 | | Mechanics of Solids
A Journal of Russian Academy of Sciences | | Founded
in January 1966
Issued 6 times a year
Print ISSN 0025-6544
Online ISSN 1934-7936 |
Archive of Issues
| Total articles in the database: | | 13088 |
| In Russian (Èçâ. ÐÀÍ. ÌÒÒ): | | 8125
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| In English (Mech. Solids): | | 4963 |
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| Yu.N. Chelnokov, "Regular Quaternion Equations of Orbital Motion in the Earth’s Gravitational Field in KS-Variables and Their Modifications. Reduction of Dimensionality, First Integrals of Equations," Mech. Solids. 60 (1), 48-64 (2025) |
| Year |
2025 |
Volume |
60 |
Number |
1 |
Pages |
48-64 |
| DOI |
10.1134/S0025654424605056 |
| Title |
Regular Quaternion Equations of Orbital Motion in the Earth’s Gravitational Field in KS-Variables and Their Modifications. Reduction of Dimensionality, First Integrals of Equations |
| Author(s) |
Yu.N. Chelnokov (Institute for Precision Mechanics and Control Problems of the Russian Academy of Sciences, Saratov, 410028 Russia, ChelnokovYuN@gmail.com) |
| Abstract |
Regular quaternion differential equations of the perturbed orbital motion of a cosmic body
(in particular, a spacecraft, an asteroid) in the Earth’s gravitational field are considered, which take
into account zonal, tesseral and sectorial harmonics of the field. These equations, unlike classical
equations, are regular (do not contain special points such as singularity (division by zero)) for perturbed orbital motion in the central gravitational field of the Earth. In these equations, the main variables are four-dimensional Kustaanheim–Stiefel variables (KS-variables) or four-dimensional variables proposed by the author of the article, in which the equations of orbital motion have a simpler and
symmetric structure compared to equations in KS-variables. Additional variables in the equations are
orbital energy and time. The new independent variable is related to time by a differential relation containing the distance from the cosmic body to the Earth’s center of mass (the Sundman differential time
transformation is used). Regular equations of perturbed orbital motion in quaternion osculating
(slowly changing) variables are proposed. The equations are convenient for using methods of nonlinear mechanics and high-precision numerical calculations, in particular, for forecasting and correcting
the orbital motion of spacecraft. In the case of orbital motion in the Earth’s gravitational field, the
description of which takes into account the central and zonal harmonics of the field, the first integrals
of the equations of orbital motion of the eighth order are given, changes of variables and transformations of these equations are considered, which made it possible to obtain closed systems of differential
equations of the sixth order for the study of orbital motion, as well as systems of differential equations
of the fourth and third orders, including a system of differential equations of the third order with
respect to the distance from the cosmic body to the center of mass of the Earth and the sine of geocentric latitude, as well as a system of two integro-differential equations of the first order with respect to
these two variables. |
| Keywords |
regular quaternion differential equations of perturbed orbital motion, Earth’s gravitational field, singularity, Kustaanheim–Stiefel variables (KS-variables), modified four-dimensional variables, energy of orbital motion, Sundmann time transformation, quaternion osculating (slowly changing) variables, first integrals of equations, distance to the Earth’s center of mass, latitude, longitude |
| Received |
04 June 2024 | Revised |
05 August 2024 | Accepted |
06 August 2024 |
| Link to Fulltext |
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