Gauge Freedom in the N-body Problem of Celestial Mechanics
Abstract
Whenever a standard system of six planetary equations(in the Lagrange, Delaunay, or other form) is employed, 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-parameters method) or implicitly (in the Hamilton-Jacobi approach). This constraint imposes the condition that the orbital elements osculate the trajectory, i.e., that both the instantaneous position and velocity be fit by a Keplerian ellipse (or hyperbola). Imposition of an 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.
Document Details
- Document Type
- Technical Report
- Publication Date
- Jul 07, 2003
- Accession Number
- ADA423238
Entities
People
- Michael Efroimsky
- Peter Goldreich
Organizations
- United States Naval Observatory