X iv : a st ro - p h / 02 12 24 5 v 3 7 F eb 2 00 3 Submitted to Celestial Mechanics and Dynamical Astronomy Equations for the orbital elements : Hidden symmetry

We revisit the Lagrange and Delaunay systems of equations for the orbital elements, and point out a previously neglected aspect of these equations: in both cases the orbit resides on a certain 9-dimensional submanifold of the 12-dimensional space spanned by the orbital elements and their time derivatives. We demonstrate that there exists a vast freedom in choosing this submanifold. This freedom of choice (=freedom of gauge fixing) reveals a symmetry hiding behind Lagrange’s and Delaunay’s systems, which is, mathematically, analogous to the gauge invariance in electrodynamics. Just like a convenient choice of gauge simplifies calculations in electrodynamics, so the freedom of choice of the submanifold may, potentially, be used to create simpler schemes of orbit integration. On the other hand, the presence of this feature may be a previously unrecognised source of numerical instability. We provide a practical example of a situation that cannot be correctly handled without the said gauge-type freedom taken into account.