Perturbation Methods in Solving the Problem of Two Bodies of Variable Masses with Application of Computer Algebra
Prokopenya A. Minglibayev M. Ibraimova A.
December 2024Multidisciplinary Digital Publishing Institute (MDPI)
Applied Sciences (Switzerland)
2024#14Issue 24
The classical many-body problem is not integrable, so perturbation theory based on an exact solution to the two-body problem is usually applied to investigate the dynamics of planetary systems. However, in the case of variable masses, the two-body problem is not integrable, in general, and application of perturbation theory is required to investigate it, as well. In the present paper, we use the perturbation theory to derive the differential equations determining the orbital elements of the relative motion of one body around the other. Two models of the perturbed aperiodic motion on conic and quasi-conic sections are considered and compared. Special attention is paid to the practically important case of small eccentricities, when the perturbing forces may be replaced by the corresponding power series expansions. The differential equations of the perturbed motion are averaged over the mean anomaly, and the evolutionary equations describing the behavior of the orbital elements over long periods of time are obtained for two models. Comparing the corresponding solutions to the evolutionary equations, we have shown that both models demonstrate similar behavior with regard to the secular perturbations of the orbital elements. However, the second model, based on the aperiodic motion on a quasi-conic section, is more appropriate for generalization to the many-body problem with variable masses. All the relevant symbolic and numerical calculations are performed with the computer algebra system Wolfram Mathematica.
evolutionary equations , Gyldén equation , perturbation methods , two-body problem , variable mass
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Institute of Information Technology, Warsaw University of Life Sciences—SGGW, Nowoursynowska Str. 159, Warsaw, 02-776, Poland
Faculty of Mechanics and Mathematics, Al-Farabi Kazakh National University, Al-Farabi av. 71, Almaty, 050040, Kazakhstan
Fesenkov Astrophysical Institute, Observatoriya 23, Almaty, 050020, Kazakhstan
Institute of Information Technology
Faculty of Mechanics and Mathematics
Fesenkov Astrophysical Institute
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