Gauge Freedom in the N-body Problem of Celestial Mechanics
NAVAL OBSERVATORY WASHINGTON DC
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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 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 9N-1-1 dimensional submanifold of the 12N-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.
- Celestial Mechanics
- Statistics and Probability