1. 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

2. 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
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3. 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
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4. 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
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5. Stability in two rotating bodies problem𝑟
𝑟 2 m1
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6. 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
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7. 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.
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8. The Hamiltonian of three bodies
Additive terms to Hamiltonian from rotation of bodies
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9. The equations of motion and stability problem of orbit
The orbital stability problem
Time derivative of orbital angular momentum
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10. The equations of motion and stability problem of orbit
Equation of motion
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11. The equations of motion and stability problem of orbit.
We consider: .
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12. 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.
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13. Conclusion .
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14. for your kind attention!
Thank you
for your kind attention!
Al Farabi Kazakh National University 14