On the theory of canonical perturbations and its application to Earth rotation

Abstract

Both orbital and rotational dynamics employ the method of variation of parameters. We express, in a non-perturbed setting, the coordinates (Cartesian, in the orbital case, or Eulerian in the rotation case) via the time and six adjustable constants called elements (orbital elements or rotational elements). If, under disturbance, we use this expression as ansatz and endow the "constants" with time dependence, then the perturbed velocity (Cartesian or angular) will consist of a partial derivative with respect to time and a so-called convective term, one that includes the time derivatives of the variable "constants." Out of sheer convenience, the so-called Lagrange constraint is often imposed. It nullifies the convective term and, thereby, guarantees that the functional dependence of the velocity upon the time and "constants" stays, under perturbation, the same as it used to be in the undisturbed setting. When the dynamical equations, written in terms of the "constants," are demanded to be symplectic (and the "constants" make conjugated pairs $ Q, P$), these "constants" are called Delaunay elements, in the orbital case, or Andoyer elements, in the rotational case. The Andoyer and Delaunay sets of elements share a feature not visible with a naked eye: in certain cases, the standard equations render these elements non-osculating. Hence, even though the Andoyer variables in the Kinoshita-Souchay theory are introduced in a precessing frame of the Earth orbit, they nevertheless return the angular velocity relative to an inertial frame.