Gauge Freedom in Orbital Mechanics
Abstract
In orbital and attitude dynamics the coordinates and the Euler angles are expressed as functions of the time and six constants called elements. Under disturbance, the constants are endowed with time dependence. The Lagrange constraint is then imposed to guarantee that the functional dependence of the perturbed velocity on the time and constants stays the same as in the undisturbed case. Constants obeying this condition are called osculating elements. The constants chosen to be canonical are called Delaunay elements, in the orbital case, or Andoyer elements, in the spin case. (As some Andoyer elements are time dependent even in the freespin case, the role of constants is played by their initial values.) The Andoyer and Delaunay sets of elements share a feature not readily apparent: in certain cases the standard equations render them nonosculating. In orbital mechanics, elements furnished by the standard planetary equations are nonosculating when perturbations depend on velocities. To preserve osculation, the equations must be amended with extra terms that are not parts of the disturbing function. In the case of Delaunay parameterisation, these terms destroy canonicity. So under velocitydependent disturbances, osculation and canonicity are incompatible. (Efroimsky and Goldreich 2003, 2004) Similarly, the Andoyer elements turn out to be nonosculating under angularvelocitydependent perturbation. Amendment of only the Hamiltonian makes the equations render nonosculating elements. To make them osculating, more terms must enter the equations (and the equations will no longer be canonical). In practical calculations, is often convenient to deliberately deviate from osculation by substituting the Lagrange constraint with a condition that gives birth to a family of nonosculating elements.
 Publication:

Annals of the New York Academy of Sciences
 Pub Date:
 December 2005
 DOI:
 10.1196/annals.1370.016
 arXiv:
 arXiv:astroph/0603092
 Bibcode:
 2005NYASA1065..346E
 Keywords:

 Astrophysics;
 Mathematics  Dynamical Systems;
 Nonlinear Sciences  Chaotic Dynamics;
 Physics  Classical Physics
 EPrint:
 Talk at the annual Princeton conference ``New Trends in Astrodynamics" 2005 http://www.math.princeton.edu/astrocon/