ОБ УРАВНЕНИЯХ ДВИЖЕНИЯ ЗАДАЧИ ДВУХ ТЕЛ В МЕХАНИКЕ ОТО

9-ші Халықаралық ғылыми конференция «Физиканың заманауи жетістіктері  
Алматы, Қазақстан, 12-14 қазан,2016 
жəне іргелі физикалық білім беру» 
______________________________________________________________________________________________________ 
 
43 



















r
dt
d
m
m
m
m
r
M
p
M
p
A






0

.                          (6) 
Производные 
r

 и  p

 получим из уравнений Гамильтона 
H



r
p


,
H

 

p
r


.                       (7) 
Для нахождения гамильтониана задачи нам понадобится её лагранжиан, который опре-
делим по стандартной формуле 
dt
ds
mc
L


.                                                 (8) 
Окончательное выражение для функции Лагранжа имеет вид 
 






















0
2
0
2
4
2
2
2
2
2
4
4
3
2
2

c
m
mU
c
m
v
Uv
U
c
m
v
U
m
mc
L
v
U


 
  
 
1
∇
 
∇
7
2
0
0
2
0
r
c
m
m




S
S


,                                       (9) 
а соответствующий гамильтониан 














0
0
2
2
3
4
2
2
2
2
2
3
8
1
2

m
mU
mU
m
Up
m
p
c
mU
m
p
mc
H
 
 
  
 
1
∇
 
∇
7
2
 
2
0
0
2
0
0
2
3
r
c
m
m
c
r







S
S
p
r
S




.     (10) 
Учет  приливного  взаимодействия  определим  добавкой  к  ньютонову  потенциалу  цен-
трального тела 
0
m  
'
0
U
U
U


, 
r
m
U
0
0


,                            (11) 
где 
'
U
 - приливный (tidal) потенциал. 
Вычисляя  необходимые  производные  и  подставляя  их  в  уравнения (5) и (6), получим 
уравнения  поступательного  движения  рассматриваемой  задачи  в  представлении  векторных 
элементов орбиты 
М

и 
А

 
 
 
 
 
'
∇
 
'
7
12
2
0
0
2
2
0
2
5
0
0
0
2
3
U
m
U
v
m
E
c
m
m
c
r
m
m
c
r









r
S
r
r
S
M
S
M
























,      (12) 


 
 
 












M
r
M
S
A
S
M
A









2
5
0
0
2
3
2
0
0
0
0
6
2
∇
6
4
c
mr
c
r
mc
U
m
m
mU
E



 
 
   
   





















 
 
 
 
2
 
5
7
6
 
0
0
0
2
0
2
2
0
2
5
0
S
r
p
M
S
r
S
M
r
r
S











r
S
c
r
m

 



  


2
0
0
2
2
0
2
 
'
∇
'
∇
'
∇
'
2
mc
U
U
m
m
mU
mv
E
mc
U
c
U







r
p
M
M














. (13) 
Литература 
1  Фок В.А. Теория пространства, времени и тяготения. – Москва, 1961. – 563 с. 
2  Абдильдин М.М. Анализ некоторых задач механики теории тяготения Эйнштейна // 
Проблема движения в теории гравитации Эйнштейна. – Алма-Ата, 1981. – С. 3-41. 
3  Абдильдин М.М., Баимбетов Ф.Б., Жусупов М.А., Кожамкулов Т.А., Рамазанов Т.С., 
Омаров М.С. Исследование проблем фундаментальных взаимодействий в теоретической фи-
зике. - Науч. изд-е. – Алматы, 1997. - 141 с.  

The 9
th
 International Conference «Modern  
achievements of physics and fundamental physical education»  
 
October , 12-14, 2016, Kazakhstan, Almaty 
______________________________________________________________________________________________________
 
 
44 
 
NONLINEAR EQUATION OF QUARK-GLUON CASCADE 
 
A.T. Temiraliev
1
, I.A. Lebedev
1
, A.K.Danlybaeva
2 
 
1
Institute of Physics and Technology, Almaty, Kazakhstan 
2
Al-Farabi Kazakh National University, Almaty, Kazakhstan 
 
Consideration of the contribution to the quark-gluon distribution of bremsstrahlung of gluons leads 
to a violation of Bjorken’s scaling and is determined by known linear evolution equations:  Dok-
shitzer-Gribov-Lipatov-Altarelli-Parisi (DGLAP) [1-3], Balitsky-Fadin-Kuraev-Lipatov (BFKL) [4-
5] and Gribov-Levin-Ryskin-Mueller-Qiu (GLR-MQ) [6-7]. Proposed many ways to the modeling 
of evolution equations with non-perturbativenonlinearities,considering the gluon recombination. In 
addition to the gluon splitting functions, the nonlinear gluon recombination processes become im-
portant. The action Yang-Mills (Y-M) already contains cubic and quartic nonlinearinteraction terms 
in the field strength tensor: 
= −
 
( )
( ). As is known, problems arise in the 
mathematical method of describing quantum chromodynamics at large distances, when perturbation 
theory for the decomposition in α
s
(Q
2
) is not applicable. Opportunities for the formation of regular 
structures associated with the effective competition of different types of interactions: mergers and 
splittings of quarks and gluons. Under the influence of the quantum fluctuations of the amplitudes 
of the processes there are structures which have some scale with complex self-organization. Con-
sideration of the contribution to the quark-gluon distribution of gluons bremsstrahlung leads to a 
violation of Bjorken’s scaling and it is determined by known linear evolution equations. There are 
different approaches to accounting for mergers with non-perturbativenonlinearitiesat the gluon re-
combination. Considering the evolution as a discrete quantum process we use the mathematical ap-
paratus of mappings within the framework of nonlinear dynamics theory. In the spirit of Feynman’s 
path integrals we propose [9] a nonlinear stochastic equation in the form of the evolution of nucleon 
structure functionF
2
(x,Q
2
), which represents the evolution nonlinear operator showing the distribu-
tion in the momentum representation: 
=
∙ ( , ).Using the method of Poincare sections 
(сhoosing the share of momentum as a one-dimensional section of the phase space of partons mo-
mentum distribution) we have an evolution equation 
=
∙ ( ). Here Bjorken’s/Feynman’s 
variable x
t
 is the momentum fraction at discrete time index (t=0,1,2…) and R is the control parame-
ter that characterizes the degree of coupling embossed parton with the totality of the remaining par-
tons in the nucleon at the certain energy √  and determines the character of observing regimes. To 
switch to continuous time allows the build, known as the Poincare section. In the framework of our 
quality approach we use the renormalization-group approach to the evolution equation, allowing to 
recreate a physical picture of the critical behavior. So for the quark-gluon cascade, we enter an 
iterative map in which a number of the quarks and gluons in (t+1)-th generation are proportional to 
the number of them in t-th generation. The number of partons are changing, but remains on total 
momenta. Thus, the probability to find a parton with a fraction of momentum x at time t+1 is de-
fined by the impulse distribution of partons in the time t.  Positive terms of hadron structure func-
tion meet the increasing of the quarks (q) and gluons (g) number at cascade: qq+g and gg+g 
and negative terms is the reduction, i.e. quark-antiquark, quark-gluon and gluon-gluon recombina-
tion. Using the method of Poincare sections (сhoosing the share of momentum as a one-dimensional 
section of the phase space of partons momentum distribution) and considering that the evolution 
operator is determined by hadron structure functions (F
2
), we use a one-dimensional map.  
Numerical solution of the nonlinear equation has shown the existence of an evolution termi-
nation in the field of small values of parameter. Small perturbations do not change the Q-G condi-
tion (R<<1). The increase in R leads at first only to the excitation stable state. With further increase 

9-ші Халықаралық ғылыми конференция «Физиканың заманауи жетістіктері  
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жəне іргелі физикалық білім беру» 
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45 
of the parameter occur repeated bifurcation (splitting) of period-doubling calculations of the quarks 
phase trajectories have shown the presence of the chaotic dynamics at R>>0 as a consequence of 
bifurcations. In a state of dynamic chaos two close orbits in phase space diverge exponentially with 
time with Lyapunov’s coefficient in the exponent ( =
| |
), which in a computer simulation, is 
calculated using parallel running of two close initial conditions and examines their divergence. By 
computer simulation the studies of the formation of stable structures inquark-gluon cascade, includ-
ing recombination processes. The nature of stability of fixed points (cycles) and the type of bifurca-
tions of mappings are determined by their multipliers. In turn, multipliers are the own numbers of 
the Jacobian matrix perturbations. The maximum value x
t+1 
 is found from dx
t+1
/dx
t
=0. The Jacobian 
is  =
 and the map is stable at a point x
0
if J(x
0
)<1. When the coupling constant α
s
(Q
2
) is 
small, the evolution is incoherent, if the relationship is strong enough that can occur spontaneous 
synchronization quark-gluon movements. Dynamic quark-gluon systems are highly sensitive to the 
initial conditions. There are nonperturbative effects associated with initial transverse momenta of 
partons inside the hadron and there are always fatal even quantum zero fluctuations. It is possible 
that a steady structure formation in nonlinear quark-gluon evolution is a mechanism of hadroniza-
tion. Arising in the quark-gluon cascade the strange attractor with a fractal self-similar structure 
display a new nonlinear phenomenon in the hadron physics is deterministic chaotic dynamics. Self-
similarity is related to the so-called power-law dependence on parameters. Dynamic quark-gluon 
systems are highly sensitive to the initial conditions. There are non-perturbative effects associated 
with initial transverse momenta of partons inside the hadron and there are always fatal even quan-
tum zero fluctuations. It is possible that a steady structure formation in nonlinear quark-gluon evo-
lution is a mechanism of hadronization. Arising in the quark-gluon cascade the strange attractor 
with a fractal self-similar structure display a new nonlinear phenomenon in the hadron physics is 
deterministic chaotic dynamics.  
 
References 
1.  Gribov V.N., Lipatov L.N. “Deep inelastic ep scattering in perturbation theory” // Sov. J. 
Nucl. Phys. 1972. – Vol. 15. – P. 438. 
2.  Докшицер Ю.Л. “Вычисление структурных функций глубоко неупругого рассеяния 
в e
+
e
-
 аннигиляции по теории возмущений КХД” // ЖЭТФ, 1977. − Т.73. − С.1216. 
3.  Altarelli G., Parisi G. Asymptotic freedom in parton language//Nucl. Phys. B126 (1977) 
298 
4.  Kuraev E.A. and Fadin V.S. «On radiative corrections to the cross section for single - 
photon annihilation of an e
+
 e
-
 - pair at high energy» // J. Nucl. Phys. 1985. Vol. 41. P.3. 
5.  Липатов  Л.Н. «Свойство  интегрируемости  в  квантовой  хромодинамике  при 
большом числе цветов» //УФН. 2004. – Т.174, №4. –С. 337-352  
6.  Mueller AH and Qiu J. Gluon Recombination and Shadowing at Small Values of x //Nucl 
Phys B268 (1986) 427  
7.  MAYURI DEVEE and J K SARMA Analytical Approach for the Solution of the 
Nonlinear GLR-MQ Equation //Proc Indian Natn Sci Acad 81 No. 1 2015, p. 16-21 
8.  Rasool, M.H., Ahmad, M.A. and Ahmad, S. «Slow Particle Production in Nucleus-
Nucleus Collisions at Relativistic Energies» //Journal of Modern Physics, 7, 51-64, 2016 
http://dx.doi.org/10.4236/jmp.2016.71006   
9.  Темиралиев А.Т., Данлыбаева А.К «Формирование структур в нелинейной кварк-
глюонной эволюции»// Известия НАН РК серия физ-мат 2014 №2. 
 
 

The 9
th
 International Conference «Modern  
achievements of physics and fundamental physical education»  
 
October , 12-14, 2016, Kazakhstan, Almaty 
______________________________________________________________________________________________________
 
 
46 
 
MAGNETIC FIELD CONSTRAINTS FROM MICROQUASAR QPOS. 
 
A. Tursunov and M. Kološ 
 
Institute of Physics and Research Centre of Theoretical Physics and Astrophysics,  
Faculty of Philosophy & Science, Silesian University in Opava,  
Bezručovo náměstí 13, CZ-74601 Opava, Czech Republic 
Arman.Tursunov@fpf.slu.cz
, 
Martin.Kolos@fpf.slu.cz
 
 
Due to the possibilities of achievement of high precisions in the measurements of the frequencies of 
receivedsignals one can get useful information about the central object and theelectromagnetic 
fields in its vicinity from the analysis of obtained frequencies. In order to test the role of large-scale 
magnetic fields in quasi-periodicoscillation (QPO) phenomena observed in microquasars, we study 
oscillatory motion of test charged particles in the vicinity of a rotating black hole immersed into an 
externalasymptotically uniform magnetic field.Investigation of the high-frequency QPOs observed 
inmany black hole or neutron star low-mass X-ray binaries can open up prospectsof understanding 
the phenomena occurring in the presence of strong gravity. Some of HF QPOs come in pairs of the 
upper and lower frequencies of twin peaksin the Fourier power spectra. We determine the funda-
mental frequenciesof small harmonic oscillations of charged test particles around stable circular or-
bitsin the equatorial plane of a magnetized black hole[1,2], and discuss the frequencies of the radial 
and latitudinal harmonic oscillations [3] in dependence on themass of the black hole and the 
strength of the magnetic field. We demonstrate thatassuming relevance of resonant phenomena of 
the radial and latitudinal oscillations ofcharged particles at their frequency ratio 3 : 2 or 2 : 3, as 
theobserved values of the twin HF QPO frequencies for the sources show clear ratio
 
: 
= 3 ∶ 2 
[4], the oscillatory frequencies of chargedparticles can be well related to the frequencies of the twin 
HF QPOs observed in the microquasars GRS 1915+105, XTE 1550-564 and GRO 1655-40. 
The procedure of fitting the charged particle oscillation frequencies to the observedfrequen-

9-ші Халықаралық ғылыми конференция «Физиканың заманауи жетістіктері  
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47 
cies is presented in the figure, for all the three microquasars. From the restrictions on the mass of 
the central objectM for each of the sources and the parameter of the rotation a, we obtain simulta-
neously restrictionson the magnetic field parameter which is proportional to the strength of the 
magnetic field and the specific charge of a test particle. According to our results the strong magnet-
ic fields are not necessary. For electrons to obtain such frequencies the strength should be of order 
of ratio
 
~ 0.1 mG, which is comparable to themagnetic field strength in the heliosphere. For 
protons 
 
~ 1 G, which is comparableto the Earth’s magnetic field at its surface, and for partially 
ionized (one electron lost) ironatom 
~ 10 G is comparable to the magnetic field strength in the 
Earth’s core. The exact values of the magnetic field differ for each of microquasars but they have 
the same order of magnitude. The inverse estimations of the mass of the test body with one electron 
lost forthe fixed values of magnetic field show that for the magnetic field of   ~ 10 G , the mass of 
the oscillating object should be of order   ~ 10
g , what is comparable with the mass of the 
cosmic dust grains. 
References 
[1] A. Tursunov, M. Kološ, and Z. Stuchlík, Circular orbits and related quasiharmonic oscilla-
tory motion of charged particles around weaklymagnetized rotating black holes, Phys. Rev. D 93, 
084012 (2016). 
[2] M. Kološ, Z. Stuchlík and A. Tursunov, Quasi-harmonic oscillatory motion of charged 
particles around a Schwarzschild black holeimmersed in a uniform magnetic field, Class. Quan. 
Grav. 32, 165009 (2015). 
[3] A. N. Aliev and D. V. Galtsov, General Relativity andGravitation 13, 899 (1981). 
[4] J. E. McClintock, R. Narayan, S. W. Davis, L. Gou,A. Kulkarni, J. A. Orosz, R. F. Penna, 
R. A. Remillard, and J. F. Steiner, Classical and Quantum Gravity 28, 114009 (2011). 
 
 
 
 
 
 

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