Gauge Freedom in the N - body Problem of Celestial

We summarise research reported in (Efroimsky 2002, 2003; Efroimsky & Goldreich 2003a,b) and develop its application to planetary equations in non-inertial frames. 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 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. For coordinate-dependent perturbations, any gauge different from that of Lagrange makes the Delaunay system non-canonical. In a more general case of disturbances dependent also upon velocities, there exists a ”generalised Lagrange gauge” wherein the Delaunay system is symplectic (and the orbital elements are osculating in the phase space). This gauge reduces to the regular Lagrange gauge for perturbations that are velocity-independent. We provide a practical example illustrating how the gauge formalism considerably simplifies the calculation of satellite motion about an oblate precessing planet.