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Светлана Скрыпник, доцент, кандидат физико-математических наук, доцент
Донецкий национальный университет экономики и торговли им. М. Туган-Барановского, Украина
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 . 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 
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
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.