About orbital stability of a trial particle

in the problem of three bodies

U.N.Zakirov

Institute of Mechanics and Engineering, Kazan Science Center of RAS

Lobachevsky, 2/31, Kazan, 420111, RUSSIA

zakirov@kfti.knc.ru

In this article the opportunities of the field theory of Kaluza-Wesson (five- dimensional theory of a relativity [1]), simulating a conservative system of binary Stars interacting among themselves are demonstrated; thus the crossflow of mass-energy of a shell of Stars is implemented through Lagrange point L₁; the ablation of energy through Lagrange point L₂ is neglected. The estimation of perturbation of the Star 2 from the Star 1 is implemented by means of the solution of the equation of a deviation by the method of M.F.Schirokov on the basis of the known metric of the perturbing Star; the Star 2 (black hole, neutron Star) with a strong gravitational field surrounded by the accretion disk is considered as a test. Thus, the mass of Stars is considered to be variable. In these requirements the orbital stability of a trial particle propellenting around the black hole, depends on the perturbation of the Star 1, supplementing the results [3].

 

Alongside with the known parameters describing the disturbing body (Star 1), are possible experimentally not found, but theoretically reasonable particles, for example, magnetic charge, the solution GRG with allowance of magnetic charge as the solution NUT (Newman, Unti, Tamburino).

According to our approach, by the method of M.F. Schirokov can be determined the perturbation of total energy, including energy of a monopole and a count in the scalar field of g₅₅, then with the help of Lagrange-Dirichlet function it will be possible to analyze the influence on orbital stability of a magnetic charge (monopole).

In some articles the research (problem 2 bodies) of orbital stability of the trial particle in frameworks GRG (4D) have been conducted; in particular, in the article the global analytical approximation of the metric of the external field of quick-moving neutron Star by the precise solution of Einstein's equations (4D) in the assignment (task) of mass of the Star of its angular momentum and quadrupole distribution of masses is used. The similar researches within the framework of the theory of Kaluza-Wesson in 5D in the present article are not affected. Our approach with the usage 5D allows to solve analytically the problem 3 and more bodies. It is easy to take into account the oblateness of the disturbing body taking into account the solution of M.F. Schirokov.

The model, offered can be interpreted as the controllable system with variable structure instituted by dimensionally of space-time; at b=0љ or g₅₅=0љ when ws=lp, l=0, +1 implements transition to 4љ dimensional metric solved with the help of nonlinear Einstein's equations are carried out R^ab- g^abR = T^ab, and equations of a deviation solved by the method of M.F. Schirokov; at the transition is recovered to 5- dimensional metrics with a requirement independent from 5 ws¹lp, b¹0 till; then the change of masses of the Star pair is determined from the conservation law of the induced tensor energy-impulse appearing from the vacuum solution of the theory of Kaluza-Wesson ⁵R_AB=0.

So, in the present article we have given the physical interpretation of the problem about orbital stability within the framework of the generalized theory of gravitation (5D), that is stimulant of the count at analysis of gravity waves and for a detail mathematical analysis of stability of the indicated physical problems within the framework of Riemannian geometry, which was born simultaneously with the genius work on stability by A.M. Lyapunov.