34th Meeting of the AAS Division on Dynamical Astronomy, May 2003
9 Standards and Gauges
Oral, Tuesday, May 6, 2003, 1:00-3:05pm,

## [9.04] Gauge Drift in Numerical Integrations of the Lagrange Planetary Equations

M. A. Murison, M. Efroimsky (U.S. Naval Observatory)

Efroimsky (2002) and Newman & Efroimsky (2003) recognized that the Lagrange and Delaunay planetary equations of celestial mechanics may be generalized to allow transformations analogous to the familiar gauge transformations in electrodynamics. As usually presented, the Lagrange equations, which are derived by the method of variation of parameters (invented by Euler and Lagrange for this very purpose), assume the Lagrange constraint, whereby a certain combination of parameter time derivatives is arbitrarily equated to zero. This particular constraint ensures an osculating orbit that is unique. The transformation of the description, as given by the (time-varying) osculating elements, into that given by the Cartesian coordinates and velocities is invertible. Relaxing the constraint enables one to substitute instead an arbitrary gauge function. This breaks the uniqueness and invertibility between the orbit instantaneously described by the orbital elements and the position and velocity components (i.e., many different orbits, precessing at different rates, can at a given instant share the same physical position and physical velocity through space). However, the orbit described by the (varying) orbital elements obeying a different gauge is no longer osculating.

In numerical calculations that integrate the traditional Lagrange and Delaunay equations, even starting off in a certain (say, Lagrange's) gauge, some fraction of the numerical errors will, nevertheless, diffuse into violation of the chosen constraint. This results in an unintended ``gauge drift''. Geometrically, numerical errors cause the trajectory in phase space to leave the gauge-defined submanifold to which the motion was constrained, so that it is then moving on a different submanifold. The method of Lagrange multipliers can be utilized to return the motion to the original submanifold (e.g., Nacozy 1971, Murison 1989). Alternatively, the accumulated gauge drift may be compensated by a gauge transformation, similarly returning the motion to the original submanifold, or at least to one that is closer to the original. In this paper, we numerically explore the gauge drift using a representative nontrivial example of two planets orbiting the Sun. The Lagrange equations written in a gauge-invariant form are integrated. We present results on (1) rates of the gauge drift and (2) experiments with gauge-motivated correctors.