On integrable cases of the Poincare problem

Ferrandiz, J.M.; Barkin, Y.V.

Авторы: Ferrandiz J.M., Barkin Y.V.
Журнал: Astronomical and Astrophysical Transactions
Том: 19
Год издания: 2001
Издательство: Cambridge Scientific Publishers
Местоположение издательства: Cambridge
Первая страница: 769
Последняя страница: 780
DOI: 10.1080/10556790108244091
Аннотация: Oscillations of a rotating liquid core in the cavity of a rigid envelope have been studied by many authors, starting from the classical works of PoincarC (1910). The basis of these studies usually contains the equations of motion in quasi-coordinates, but in recent years a new approach to the classical problem has been suggested, based on the equations of motion in Andoyer variables (Sevilla and Romero, 1987; Getino and Ferrandiz, 1997; etc.). Andoyer variables and angle-action variables (for the Euler-Poinsot problem) were effectively used for studies of integrability of the native Kirgoff problem (Barkin and Borisov, 1989).The papers of Sevilla and Romero (1987)’ Getino and Ferrandiz (1997) and others were directed to the problems of the Earth’s rotation. The principal effects of the liquid core in the Earth’s rotation were studied, and amplitudes of lunar and solar perturbations, corrected by the liquidcore iduence, were constructed (Getino and Ferrandiz, 1997). The purpose of the above-mentioned papers was to give an analytical description of the Earth’s rotation effects and the authors naturally used some simplifications of the equations of motion. In this work we consider an exact treatment of the classical PoincarC problem in Andoyer variables to study integrable cases of this problem.The canonical equations of the Poincare problem in Andoyer variables were obtained. Three cases of the integrability of these equations have been identified and studied for which the equations of the problem are reduced to a one-degree-ofkeedom system. The full system of first integrals and quadratures was obtained.One of them is the Poincarb case of an axysimmetric body and liquid core and the other two are generated cases, for which the ellipsoidal cavity is a thin disk.
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On integrable cases of the Poincare problemстатья

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