The theory of canonical perturbations applied to attitude dynamics and to the Earth rotation. Osculating and nonosculating Andoyer variables |
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| Authors: | Michael Efroimsky Alberto Escapa |
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| Institution: | (1) US Naval Observatory, Washington, DC 20392, USA;(2) Departamento de Matemática Aplicada, Universidad de Alicante, Alicante, E-03080, Spain |
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| Abstract: | In the method of variation of parameters we express the Cartesian coordinates or the Euler angles as functions of the time
and six constants. If, under disturbance, we endow the “constants” with time dependence, the perturbed orbital or angular
velocity will consist of a partial time derivative and a convective term that includes time derivatives of the “constants”.
The Lagrange constraint, often imposed for convenience, nullifies the convective term and thereby guarantees that the functional
dependence of the velocity on the time and “constants” stays unaltered under disturbance. “Constants” satisfying this constraint
are called osculating elements. Otherwise, they are simply termed orbital or rotational elements. When the equations for the
elements are required to be canonical, it is normally the Delaunay variables that are chosen to be the orbital elements, and
it is the Andoyer variables that are typically chosen to play the role of rotational elements. (Since some of the Andoyer
elements are time-dependent even in the unperturbed setting, the role of “constants” is actually played by their initial values.)
The Delaunay and Andoyer sets of variables share a subtle peculiarity: under certain circumstances the standard equations
render the elements nonosculating. In the theory of orbits, the planetary equations yield nonosculating elements when perturbations
depend on velocities. To keep the elements osculating, the equations must be amended with extra terms that are not parts of the disturbing function Efroimsky, M., Goldreich, P.: J. Math. Phys. 44, 5958–5977 (2003); Astron. Astrophys. 415, 1187–1199 (2004); Efroimsky, M.: Celest. Mech. Dyn. Astron. 91, 75–108 (2005); Ann. New York Acad. Sci. 1065, 346–374 (2006)]. It complicates both the Lagrange- and Delaunay-type planetary equations and makes the Delaunay equations
noncanonical. In attitude dynamics, whenever a perturbation depends upon the angular velocity (like a switch to a noninertial
frame), a mere amendment of the Hamiltonian makes the equations yield nonosculating Andoyer elements. To make them osculating,
extra terms should be added to the equations (but then the equations will no longer be canonical). Calculations in nonosculating
variables are mathematically valid, but their physical interpretation is not easy. Nonosculating orbital elements parameterise
instantaneous conics not tangent to the orbit. (A nonosculating i may differ much from the real inclination of the orbit, given by the osculating i.) Nonosculating Andoyer elements correctly describe perturbed attitude, but their interconnection with the angular velocity
is a nontrivial issue. The Kinoshita–Souchay theory tacitly employs nonosculating Andoyer elements. For this reason, even
though the elements are introduced in a precessing frame, they nevertheless return the inertial velocity, not the velocity
relative to the precessing frame. To amend the Kinoshita–Souchay theory, we derive the precessing-frame-related directional
angles of the angular velocity relative to the precessing frame. The loss of osculation should not necessarily be considered
a flaw of the Kinoshita–Souchay theory, because in some situations it is the inertial, not the relative, angular velocity
that is measurable Schreiber, K. U. et al.: J. Geophys. Res. 109, B06405 (2004); Petrov, L.: Astron. Astrophys. 467, 359–369 (2007)]. Under these circumstances, the Kinoshita–Souchay formulae for the angular velocity should be employed (as
long as they are rightly identified as the formulae for the inertial angular velocity). |
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| Keywords: | Earth rotation Attitude mechanics Attitude dynamics Kinoshita theory of the Earth rotation Kinoshita– Souchay theory of the Earth rotation Hamiltonian theory of the Earth rotation Andoyer variables Andoyer elemets Osculation Nonosculation Canonical perturbation theory Rigid-body rotation Right-body dynamics Rigid-body mechanics Poinsot problem Euler– Poinsot problem |
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