Investigation of a test particle motion in the equatorial plane of the axially symmetric gravitational field in terms of the adiabatic theory
Keywords:
Vector elements, Schwarzschild’s metric, quadrupole moment, equations of motion, Lagrangian formalism, Hamiltonian formalism, adiabatic theoryAbstract
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.
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