Efroimsky, M. and Goldreich, P. (2004) Gauge freedom in the Nbody problem of celestial mechanics. Astronomy and Astrophysics, 415 (3). pp. 11871199. ISSN 00046361. doi:10.1051/00046361:20034058. https://resolver.caltech.edu/CaltechAUTHORS:20130226132906363

PDF
 Published Version
See Usage Policy. 223kB 
Use this Persistent URL to link to this item: https://resolver.caltech.edu/CaltechAUTHORS:20130226132906363
Abstract
The goal of this paper is to demonstrate how the internal symmetry of the Nbody celestialmechanics 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 noninertial frames. This class of problems is treated by the variationofconstants 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(N1)dimensional submanifold of the 12(N1)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 variationofconstants method) or implicitly (in the HamiltonJacobi 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 coordinatedependent perturbations, any gauge different from that of Lagrange makes the Delaunay system noncanonical. 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 velocityindependent. Finally, we provide a practical example illustrating how the gauge formalism considerably simplifies the calculation of satellite motion about an oblate precessing planet.
Item Type:  Article  

Related URLs: 
 
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 variationofconstants method. Research by ME was supported by NASA grant W19948. Research by PG was partially supported by NSF grant AST 0098301.  
Funders: 
 
Subject Keywords:  celestial mechanics; reference systems; solar system: general; methods: Nbody simulations; methods: analytical – methods: numerical  
Issue or Number:  3  
DOI:  10.1051/00046361:20034058  
Record Number:  CaltechAUTHORS:20130226132906363  
Persistent URL:  https://resolver.caltech.edu/CaltechAUTHORS:20130226132906363  
Official Citation:  Gauge freedom in the Nbody problem of celestial mechanics M. Efroimsky and P. Goldreich A&A 415 (3) 11871199 (2004) DOI: http://dx.doi.org/10.1051/00046361:20034058  
Usage Policy:  No commercial reproduction, distribution, display or performance rights in this work are provided.  
ID Code:  37149  
Collection:  CaltechAUTHORS  
Deposited By:  Tony Diaz  
Deposited On:  27 Feb 2013 00:04  
Last Modified:  09 Nov 2021 23:27 
Repository Staff Only: item control page