Equations for the Keplerian elements : Hidden symmetry

We revisit the Lagrange and Delaunay systems of equations for the six osculating orbital elements, and point out a previously neglected aspect of these equations. A careful re-examination of the derivation of these systems shows that in both cases the orbit resides on a certain 9-dimensional submanifold of the 12-dimensional space spanned by the osculating elements and their time derivatives. We demonstrate that there exists a certain amount of freedom in choosing this submanifold. The choice is mathematically analogous to gauge fixing in electrodynamics. This freedom of choice (=freedom of gauge fixing) reveals a symmetry hiding behind Lagrange’s and Delaunay’s systems, which i s similar 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 error and instability.