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

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. The variable “constants” obeying this condition are called osculating elements. Otherwise, they are simply called orbital or rotational elements. 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 Serret-Andoyer elements, in the rotational case. The Serret-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. In orbital mechanics, the elements, calculated via the standard planetary equations, come out non-osculating when perturbations depend on velocities. This complication often arises but seldom gets noticed. To keep elements osculating under such perturbations, extra terms must enter the equations, terms that will not be parts of the disturbing function (Efroimsky & Goldreich 2003, 2004). In the case of parametrisation through the Kepler elements, this will merely complicate the equations. In the case of Delaunay parametrisation, these extra terms will not only complicate the Delaunay equations, but will also destroy their canonicity. Under velocity-dependent disturbances, the osculation and canonicity conditions are incompatible. Similarly, in rotational dynamics, the Serret-Andoyer elements come out non-osculating when the perturbation depends upon the angular velocity of the top. Since a switch to a non-inertial frame is an angular-velocity-dependent perturbation, then amendment of the dynamical equations by only adding extra terms to the Hamiltonian makes these equations