Gauge Symmetry of the N-body Problem in the Hamilton-Jacobi Approach

Abstract

In most books the Delaunay and Lagrange equations for the orbital elements are derived by the Hamilton-Jacobi method: one begins with the 2-body Hamilton equations, performs a canonical transformation to the orbital elements, and obtains the Delaunay system. A standard trick is then used to generalise the approach to the N-body case. We re-examine 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 so-called Lagrange constraint. 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 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 of the celestial-mechanics equations and 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 non-canonical. In a more general setting, when the disturbance depends not only upon positions but also upon velocities, there is a "generalised Lagrange gauge" wherein the Delaunay system is symplectic. This gauge renders orbital elements that are osculating in the phase space. It coincides with the regular Lagrange gauge when the perturbation is velocity-independent.