A&A 415, 1187-1199 (2004)

DOI: 10.1051/0004-6361:20034058

## Gauge freedom
in the *N*-body problem of celestial mechanics

**M. Efroimsky**

^{1}and P. Goldreich^{2}^{1}US Naval Observatory, Washington, DC 20392, USA

e-mail: efroimsk@ima.umn.edu

^{2}IAS, Princeton NJ 08540 & CalTech, Pasadena, CA 91125, USA

e-mail: pmg@tapir.caltech.edu

(Received 8 July 2003 / Accepted 22 October 2003)

** Abstract **

The goal of this paper is to demonstrate how the internal symmetry
of the
*N*-body celestial-mechanics problem can be exploited in
orbit calculation.

We start with summarising research reported in (Efroimsky [CITE],
[CITE]; Newman & Efroimsky [CITE]; Efroimsky & Goldreich [CITE]) and
develop its application to planetary equations in non-inertial
frames. This class of problems is treated by the
variation-of-constants method. As explained in the previous
publications, whenever a standard system of six planetary
equations (in the Lagrange, Delaunay, or other form) is employed
for
*N* objects, the trajectory resides on a 9(
*N*-1)-dimensional
submanifold of the 12(
*N*-1)-dimensional space spanned by the
orbital elements and their time derivatives. The freedom in
choosing this submanifold reveals an internal symmetry inherent in
the description of the trajectory by orbital elements. This
freedom is analogous to the gauge invariance of electrodynamics.
In traditional derivations of the planetary equations this freedom
is removed by hand through the introduction of the Lagrange
constraint, either explicitly (in the variation-of-constants
method) or implicitly (in the Hamilton-Jacobi approach). This
constraint imposes the condition (called "osculation condition")
that both
the instantaneous position and velocity be fit by a Keplerian
ellipse (or hyperbola), i.e., that the instantaneous Keplerian
ellipse (or hyperbola) be tangential to the trajectory. Imposition
of any supplementary constraint different from that of Lagrange
(but compatible with the equations of motion) would alter the
mathematical form of the planetary equations without affecting the
physical trajectory.

However, for coordinate-dependent perturbations, any gauge different from that of Lagrange makes the Delaunay system non-canonical. Still, it turns out that in a more general case of disturbances dependent also upon velocities, there exists a "generalised Lagrange gauge", i.e., a constraint under which the Delaunay system is canonical (and the orbital elements are osculating in the phase space). This gauge reduces to the regular Lagrange gauge for perturbations that are velocity-independent.

Finally, we provide a practical example illustrating how the gauge formalism considerably simplifies the calculation of satellite motion about an oblate precessing planet.

**Key words:**celestial mechanics

**--**reference systems

**--**solar system: general

**--**methods:

*N*-body simulations

**--**methods: analytical

**--**methods: numerical

Offprint request: M. Efroimsky, efroimsk@ima.umn.edu

**©**

*ESO 2004*