Gauge symmetry of the N-body problem in the Hamilton–Jacobi approach

In most books the Delaunay and Lagrange equations for the orbital elements are derived by the Hamilton–Jacobi method: one begins with the two-body Hamilton equations in spherical coordinates, performs a canonical transformation to the orbital elements, and obtains the Delaunay system. A standard trick is then used to generalize the approach to the N-body case. We reexamine this step and demonstrate that it contains an implicit condition which restricts the dynamics to a 9(N−1)-dimensional submanifold of the 12(N−1)-dimensional space spanned by the elements and their time derivatives. The tacit condition is equivalent to the constraint that Lagrange imposed “by hand” to remove the excessive freedom, when he was deriving his system of equations by variation of parameters. It is the condition of the orbital elements being osculating, i.e., of the instantaneous ellipse (or hyperbola) being always tangential to the physical velocity. Imposure of any supplementary condition different from the Lagrange constraint (but compatible with the equations of motion) is legitimate and will not alter the physical trajectory or velocity (though will alter the mathematical form of the planetary equations). This freedom of nomination of the supplementary constraint reveals a gauge-type internal symmetry instilled into the equations of celestial mechanics. Existence of this internal symmetry has consequences for the stability of numerical integrators. Another important aspect of this freedom is that any gauge different from that of Lagrange makes the Delaunay system noncanonical. In a more general setting, when the disturbance depends not only upon positions but also upon velocities, there is a “generalized Lagrange gauge” wherein the Delaunay system is symplectic. This special gauge renders orbital elements that are osculating in the phase space. It coincides with the regular Lagrange gauge when the perturbation is velocity independent.