# Investigation of a test particle motion in the equatorial plane of the axially symmetric gravitational field in terms of the adiabatic theory

### Abstract

In this paper the motion of a test particle has been investigated in the gravitational field of a spherically symmetric central body employing the vector elements of orbits within general theory of relativity. In the literature this problem is known as the Schwarzschild problem. In order to solve this problem, we have used the Lagrange’s formalism, Hamilton’s formalism, averaging method, perturbation theory and adiabatic theory.The motion of the test particle has also been studied in an axially symmetric gravitational field. As a result the expression for the perihelion shift of planets’ orbit has been generalized by the quadruple moment of the central body. It was shown that the quadruple moment has contribution to the classical and relativistic corrections. All calculations have been conducted in approximations of (where c is the speed of light) and (quadruple moment).

In order to calculate the perihelion shift for the axially symmetric metric two different methods were used. In the first case, Hamilton’s canonical expressions have been used directly to obtain the equations of motion, and in the second case the theory of adiabatic invariants has been used. The adiabatic theory of bodies motion is a method for study the evolutionary motions in the mechanics of general theory of relativity. As a result the expressions obtained by two different ways coincided with each other and it was clearly shown that the adiabatic theory is more efficient than the first method.

The article is dedicated to the 80th anniversary of academician Meirhan Abdildin’s birth.

### References

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2. C.M. Will, Theory and experiment in gravitational physics. Revised edition, (Cambridge University Press, 1993).

3. R.M. Wald, General Relativity, (The University of Chicago Press, 1984), 473 p.

4. M.P. Hobson, G.P. Efstathio U and A.N.Lazenby, General Relativity, An Introduction for Physicists,(Cambridge University Press, 2006), 592 p.

5. L. Ryder, Introduction to General Relativity, (Cambridge University Press, 2009), 460 p.

6. B.F. Schutz, A First course in General Relativity, (Cambridge University Press, 2009), 412 p.

7. L.D. Landau and Е.М. Livshitz, Teoriya Polya. (М.: Fizmatlit, 2003). 536 p. (in Russ).

8. V.А. Fock, Teoriya prostranstva, vremeni i tyagotenya, (М.: Nauka, 1961), (in Russ).

9. М.М. Abdildin, Mechanica teorii gravitacii Einshteina, (Аlma-Аta: Nauka, 1988), 200 p. (in Russ).

10. М.М. Abdildin, Problema dvizhenya tel v obshei teorii otnositelnosti, (Almaty: Kazakh Universiteti, 2006). 132 p. (in Russ).

11. V.А. Brumberg, Reliativistskaya nebesnaya mechanica, (М.: Nauka, 1972).

12. М.М. Abdildin, F.B. Baimbetov, М.А. Jusupov, Т.А. Kojamkulov, Т.S. Ramazanov and М.S. Оmarov. Issledovanie problem fundamentalnyh vzaimodeistvi v teoreticheskoi physice, (Аlmaty,1997), 141 p.

13. L.D. Landau and Е.М. Livchitz, Mechanics (М.: Fizmatlit, 2004). 224 p. (in Russ).

14. H. Stephani, D. Kramer, M. A. H. MacCallum, C. Hoenselaers, and E. Herlt, Exact Solutions of Einstein’s Field Equations, (Cambridge University Press, Cambridge, UK, 2003).

15. K. Boshkayev, H. Quevedo and R. Ruffini, Physical Review D 86, 064043 (2012).

16. H.Queverdo and B.Mashhoon, Physics Letters A, 109 (1, 2), 13-18 (1985).

17. H.Queverdo and B.Mashhoon, Physics Letters A 148,149 (1990).

18. L.A. Pachon, J.A. Rueda and J.D.Sanabria-Gomez, Phys.Rev.D.73, 104038 (2016).

19. V.S.Manko, J.Martin and E. Ruiz, J.Math.Phys. 36, 3063 (1995).

20. V.S. Manko, J.D. Sanabria-Gomez and O.V.Manko, Phys.Rev.D. 62, 044048 (2000).

21. D. Bini, A. Geralico, O. Luongo and H. Quevedo, Classical and Quantum Gravity 26, 225006 (2009).

22. D. Bini, K. Boshkayev and A. Geralico, Classical and Quantum Gravity 29, 145003 (2012).

23. D.Bini, K. Boshkayev, R. Ruffini and I. Siutsou, Il Nuovo Cimento 36C, 1, 31-36 (2013).

24. H. Quevedo and L. Parkes, General Relativity and Gravitation 21, 1047 (1989).

25. K.A.Boshkaev, H.Quevedo, M.S.Abutalip, Zh.A.Kalymova and Sh.S.Suleymanova, IJMA. 31(2,3), 1641006 (2016).

Published

2018-03-24

How to Cite

BOSHKAYEV, K.A. et al.
Investigation of a test particle motion in the equatorial plane of the axially symmetric gravitational field in terms of the adiabatic theory.

**Recent Contributions to Physics (Rec.Contr.Phys.)**, [S.l.], v. 64, n. 1, p. 67-80, mar. 2018. ISSN 2663-2276. Available at: <https://bph.kaznu.kz/index.php/zhuzhu/article/view/858>. Date accessed: 01 nov. 2020.
Section

Theoretical Physics. Nuclear and Elementary Particle Physics. Astrophysics