Gauge Freedom in the N-body Problem of Celestial Mechanics
Abstract
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 summarizing research reported in (Efroimsky 2002, 2003; Newman & Efroimsky 2003; Efroimsky & Goldreich 2003) and develop its application to planetary equations in non-inertial frames. This class of problems is treated by the variations of-constraints 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)-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-constraints methods 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 Keplerlan 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.
Document Details
- Document Type
- Technical Report
- Publication Date
- Oct 22, 2003
- Accession Number
- ADA422861
Entities
People
- M. Efroimsky
- P. Goidreich
Organizations
- United States Naval Observatory