Қазақ ұлттық қыздар педагогикалық университеті / Казахский национальный женский педагогический университет

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Автор: search.filters.author.Ayazhan OrazymbetЕсть файлы: Да

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    NUMERICAL STUDY OF ORBITAL STABILITY IN THE RELATIVISTIC RESTRICTED THREE-BODY PROBLEM
    (2026-03-29)
    Ayazhan Orazymbet
    ;
    Aliya Taukenova
    ;
    Daniya Utepova
    ;
    Nurzada Beissen
    ;
    N. Sandibayeva
    This paper presents a numerical study of the relativistic restricted three-body problem within the framework of General Relativity. Using the Lagrangian and Hamiltonian formalisms, the equations of motion with relativistic corrections up to the order of 1/c2 were derived and solved numerically in Wolfram Mathematica. The developed model allows one to analyze orbital stability under small relativistic perturbations. Numerical simulations were performed using the Runge–Kutta integration method for three systems: “Earth–Sun–Moon,” “Earth–Sun– Mercury,” and an equal-mass configuration. The results confirm the stability of circular orbits and reproduce the observed relativistic precession of Mercury’s perihelion. For the equal-mass system, the calculations reveal a transition from quasi-periodic to chaotic motion, depending on the initial conditions. The study demonstrates the reliability and efficiency of the Mathematica environment for modeling nonlinear relativistic dynamics and shows that the proposed approach can be useful for further research in celestial mechanics and gravitational physics.
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    Vertical epicyclic stability and dissipative settling in a weakly deformed q-metric spacetime
    (L. N. Gumilyov Eurasian National University, 2026-09-30)
    Ayazhan Orazymbet
    ;
    Aray Muratkhan
    ;
    Aliya Taukenova
    ;
    A. Dalelkhankyzy
    We investigate the motion of massive test particles close to the equatorial plane of a weakly deformed q-metric spacetime. Working to first order in the dimensionless quadrupole parameter q, we derive the orbital angular velocity and the vertical epicyclic frequency for equatorial circular orbits while keeping the physical monopole mass fixed. The Schwarzschild limit is recovered at q = 0, whereas a nonzero quadrupole removes the degeneracy between the vertical and azimuthal frequencies already at O(q). A small phenomenological damping term is then introduced into the linearized vertical equation to describe the gradual approach of a perturbed orbit toward the equatorial plane. In the conservative case, the particle continues to oscillate about this plane without any decrease in amplitude. With damping, the amplitude decays exponentially. The quadrupolar deformation changes the vertical oscillation frequency and, consequently, the number of cycles completed before a specified settling threshold is reached. It does not, however, set the decay timescale, which follows from the adopted damping prescription. Since the coordinate-time damping coefficient is kept fixed in our numerical examples, the trajectories share the same exponential envelope but oscillate at different frequencies. The comparison with the exact q-metric also provides a direct check of the firstorder approximation. Over the sampled domain |q| ≤ 0.1 and 6 ≤ r/M ≤ 20, the relative error in the vertical frequency remains below 0.34%, while the corresponding frequency ratio is reproduced to better than 0.10%. These results show explicitly how the quadrupolar geometry modifies vertical orbital dynamics within a model in which dissipative settling is introduced independently.