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ON THE THREE INVARIANT RELATIONS OF MOTION’S EQUATIONS OF THE SYMMETRIC GYROSTAT IN A MAGNETIC FIELD

ON THE THREE INVARIANT RELATIONS OF MOTION’S EQUATIONS OF THE SYMMETRIC GYROSTAT IN A MAGNETIC FIELD
Shchetinina Elena, professor, doctor of mathematics and physics, full professor

Svetlana Skripnik, associate professor, candidate of mathematics and physics, associate professor

Donetsk National University of Economics and Trade named after M. Tugan-Baranovsky, Ukraine

Conference participant

УДК531.38

The problem of the gyrostat motion in a magnetic field with the effect of Barnett-London is considered. It is assumed that gyrostatic moment depends on the time. The conditions for the existence of the motion equations of three invariant relations of a special kind are defined. These solutions of the  motion equations are characterized by the elliptic functions of time.

Keywords: symmetric gyrostat, invariant relation,  magnetic field.

 

The problem of a gyrostat motion in a magnetic field with the Barnett-London effect describes the motion of neutral ferromagnet (not initially magnetized) in a magnetic field, which is the rotation of the magnetization along the axis of rotation, which is magnetized along the rotation axis in the rotation [1]. This effect is called the Barnett effect and is characterized by the appearance of the magnetic moment, which depends on the angular velocity. A similar phenomenon appears when rotating superconducting solid at fast rotation in the magnetic field (the London effect). The mechanism of magnetization in both cases due to various reasons, but the motion equations can be presented in the same form [1–3].

The motion equations of a gyrostat with variable gyrostatic moment in a magnetic field with the effect of Barnett-London have the form [1]

Here we introduce the notation: x=(x1, x2, x3)– a moment-of-momentum body-transmitter; v=(v1,v2,v3) – an unit vector indicating the direction of the magnetic field; ?=?(t) – the value gyrostatic moment ?(t)?; ?=(?1,?2,?3)  – constant unit vector; ?=(?ij)– gyration tensor; s=(s1,s2,s3) – a constant vector; B=(Bij), C=(Cij) – constant symmetric matrix of the third order; point above the variables denotes differentiation at time.

A case is examined when

Then from equations (1) will get expressions

Equation (2) allows to specify the first integral of (2)–(5)

here ?0 – an arbitrary constant.

Define invariant relations of equations (2)–(5)  as  

Equality on the parameters of equations (1) and invariant relations (7)

are the conditions for the existence of invariant relations (7).

Consideration of the case  leads to the relations

Function v1(t)t is found from integral

which is reduced to elliptic integral in the Legandre form. Thus, v1=v1(t)  is an elliptic function of time. According to the formula

?(t) is also an elliptic function of time.The other variables of the problem   can be determined respectively from (13), (7). The above functions describe a new solution of system  (1).

To reduce the problem of integrating the motion equations  in case  to quadrature we introduce new variables ? and ? instead vi

The equation

is obtained for the finding of function ?=?(t). On the basis (16) one can determine a function ?(t).

   

The functions  in (17), (18) can use in (7), (16) to obtain the dependence main variables  from time. Because of the structure of formula (17), all of the functions are elliptic functions of time.

Thus, in the paper, we obtained conditions for the existence of the equations (1) of three linear invariant relations of a special form (7). Two classes partial solutions of  motion equations, which are expressed in the form of elliptic functions of time, are specified.

 

References:

  1. Gorr G.V., Маznev А.V. The dynamics of a gyrostat with a fixed point. – Donetsk: DonNU, 2009. – 222 p. (in Russian)
  2. Volkova О.S., Gashenenko I.N. The pendulum motion of a heavy gyrostat with variable gyrostatic moment // Mechanics of rigid body. – 2009. – V. 39. – P. 42 – 49. (inRussian)
  3. Маznev А.V. Precessional motion of a gyrostat with variable gyrostatic moment under the action of potential and gyroscopic forces // Mechanics of rigid body. – 2010. – V. 40. – P. 91 – 104. (in Russian)
Comments: 3

Elena Artamonova

This article is interesting.In the paper, we obtained conditions for the existence of the equations of a gyrostat with variable gyrostatic moment in a magnetic field with the effect of Barnett-London.

Ivanova Tatiana Alecsandrovna

Немного неясно в тригонометрических функциях синуса и косинуса значения в градусах или радианах?И диференциал тау ,что дает в делимом формулы17.Такое поистине замечательное название статьи.Так хочется наглядного,графического представления движения. Всю глубину статьи я оенить так и не смогла.Что-то у меня с проверкой работ не ладится....Так то бывает.

Travnikov Yevgeniy Nikolayevich

Отличная статья, но надо было выделить выводы. Желаю дальнейших успехов, Доц. КПИ, гранд. конструктор ВПК СССР..
Comments: 3

Elena Artamonova

This article is interesting.In the paper, we obtained conditions for the existence of the equations of a gyrostat with variable gyrostatic moment in a magnetic field with the effect of Barnett-London.

Ivanova Tatiana Alecsandrovna

Немного неясно в тригонометрических функциях синуса и косинуса значения в градусах или радианах?И диференциал тау ,что дает в делимом формулы17.Такое поистине замечательное название статьи.Так хочется наглядного,графического представления движения. Всю глубину статьи я оенить так и не смогла.Что-то у меня с проверкой работ не ладится....Так то бывает.

Travnikov Yevgeniy Nikolayevich

Отличная статья, но надо было выделить выводы. Желаю дальнейших успехов, Доц. КПИ, гранд. конструктор ВПК СССР..
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