  Journal of Applied Mathematics and Mechanics Russian Academy of Sciences   Founded
in January 1936
(Translated from 1958)
Issued 6 times a year
ISSN 00218928 (print version) 
Archive of Issues
Total articles in the database:   2048 
In Russian (ÏÌÌ):   1249

In English (J. Appl. Math. Mech.):   799 

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A.V. Shatina and Ye.V. Sherstnev, "The motion of a satellite in the gravitational field of a viscoelastic planet," J. Appl. Math. Mech. 76 (6), 658665 (2012) 
Year 
2012 
Volume 
76 
Issue 
6 
Pages 
658665 
Title 
The motion of a satellite in the gravitational field of a viscoelastic planet 
Author(s) 
A.V. Shatina (Moscow, Russia, shatina_av@mail.ru)
Ye.V. Sherstnev (Moscow, Russia) 
Abstract 
The motion of a satellite in the gravitational field of a massive deformable planet is investigated. The planet is modelled by a homogeneous isotropic viscoelastic body of spherical shape in the natural undeformed state, while the satellite is modelled by a point mass. A potential energy functional of elastic deformations is introduced in accordance with the classical theory of elasticity of small deformations, while the dissipative force functional corresponds to the KelvinVoigt model. A system of integrodifferential equations of motion of the system is derived from the d’AlembertLagrange variational principle. Assuming that the stiffness of the viscoelastic planet is high, a small parameter is introduced, inversely proportional to Young's modulus, and an approximate system of ordinary differential equations is constructed in vector form by the method of separation of motions, describing the translationalrotational motion of the planetsatellite system, when perturbations due to elasticity and dissipation are taken into account. The system of equations obtained has a stationary solution, corresponding to the motion of the satellite in a circular orbit in a plane orthogonal to the constant vector, in which the number of stationary orbits cannot exceed two. It is shown that, in the case when two stationary orbits exist, the stationary solution, corresponding to the motion in an orbit of greater radius, is asymptotically stable, while in an orbit of smaller radius it is unstable. An evolutionary system of equations of motion of the satellite in Delaunay variables is obtained, which describes the change in the orbit parameters. Averaging was carried out over the "fast" angular variable  the mean anomaly. 
Received 
29 March 2011 
Link to Fulltext 
http://www.sciencedirect.com/science/article/pii/S0021892813000257 
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