by hand through the introduction of the Lagrange constraint, either explicitly (in the variation-of-constants method) or implicitly (in the Hamilton-Jacobi approach). This constraint imposes the condition (called "osculation condition") that both the instantaneous position and velocity be fit by a Keplerian ellipse (or hyperbola), i.e., that the instantaneous Keplerian ellipse (or hyperbola) be tangential to the trajectory. Imposition of any supplementary constraint different from that of Lagrange (but compatible with the equations of motion) would alter the mathematical form of the planetary equations without affecting the physical trajectory. However, for coordinate-dependent perturbations, any gauge different from that of Lagrange makes the Delaunay system non-canonical. Still, it turns out that in a more general case of disturbances dependent also upon velocities, there exists a "generalised Lagrange gauge", i.e., a constraint under which the Delaunay system is canonical (and the orbital elements are osculating in the phase space). This gauge reduces to the regular Lagrange gauge for perturbations that are velocity-independent. Finally, we provide a practical example illustrating how the gauge formalism considerably simplifies the calculation of satellite motion about an oblate precessing planet.

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Additional Information:© 2004 ESO. Received 8 July 2003; Accepted 22 October 2003. The authors are grateful to William Newman and Victor Slabinski for their help in improving the manuscript, to Sergei Klioner for drawing the authors’ attention to the paper by Brumberg et al., and to Michael Nauenberg for the information on Newton’s priority in developing the variation-of-constants method. Research by ME was supported by NASA grant W-19948. Research by PG was partially supported by NSF grant AST 00-98301.
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NSFAST 00-98301
Subject Keywords:celestial mechanics; reference systems; solar system: general; methods: N-body simulations; methods: analytical – methods: numerical
Issue or Number:3
Record Number:CaltechAUTHORS:20130226-132906363
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Official Citation:Gauge freedom in the N-body problem of celestial mechanics M. Efroimsky and P. Goldreich A&A 415 (3) 1187-1199 (2004) DOI:
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:37149
Deposited By: Tony Diaz
Deposited On:27 Feb 2013 00:04
Last Modified:09 Nov 2021 23:27

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