The orbital stability of a test particle motion in the field of two massive rotating bodies M. Abishev, S.Toktarbay, A. Abylayeva and A. Talkhat Al Farabi Kazakh National University Department of Theoretical and Nuclear physics Research Institute of Experimental and Theoretical Physics ICGAC-XIII+IK15, Seoul, July 03-July 07, 2017 1

Outline Adiabatic theory of motion in GR mechanics Circular restricted three-body problem Stability in two rotating bodies problem Rotating three bodies problem in GR mechanics 2/11 Al Farabi Kazakh National University

Adiabatic theory of motion in GR mechanics The adiabatic theory of motion is developed by M. M. Abdildin for the study of evolutionary motion in the mechanics of general relativity It is based on the use of vector elements for the description of motion, on the asymptotic methods of the theory of nonlinear oscillations, and on the method of adiabatic invariants. Knowledge of the angular velocity makes it possible to calculate the majority of known relativistic effects without solving the relativistic equations of motion M - orbital moment A - Laplace vector 2/11 Al Farabi Kazakh National University

Circular restricted three-body problem m1 >> m2 >> m Circular orbit m 𝑟 2 − 𝑟 m2 𝑟 𝑟 2 m1 where m1 and m2 are the masses of bodies, m is the mass of the test body are the radius vectors of the corresponding bodies 3/11 Al Farabi Kazakh National University

Stability in two rotating bodies problem 𝑟 𝑟 2 m1 4/11 Al Farabi Kazakh National University

Stability in two rotating bodies problem Orbital stability conditions for the two rotating bodies: when the angular momentum of the test body is parallel or antiparallel to the angular momentum of the central body m2 𝑟 𝑟 2 m1 4/11 Al Farabi Kazakh National University

The rotating three bodies problem in GR mechanics Three body problem: Three point-like objects (1) Black – classical terms Blue – relativistic kinematics terms Violet- orbital moments interaction Red – other curved space-time effects M. E. Abishev, S. Toktarbay and B. A. Zhami. Stability in the restricted three body problem in GR. Gravitation and Cosmology, 2014, Vol. 20, No. 3, pp. 149–151. Three body problem: Three rotating objects (2) Lagrangian Hamiltonian (3) Where and for the three point masses and - for the own rotation of bodies. 5/11 Al Farabi Kazakh National University

The Hamiltonian of three bodies Additive terms to Hamiltonian from rotation of bodies 6/11 Al Farabi Kazakh National University

The equations of motion and stability problem of orbit The orbital stability problem Time derivative of orbital angular momentum 7/11 Al Farabi Kazakh National University

The equations of motion and stability problem of orbit Equation of motion 8/11 Al Farabi Kazakh National University

The equations of motion and stability problem of orbit . We consider: . 9/11 Al Farabi Kazakh National University

Averaged equations To obtain the evolutionary equations of motion, . To obtain the evolutionary equations of motion, one needs to integrate For the repetition period of the system configurations The circular orbit of a test body is stable. 10/11 Al Farabi Kazakh National University

Conclusion . 11/11 Al Farabi Kazakh National University

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