# Approximate solution of the Liouville equations in action-angle variables for the Euler-Poinsot problem

### Аuthors

Barkin M. Y.

Moscow Aviation Institute (National Research University), 4, Volokolamskoe shosse, Moscow, А-80, GSP-3, 125993, Russia

e-mail: barkin@yandex.ru

### Abstract

Construct an approximate solution of the Liouville problem about free rotational motion of an isolated celestial body with weakly variable geometry of mass, based on the equations of motion in action — angle variables for the Euler — Poinsot problem.
In this paper the new form of the equations of rotational motion of a celestial body in the action-angle variables for the Euler-Poinsot problem have been obtained. This allowed effectively applied the small parameter method and derive analytical formulas for the perturbations of action-angle variables of the first order. An important role herewith were played by the Fourier series for the first degrees, squares and mixed product of the direction cosines of the axes of inertia of the body in the basic coordinate system.
Perturbation theory of rotational motion of a celestial body contains effects that have not been described previously. And unperturbed motion accepted as free motion of a rigid body by Euler-Poinsot law, whose parameters can be arbitrary (any polhode except separatrices). As a consequence of this generalization (for triaxial planet) a description of amplitudes perturbations and spectrum perturbations is given by using the apparatus of elliptic functions and elliptic integrals.
The obtained results represent an important interest for research in celestial mechanics and geodynamics. They allow identify new effects in pole motion and in the diurnal rotation of the planets and asteroids. So the formulas for the first-order perturbation contain variations with new periods (compared to perturbations, obtained by classical method based on the Euler-Liouville equations).
Researchers studying perturbed rotational motions of the celestial bodies, have received new opportunities with using equations of motion, constructed in action -angle variables. The developed approach allows the direct use of data of space geodesy about variations of the Earth’s mass geometry directly from the observed variations of the geopotential coefficients. Thereby methods of space geodesy and study methods of perturbed motions of the Earth’s pole and variations of its axial rotation, act here like a single tandem and allow obtaining new results. Primarily, these results are of interest to study the effect of redistribution of mass of celestial bodies on movement of their poles and on their diurnal axial rotation. Obtained form of the equations of Liouville problem in action-angle variables, will be widely used in subsequent studies.
The results of this paper can be used in the basic and special courses of lectures for undergraduate and postgraduate students in the departments of theoretical mechanics, theoretical astronomy and celestial mechanics, etc.

### Keywords:

action-angle variables, the unperturbed motion Euler – Poinsot problem, Liouville problem, Fourier series, elliptic integral

### References

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