On the Qualitative Analysis of the Equations of Motion of a Rigid Body in a Magnetic Field

Abstract

In the problem on the motion of a rigid body with a fixed point under the influence of a magnetic field generated by the Barnett–London effect as well as potential forces, the particular cases of existence of additional quadratic integrals are presented and the qualitative analysis of the equations of motion of the body is carried out in one of these cases.

INTRODUCTION

The problem on the rotation of a rigid body about a fixed point under the action of forces of various nature (and also in the absence of forces) has a long history, but the interest in this problem still persists. An absolutely rigid body is used as a model for describing the motion of many complex technical devices, including spacecraft, industrial robots, rockets, etc. In the present paper, we consider the problem on the rotation of a rigid body with a fixed point in a uniform magnetic field taking into account the London–Barnett effect and potential forces. It is known that a “neutral” ferromagnet becomes magnetized along the rotation axis during rotation (the Barnett effect [1]). A similar phenomenon also takes place during the rotation of a superconducting solid body (the London effect [2]). The magnetic moment $\mathbf {B}$ is related to the angular velocity $\boldsymbol \omega$ by the formula $\mathbf B = B\boldsymbol \omega$, where $B$ is a symmetric linear operator.

The motion of the body is described by Euler–Poisson equations of the form

$$ A \dot {\boldsymbol \omega } = A \boldsymbol \omega \times \boldsymbol \omega + B \boldsymbol \omega \times \boldsymbol \gamma + \boldsymbol \gamma \times (C \boldsymbol \gamma - \mathbf {s}), \quad \dot {\boldsymbol \gamma } = \boldsymbol \gamma \times \boldsymbol \omega , $$

(1)

where $\boldsymbol \omega = (\omega _1, \omega _2, \omega _3)$ is the angular velocity of the body, $\boldsymbol \gamma = (\gamma _1,\gamma _2, \gamma _3)$ is a unit vector characterizing the direction of gravity, $\mathbf {s} = (s_1, s_2, s_3)$ is the vector of the center of mass of the body, and $A$, $B$, and $C$ are symmetric $3\times 3$ matrices: $A$ is the body tensor of inertia relative to the fixed point, $B$ is a matrix characterizing the body magnetic moment, and $C$ is a matrix characterizing the effect of potential forces on the body.

For $C_i = \nu A_i$, $i = 1, 2, 3$, where $\nu$ is the gravitational constant, the differential equations (1) describe the motion of a body in a magnetic and a central Newtonian field.

Equations (1) admit two common first integrals

$$ V_1 = A \boldsymbol \omega \cdot \boldsymbol \gamma = \varkappa , \quad V_2 = \boldsymbol \gamma \cdot \boldsymbol \gamma = 1 $$

(2)

and are nonintegrable in the general case.

There are quite a few papers studying the influence of the Barnett–London effect on the motion of a body in various aspects. Similar problems arise in many applications, for example, in cosmodynamics [3] and when designing devices using noncontact suspension [4]. The analysis of Eqs. (1) from the viewpoint of their integrability and the search for particular solutions is carried out, for example, in [5,6,7,8]. A linear invariant relation of the Hess type [9] for Eqs. (1) was found in [5]. In [6, 7], the cases of their integrability are indicated where $A$ and $B$ are diagonal matrices and there are no potential forces. It was shown [7] that for $B = \lambda E$ $(\lambda = \mathrm {const})$ Eqs. (1) are reduced to the Kirchhoff equations describing the motion of a rigid body in an ideal fluid.

In the present paper, it is assumed that

$$ A =\mathrm {diag}\thinspace (A_1, A_2, A_3),\quad C = \mathrm {diag}\thinspace (C_1, C_2, C_3) $$

in Eqs. (1). A qualitative analysis of these equations is carried out in special cases where there exist additional first integrals. Linear invariant relations for Eqs. (1) were obtained in [10] using the method of indeterminate coefficients in conjunction with the method of Gröbner bases [11]. In the same way, the following quadratic integrals of the equations in question are found under certain restrictions on the problem parameters:

• 1. For $A_1 = A_2$, $B_{13} = B_{23} = 0$, $B_{33} = B_{11} + B_{22}$, $C_1 = C_3 = C_2$, and $s_1 = s_2 = s_3 = 0$, one has the integral

  $$ \begin {aligned} K_1 &= \omega _1^2 + \omega _2^2 + \frac { A_3}{A_2^2} (2 A_2 - A_3) \omega _3^2 - \frac {1}{A_2^2} \big (2 A_2 (B_{11} - B_{22}) \omega _1 \gamma _1 + 2 A_2 B_{12}\big (\omega _1 \gamma _2 + \omega _2\gamma _1) \\ &\qquad {}+2 A_3 B_{11} \omega _3 \gamma _3 - \big (B_{22}^2 - B_{11}^2\big ) \gamma _1^2 + 2 B_{12} (B_{11} + B_{22})\gamma _1 \gamma _2 + (B_{11}^2 + B_{12}^2) \gamma _3^2 \big ). \end {aligned} $$

  (3)

• 2. For $A_1 = A_3$, $B_{12} = B_{23} = 0$, $B_{22} = B_{11} +B_{33}$, $C_1 = C_2 = C_3$, and $s_1 = s_2 = s_3 = 0$, one has the integral

  $$ \begin {aligned} K_2 &= \omega _1^2 - \frac {A_2}{A_3^2} (A_2 - 2 A_3) \omega _2^2 + \omega _3^2 + \frac {1}{A_3^2} \Big (2 A_3 \big (B_{33} \omega _1 \gamma _1 + B_{11} \omega _3 \gamma _3 - B_{13} (\omega _1 \gamma _3 + \omega _3 \gamma _1)\big ) \\ &\qquad {}+ \big (B_{33}^2 - B_{11}^2\big ) \gamma _1^2 - \big (B_{11}^2 + B_{13}^2\big ) \gamma _2^2 - 2 B_{13} (B_{11} + B_{33}) \gamma _1 \gamma _3\Big ). \end {aligned} $$

• 3. For $A_2 = A_3$, $B_{12} = B_{13} = 0$, $B_{33} = B_{11} -B_{22}$, $C_1 = C_2 = C_3$, and $s_1 = s_2 = s_3 = 0$, one has the integral

  $$ \begin {aligned} K_3 &= \omega _1^2 - \frac {A_3^2 \big (\omega _2^2 + \omega _3^2\big )}{A_1 (A_1 - 2 A_3)} + \frac {1}{A_1 (A_1 - 2 A_3)} \big (2 A_1 B_{22} \omega _1 \gamma _1 - 2 A_3 (B_{11} - 2 B_{22}) \omega _2 \gamma _2 \\ &\qquad {}+ 2 A_3 B_{23} (\omega _2 \gamma _3 + \omega _3 \gamma _2) + \big (B_{22}^2 + B_{23}^2\big ) \gamma _1^2 - B_{11} (B_{11} - 2 B_{22}) \gamma _2^2 + 2 B_{11} B_{23} \gamma _2 \gamma _3\big ). \end {aligned} $$

As can be seen, the integrals exist under the condition of dynamic symmetry of the body, and the coordinates of the center of mass of the body coincide with the coordinates of the fixed point.

Further, using generalizations of the Routh–Lyapunov method [12], a qualitative analysis of Eqs. (1) is carried out in the case where these equations admit one of the above quadratic integrals.

1. STATEMENT OF THE PROBLEM

Consider the differential equations (1) for the case in which they admit the integral $K_1$ in (3). The equations are written as follows:

$$ \begin {gathered} \begin {aligned} A_2 \dot \omega _1 &= (B_{12} \omega _1 + B_{22} \omega _2) \gamma _3 - \big ((B_{11} + B_{22}) \gamma _2 + (A_3 - A_2) \omega _2\big ) \omega _3, \\ A_2 \dot \omega _2 &= \big ((B_{11} + B_{22}) \gamma _1 + (A_3 - A_2) \omega _1\big ) \omega _3 - (B_{11} \omega _1 + B_{12} \omega _2) \gamma _3, \\ A_3 \dot \omega _3 &= (B_{11} \omega _1 + B_{12} \omega _2) \thinspace \gamma _2 - (B_{12} \omega _1 + B_{22} \omega _2) \gamma _1, \end {aligned} \\ \dot \gamma _1 = \omega _3 \gamma _2 - \omega _2 \gamma _3, \quad \dot \gamma _2 = \omega _1 \gamma _3 - \omega _3 \gamma _1, \quad \dot \gamma _3 = \omega _2 \gamma _1 - \omega _1 \gamma _2. \end {gathered} $$

(4)

The integrals (2) become

$$ \widetilde V_1 = A_2 ( \omega _1 \gamma _1 + \omega _2 \gamma _2) + A_3 \omega _3 \gamma _3 = \varkappa , \quad V_2 = \sum _{i=1}^2 \gamma _i^2 = 1. $$

(5)

Let us pose the problem of qualitative analysis of this system. Based on the necessary conditions for the extremum of the first integrals of the problem (or some combination of them), special sets of differential equations will be found and their stability in the sense of Lyapunov will be studied. With the chosen method of analysis, the special sets are defined as stationary sets [12], i.e., sets of any finite dimension on which necessary conditions for the extremum of elements of the algebra of first integrals of the problem are satisfied. Stationary sets of dimension zero are traditionally called stationary solutions, and stationary sets of nonzero dimension are called stationary invariant manifolds.

2. ISOLATION OF STATIONARY SOLUTIONS AND INVARIANT MANIFOLDS

In accordance with the above-indicated method, we take a linear combination

$$ 2 \Omega = \lambda _0 K_1 - 2 \lambda _1 \widetilde V_{1} -\lambda _2 V_{2} $$

(6)

of the first integrals and write necessary conditions for the extremum of $\Omega$ with respect to the variables $\omega _i$ and $\gamma _i$,

$$ \begin {aligned} \frac {\partial \Omega }{\partial \omega _1} &= \lambda _0 (\omega _1 - A_2^{-1} ((B_{11} - B_{22}) \gamma _1 + B_{12} \gamma _2)) - \lambda _1 A_2 \gamma _1 = 0,\\ \frac {\partial \Omega }{\partial \omega _2} &= \lambda _0 \big (\omega _2 - A_2^{-1} B_{12} \gamma _1\big ) - \lambda _1 A_2 \gamma _2 = 0, \\ \frac {\partial \Omega }{\partial \omega _3} &= \lambda _0 A_2^{-2} A_3 \big ((2 A_2 - A_3) \omega _3 - B_{11} \gamma _3\big ) - \lambda _1 A_3\gamma _3 = 0, \\ \frac {\partial \Omega }{\partial \gamma _1} &= - \lambda _0 A_2^{-2} \big (A_2 (B_{11} - B_{22}) \omega _1 + A_2 B_{12} \omega _2 + (B_{11}^2 - B_{22}^2) \gamma _1 \\ &\qquad {}+ B_{12} (B_{11} + B_{22}) \gamma _2\big ) - \lambda _1 A_2 \omega _1 - \lambda _2 \gamma _1 = 0, \\ \frac {\partial \Omega }{\partial \gamma _2} &= - \lambda _0 A_2^{-2} B_{12}\big (A_2 \omega _1 + (B_{11} + B_{22}) \gamma _1\big ) - \lambda _1 A_2 \omega _2 - \lambda _2 \gamma _2 = 0, \\ \frac {\partial \Omega }{\partial \gamma _3} &= - \lambda _0 A_2^{-2} \big (A_3 B_{11} \omega _3 + (B_{11}^2 + B_{12}^2) \gamma _3\big ) - \lambda _1 A_3 \omega _3 - \lambda _2 \gamma _3 = 0. \end {aligned} $$

(7)

Here the $\lambda _i$ are the parameters of the family of integrals $\Omega$.

In the case of dependent equations, the solutions of system (7) permit one to determine the invariant manifolds of the differential equations (4) corresponding to the family of first integrals $\Omega$. To find the solutions, we use the system of computer algebra Wolfram Mathematica.

As can be seen, Eqs. (7) can be separated in the variables. Let us construct a lexicographic Gröbner basis with respect to $\lambda _1 > \lambda _2> \gamma _1 > \omega _1$ for the left-hand sides depending on $\omega _1$, $\omega _2$, and $\gamma _1$, $\gamma _2$. As a result, we obtain a system of equations splitting into two subsystems. Below we give the lexicographic bases of these subsystems:

$$ \begin {aligned} \;\quad \big (B_{12}^2 - B_{11} B_{22}\big ) \gamma _2 - A_2 (B_{12} \omega _1 + B_{22} \omega _2) &= 0, \\ -B_{12} \gamma _1 + B_{11} \gamma _2 + A_2 \omega _2 &= 0, \\ \lambda _0 \big (B_{11}^2 + B_{12}^2\big ) + \lambda _2 A_2^2 &= 0, \\ -\lambda _0 B_{11} - \lambda _1 A_2^2 &= 0; \end {aligned} $$

(8)

$$ \begin {aligned} -B_{12} \omega _1^2 + ((B_{11} - B_{22}) \omega _1 + B_{12} \omega _2) \omega _2 &= 0, \\ - \omega _1 \gamma _2 + \omega _2 \gamma _1 &= 0, \\ -\big (A_2^2 \omega _2^2 + B_{12} (B_{11} + B_{22}) \omega _1 \gamma _2^2 \big ) \lambda _0 - A_2^2 \omega _2 \gamma _2^2 \lambda _2 &= 0,\quad \; \\ \big (B_{12} \omega _1 \gamma _2 - A_2 \omega _2^2\big ) \lambda _0 + A_2^2 \omega _2 \gamma _2 \lambda _1 &= 0. \end {aligned} $$

(9)

From the last two equations in (8), we find

$$ \lambda _1 = -\frac {B_{11}}{A_2^2} \lambda _0, \quad \lambda _2 = -\frac {B_{11}^2 + B_{12}^2}{A_2^2} \lambda _0 $$

(10)

and substitute them into the remaining equations (7) (depending on $\omega _3$ and $\gamma _3$). These equations are reduced to one equation $\omega _3 =0$. It can be verified by a straightforward calculation according to the definition of invariant manifold that the equations

$$ \begin {aligned} \big (B_{12}^2 - B_{11} B_{22}\big ) \thinspace \gamma _2 - A_2 (B_{12} \omega _1 + B_{22} \omega _2)&= 0, \\ A_2 \omega _2 - B_{12} \gamma _1 + B_{11} \gamma _2 &= 0, \\ \omega _3 &= 0 \end {aligned} $$

(11)

define an invariant manifold of codimension 3 for the equations of motion (4).

The differential equations on this invariant manifold are written in the form

$$ \begin {aligned} \dot \omega _1 &= \frac {B_{12}^2 - B_{11} B_{22}}{A_2^2} \gamma _2 \gamma _3, \\ \dot \gamma _2 &= \omega _1 \gamma _3, \\ \dot \gamma _3 &= - \omega _1 \gamma _2 + ( A_2 \omega _1 - B_{12} \gamma _2) \bigg (\frac { B_{12}}{B_{22}^2} \omega _1 + \frac {B_{11} B_{22} - B_{12}^2}{A_2 B_{22}^2}\gamma _2 \bigg ) \end {aligned} $$

(12)

and describe pendulum-like oscillations of the body.

Let us substitute $\lambda _1$ and $\lambda _2$ given by (10) into (6). By a straightforward calculation it can also be verified that the integral

$$ 2 \Omega _1 = K_1 + \frac {2 B_{11}}{A_2^2} \widetilde V_{1} + \frac {B_{11}^2 + B_{12}^2}{A_2^2} \thinspace V_{2} $$

(13)

takes a stationary value on the invariant manifold (11).

In a similar way, based on Eqs. (9), we obtain the equations

$$ \begin {aligned} -B_{12} \omega _1^2 + \big ((B_{11} - B_{22})\omega _1 + B_{12}\omega _2\big ) \omega _2 &= 0, \\ - \omega _1 \gamma _2 + \omega _2 \gamma _1 &= 0, \\ \omega _3 &= 0, \\ \gamma _3 &= 0 \end {aligned} $$

(14)

defining an invariant manifold of codimension 4.

The differential equations $\dot \omega _2 = 0$ and $\dot \gamma _2 = 0$ on this invariant manifold have the following family of solutions:

$$ \begin {aligned} \omega _2 = \omega _2^0 &= \mathrm {const}, \\ \gamma _2 = \gamma _2^0 &= \mathrm {const}. \end {aligned} $$

Thus, from the geometrical viewpoint, the invariant manifold (14) in the space $\mathbb {R}^6$ is associated with a surface whose each point is a fixed point in the phase space.

Using the maps of some atlas on the invariant manifold (14), one can readily show that the integral

$$ \Omega _2 = -\frac {1}{4 A_2^2 V_2} \Big (2 \widetilde V_1^2 - 2 z_1 \widetilde V_1 V_2 - V_2 \big ((B_{11} + B_{22}) z_1 V_2 + 2 A_2^2 K_1\big ) \Big ), $$

(15)

where $z_1 = B_{11} - B_{22} - D$ and $D = \sqrt {(B_{11} - B_{22})^2 + 4 B_{12}^2}$, takes a stationary value on the invariant manifold (14) in the map

$$ \begin {aligned} \omega _1 &= \frac {z_1 \omega _2}{2 B_{12}}, &\quad \omega _3 &= 0, \\ \gamma _1 &= \frac {z_1 \gamma _2}{2 B_{12}}, &\quad \gamma _3 &= 0, \end {aligned} $$

(16)

and the integral

$$ \Omega _3 = -\frac {1}{4 A_2^2 V_2} \Big (2 \widetilde V_1^2 - 2 z_2 \widetilde V_1 V_2 - V_2 \big ((B_{11} + B_{22}) z_2 V_2 + 2 A_2^2 K_1\big ) \Big ), $$

(17)

in the map

$$ \begin {aligned} \omega _1 &= \frac {z_2 \omega _2}{2 B_{12}}, &\quad \omega _3 &= 0, \\ \gamma _1 &= \frac {z_2 \gamma _2}{2 B_{12}}, &\quad \gamma _3 &= 0. \end {aligned} $$

(18)

Here $z_2 = B_{11} - B_{22} + D$.

Two more invariant manifolds different from the ones above and a condition on the parameters $\lambda _1$ and $\lambda _2$ under which the integral $\Omega$ (6) takes a stationary value on these invariant manifolds can be obtained by constructing a lexicographic basis for the polynomials of the entire system (7) for $\omega _3 > \omega _1 > \omega _2 >\gamma _1 > \lambda _2 > \lambda _1$. The equations of the invariant manifold are written in the form

$$ \begin {aligned} 2 B_{12} \gamma _1 - (B_{11} - B_{22} \pm D) \gamma _2 &= 0, \\ 2 (A_2 - A_3) \omega _2 - (B_{11} + B_{22} \mp D) \gamma _2 &= 0,\\ 2 (A_2 - A_3) B_{12} \omega _1 + \big (2 B_{12}^2 - B_{22} (B_{11} - B_{22} \pm D)\big ) \gamma _2 &= 0, \\ 2 (A_2 - A_3) \omega _3 - (B_{11} + B_{22} \mp D) \gamma _3 &= 0. \end {aligned} $$

(19)

The differential equations on these invariant manifolds are similar to the equations on the invariant manifolds (14),

$$ \begin {aligned} \dot \gamma _2 &= 0, \\ \dot \gamma _3 &= 0. \end {aligned} $$

The integral $\Omega$ in (6) takes a stationary value on the invariant manifold (19) for

$$ \begin {aligned} \lambda _1 &= \frac {\lambda _0}{2 A_2^2 (A_2 - A_3)}\big (A_3 (B_{11} - B_{22}) + 2 A_2 B_{22} \mp (2 A_2 - A_3) D\big ), \\ \lambda _2 &= -\frac {\lambda _0}{2 A_2^2 (A_2 - A_3)^2} \big (2 A_2^2 (B_{11}^2 + B_{12}^2) - (2 A_2 - A_3) A_3 (B_{11} + B_{22}) (B_{11} - B_{22} \pm D) \big ). \end {aligned} $$

Let us study the relationship between the manifolds. We find the intersection of the invariant manifolds (11) and (14). To this end, for the polynomials of the system obtained by combining Eqs. (11) and (14) we construct the Gröbner lexicographic basis with respect to $\gamma _1 > \gamma _3 > \omega _1 > \omega _2 >\omega _3$,

$$ \begin {aligned} \omega _3 &= 0, \\ (B_{12}^2 - B_{11} B_{22}) \gamma _2^2 - A_2 (B_{11} + B_{22}) \omega _2 \gamma _2 - A_2^2 \omega _2^2 &= 0, \\ (B_{11} B_{22} - B_{12}^2) \gamma _2 + A_2 B_{12} \omega _1 + A_2 B_{22} \omega _2 &= 0, \\ \gamma _3 &= 0, \\ B_{12} \gamma _1 - B_{11} \gamma _2 - A_2 \omega _2 &= 0. \end {aligned} $$

(20)

Equations (20), together with the integral $V_2 = 1$, determine the following solutions of the differential equations (4):

$$ \begin {aligned} \gamma _1 &= \mp \frac {\sqrt {2} B_{12}}{\sqrt { DD_1}}, &\enspace \gamma _2 &= \pm \frac {D_1}{\sqrt {2 D}}, &\enspace \gamma _3 &= 0, \\ \omega _1 &= \pm \frac {D_1 \big (2 B_{12}^2 - B_{22} (B_{11} - B_{22} - D)\big )}{2 \sqrt {2 D} A_2 B_{12}}, &\enspace \omega _2 &= \mp \frac {D_1 (B_{11} + B_{22} + D)}{2 \sqrt {2 D} A_2}, &\enspace \omega _3 &= 0; \end {aligned} $$

(21)

$$ \begin {aligned} \gamma _1 &= \pm \frac {\sqrt {2} B_{12}}{\sqrt {D D_2}}, &\enspace \gamma _2 &= \pm \frac { D_2}{\sqrt {2 D}}, &\enspace \gamma _3 &= 0, \\ \omega _1 &= \pm \frac {D_2\big (2 B_{12}^2 - B_{22} (B_{11} - B_{22} + D)\big ) }{2\sqrt {2 D} A_2 B_{12}}, &\enspace \omega _2 &= \mp \frac {(B_{11} + B_{22} - D) D_2}{2 \sqrt {2 D}A_2}, &\enspace \omega _3 &= 0. \end {aligned} $$

(22)

Here and in the following, $D_1 = \sqrt {B_{11} - B_{22} + D}$ and $D_2 =\sqrt {B_{22} -B_{11} + D}$.

From the mechanical viewpoint, the solutions (21), (22) correspond to permanent rotations of a body around an axis located in the $Oxy$-plane (in the system of axes associated with the body) with the angular velocity $\omega ^2 = \big (B_{11}^2 + 2 B_{12}^2 \pm B_{11} D + B_{22} (B_{22} \pm D)\big )/(2 A_2^2)$.

It can be verified by a straightforward calculation that the integral

$$ 2 \Omega _4 = K_1 + \frac {1}{A_2^2} \big (2 B_{11} \widetilde V_{1}+ (B_{11}^2 + B_{12}^2) V_{2} \big ) $$

takes a stationary value on the solutions (21) and (22).

In a similar way, it can be shown that the intersection of the invariant manifold (14) with each of the two invariant manifolds (19) is nonempty. They have common points that also correspond to permanent rotations of the body.

3. ON THE STABILITY OF STATIONARY SOLUTIONS AND INVARIANT MANIFOLDS

Let us study the stability of the invariant manifold (11) using the integral $\Omega _1$ (13) to obtain sufficient conditions.

We introduce the deviations

$$ \begin {aligned} y_{1}&= \omega _1 + \frac {1}{A_2 B_{12}} \big (A_2 B_{22} \omega _2 + (B_{11} B_{22} - B_{12}^2) \gamma _2\big ), \\ y_{2}&= \gamma _1 - \frac {1}{B_{12}} (A_2 \omega _2 + B_{11} \gamma _2), \\ y_{3}&= \omega _3\end {aligned} $$

and write the variation of the integral $\Omega _1$ in a neighborhood of the solution to be studied,

$$ 2 \Delta \Omega _1 = \frac {1}{A_2^2} \big (B_{12}^2 y_2^2 + (A_2 y_1 +B_{22} y_2)^2 + (2 A_2 - A_3) A_3 y_3^2\big ). $$

Consider the restriction of $\Delta \Omega _1$ to the set defined by the first variation of the integral $\widetilde V_1$,

$$ \delta \widetilde V_1 = \frac {2}{B_{12}} (A_2 \omega _2 + B_{11} \gamma _2) y_2 = 0. $$

On this set, $\Delta \Omega _1$ acquires the form

$$ 2 \Delta \widetilde \Omega _1 = y_1^2 + \frac {(2 A_2 - A_3) A_3}{A_2^2} y_3^2. $$

The condition

$$ 2 A_2 > A_3 $$

(23)

of positive definiteness of the quadratic form $\Delta \widetilde \Omega _1$ is sufficient for the stability of the invariant manifold studied.

Now let us analyze the stability of the invariant manifold (14) using the integral $\Omega _2$ (15) to obtain sufficient conditions. The analysis is carried out in the map (16) on this invariant manifold.

We introduce the deviations

$$ y_{1}= \gamma _1 - \frac {z_1 \gamma _2}{2 B_{12}}, \quad y_{2}= \omega _1- \frac {z_1 \omega _2}{2 B_{12}}, \quad y_{3}= \gamma _3, \quad y_{4}=\omega _3. $$

The second variation of $\Omega _2$ on the set defined by the first variations

$$ \begin {aligned} \delta \widetilde V_1 &= \frac {A_2 z_1}{2 B_{12}} (\omega _2 y_1 + \gamma _2 y_2) = 0, \\ \delta V_2 &= \frac {\gamma _2 z_1}{B_{12}} y_1 = 0\end {aligned} $$

of the conditional integrals is written in the form

$$ \begin {aligned} 2 \delta ^{2} \Omega _2 &= \frac {1}{4 A_2^2} \Bigg [2 (2 A_2 - A_3) A_3 y_4^2 - 2 A_3 \bigg ((B_{11} + B_{22} + D) + 2 A_2 \frac {\omega _2}{\gamma _2} \bigg ) y_3 y_4 \\ &\qquad \qquad \qquad \qquad {}+ \bigg (2 A_2^2 \frac {\omega _2^2}{\gamma _2^2} -\big (B_{11} (B_{11} + D) + B_{22} (B_{22} + D) + 2 B_{12}^2\big ) \bigg ) y_3^2 \Bigg ].\end {aligned} $$

The condition

$$ \begin {aligned} (A_2 - A_3) \frac {\omega _2^2}{\gamma _2^2} - A_3 (B_{11} + B_{22} + D) \frac {\omega _2}{\gamma _2} - B_{11} (B_{11} + D) - B_{22} (B_{22} + D) - 2 B_{12}^2 &> 0, \\ 2 A_2 - A_3 &> 0\end {aligned} $$

of sign definiteness of the quadratic form $\delta ^{2} \Omega _2$ will be sufficient for the stability of the invariant manifold in question.

Since the ratio $\Phi = \widetilde V_1/V_2$ of integrals on the invariant manifold (14) acquires the form $\Phi \vert _0 = A_2 \omega _2/\gamma _2 =c = \mathrm {const}$, we see that the last inequalities are satisfied, in particular, for

$$ \begin {aligned} &(B_{11} > 0, B_{12} > 0, B_{22} > 0) \\ &\qquad {}\wedge \Bigg \{ \bigg [\bigg (\frac {A_3}{2} < A_2 < A_3 \bigg ) \wedge \bigg (\frac {A_2 \bar z_1}{A_2 - A_3} < 2 c < -\bar z_2 \bigg ) \bigg ] \vee ( (A_2 = A_3)\wedge (2 c < -\bar z_2))\\ &\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad {}\vee \bigg [(A_2 > A_3)\wedge \bigg ( (2c < -\bar z_2) \vee \bigg ( 2 c > \frac {A_2 \bar z_2}{A_2 - A_3} \bigg ) \bigg ) \bigg ]\Bigg \}, \end {aligned} $$

(24)

where $\bar z_1 = B_{11} + B_{22} - D$ and $\bar z_2 = B_{11} + B_{22}+ D$.

In the map (18), the sufficient conditions for the stability of the invariant manifold (14) become more restrictive. Under the condition of positivity of $B_{11}$, $B_{12}$, and $B_{22}$, they contain additional constraints on $B_{11}$, say, $0 < B_{11} < B_{12}^2/B_{22}$ or $B_{11} > B_{12}^2/B_{22}$.

Let us study the stability of the solutions (21). We introduce the deviations from the unperturbed solution,

$$ \begin {aligned} y_1 &= \gamma _1 \pm \frac {\sqrt {2} B_{12}}{\sqrt {D (B_{11} -B_{22} + D)}}, \\ y_2 &= \gamma _2 \mp \frac {\sqrt {B_{11} - B_{22} + D}}{\sqrt { 2 D}}, \\ y_3 &= \gamma _3, \\ y_4 &= \omega _1 \mp \frac {\sqrt {B_{11} - B_{22} + D} \big (2 B_{12}^2 + B_{22} (B_{22} - B_{11} + D)\big )} {2 \sqrt {2} A_2 B_{12} \sqrt {D}}, \\ y_5 &= \omega _2 \pm \frac {\sqrt {B_{11} - B_{22} + D} (B_{11} + B_{22} + D)}{2 \sqrt {2} A_2 \sqrt {D}}, \\ y_6 &= \omega _3.\end {aligned} $$

The variation of the integral $\Omega _4$ in deviations on the set

$$ \delta V_2 = \mp \frac {\sqrt {2} \big (2 B_{12} y_1 - (B_{11} - B_{22} + D) y_2 \big )}{\sqrt {D(B_{11} - B_{22} + D)}} = 0 $$

is written as

$$ 2 \Delta \Omega _4 = \bigg ( \frac {B_{11} + B_{22} - D}{2 A_2} y_1 + y_4 \bigg )^2 + \bigg ( \frac {B_{12}}{A_2} \bigg ( \frac {2 B_{11}}{B_{11} - B_{22} + D} - 1 \bigg ) y_1 + y_5 \bigg )^2 \\ + \frac {(2 A_2 - A_3) A_3}{A_2^2} y_6^2. $$

Let us introduce the variables

$$ \begin {aligned} \zeta _1 &= \frac {B_{11} + B_{22} - D}{2 A_2} y_1 + y_4, \\ \zeta _2 &= \frac {B_{12}}{A_2} \bigg ( \frac {2 B_{11}}{B_{11} - B_{22} + D} - 1 \bigg ) y_1 + y_5.\end {aligned} $$

In terms of the new variables, $\Delta \Omega _4$ acquires the form

$$ 2 \Delta \widetilde \Omega _4 = \zeta _1^2 + \zeta _2^2 + \frac {(2 A_2 -A_3) A_3}{A_2^2} y_6^2. $$

Since the quadratic form $\Delta \widetilde \Omega _4$ is sign definite with respect to the variables occurring in it for $A_2 > A_3/2$, we conclude that the solutions studied here are stable with respect to the variables

$$ \begin {gathered} \frac {1}{2 A_2} \bigg (2 A_2 \omega _1 + (B_{11} +B_{22} - D ) \gamma _1 \mp \frac {2 \sqrt {2 D}B_{12}}{\sqrt {B_{11} - B_{22} +D}} \bigg ),\\[.5em] \omega _2 - \left (\frac {B_{12} \thinspace \gamma _1}{A_2} \pm \frac {B_{11} (B_{11} - B_{22} - D) \left (\sqrt {2 D} \sqrt {B_{11} - B_{22} + D} \pm 2 B_{12} \gamma _1\right )}{4 A_2 B_{12}^2} \right ), \\ \omega _3. \end {gathered} $$

(25)

A similar result can be obtained for the solutions (22).

The conditions for the stability of the invariant manifold (19) coincide with the condition for the stability of the invariant manifold (11).

4. ON SOLUTIONS ON A MANIFOLD

Consider the problem of finding stationary solutions and invariant manifolds of the differential equations (12). We use the same approach as in Sec. 2.

The first integrals of Eqs. (12) can be obtained by eliminating the variables $\omega _2$, $\omega _3$, and $\gamma _1$ from the original integrals $K_1$, $\widetilde V_1$, and $V_2$ by using Eqs. (11). They have the form

$$ \begin {aligned} \overline K_1 &= B_{22}^2 \big [B_{12}^2 (\gamma _2^2 + \gamma _3^2) - B_{11}^2 (\gamma _2^2 - \gamma _3^2)\big ] \\ &\qquad {}+ (B_{12} \gamma _2 - A_2 \omega _1) \Big [(B_{11}^2 + B_{12}^2)(B_{12} \gamma _2 - A_2 \omega _1) + 2 B_{11} B_{22} (B_{12} \gamma _2 + A_2\omega _1)\Big ] = \overline c_1 = \mathrm {const}, \\ \overline V_1 &= \big (B_{12}^2 - B_{11} B_{22}\big ) \gamma _2^2 - A_2^2 \omega _1^2 = \overline c_2 = \mathrm {const}, \\ \overline V_2 &= \frac {(B_{12} \gamma _2 - A_2 \omega _1)^2}{B_{22}^2} + \gamma _2^2 + \gamma _3^2 = 1.\end {aligned} $$

Let us choose independent integrals from these integrals (for example, $\overline K_1$ and $\overline V_2$), form their linear combination $2 \overline \Omega = 2 \mu _0 \overline K_1 - \mu _1 \overline V_{2}$, and write the necessary conditions for the extremum of $\overline \Omega$ in the variables $\omega _1$, $\gamma _2$, and $\gamma _3$,

$$ \begin {aligned} \frac {\partial \overline \Omega }{\omega _1} &= \frac {1}{B_{22}^2} \bigg (\frac {B_{12} z}{A_2} \gamma _2 - (z - 2 B_{11} B_{22} \mu _0) \omega _1\bigg ) = 0,\\ \frac {\partial \overline \Omega }{\gamma _2} &= \frac {1}{A_2 B_{22}} \bigg (\frac {B_{12} z}{B_{22}} \omega _1 + \frac {1}{A_2} \bigg (2 B_{11} \big (B_{11} B_{22} - B_{12}^2\big ) \mu _0 - \frac {(B_{12}^2 + B_{22}^2) \thinspace z}{B_{22}} \bigg ) \gamma _2 \bigg ) = 0, \\ \frac {\partial \overline \Omega }{\gamma _2} &= -\frac {z}{A_2^2} \gamma _3= 0. \end {aligned} $$

(26)

Here $\mu _0$ and $\mu _1$ are parameters of the family of integrals $\overline \Omega$, and $z = \big (B_{11}^2 + B_{12}^2\big ) \mu _0 + A_2^2 \mu _1$.

Obviously, for $\mu _1 = -\big (\big (B_{11}^2 + B_{12}^2\big ) \mu _0\big )/A_2^2$ Eqs. (26) have the solution

$$ \omega _1 = 0, \quad \gamma _2 = 0. $$

(27)

A straightforward calculation using the definition of invariant manifold shows that relations (27) determine invariant manifolds of codimension 2 of the differential equations (12).

Another invariant manifold of codimension 2 was obtained by constructing a lexicographic basis for the polynomials of system (26) with respect to $\mu _1 > \omega _1 > \gamma _3$. The equations of the invariant manifold are written in the form

$$ \begin {aligned} \gamma _3 &= 0, \\ A_2^2 B_{12} \omega _1^2 - A_2 \big (2 B_{12}^2 - B_{22} (B_{11} - B_{22})\big ) \omega _1 \gamma _2 + B_{12} (B_{12}^2 - B_{11} B_{22})\gamma _2^2 &= 0. \end {aligned} $$

(28)

The differential equations $\dot \gamma _3 = 0$ $(\dot \gamma _2 =0)$ on the invariant manifold (27) (the invariant manifold (28)) have the families of solutions $\gamma _3 = \gamma _3^0 = \mathrm {const}$ ($\gamma _2 = \gamma _2^0 = \mathrm {const}$). Thus, from the geometrical viewpoint, the invariant manifolds found in the space $\mathbb {R}^3$ are associated with curves whose each point is a fixed point in the phase space of system (12).

Let us supplement equations (26) by the relation $\overline V_2 = 1$ and construct a lexicographic basis with respect to $\omega _1 > \gamma _2 > \gamma _3 > \mu _1$ for the polynomials of the resulting system. The result will be a system of equations that permits one to obtain the following solutions of the differential equations (12):

$$ \omega _1 = \gamma _2 = 0, \quad \gamma _3 = \pm 1, $$

(29)

$$ \omega _1 = \pm \frac {D_1 \big (2 B_{12}^2 - B_{22} (B_{11} - B_{22} - D)\big )}{ 2 \sqrt {2 D} A_2 B_{12}}, \quad \gamma _2 = \pm \frac {D_1}{\sqrt {2 D}},\quad \gamma _3 = 0, $$

(30)

$$ \omega _1 = \pm \frac {D_2 \big (2 B_{12}^2 - B_{22} (B_{11} - B_{22} + D)\big )}{ 2 \sqrt {2 D} A_2 B_{12}}, \quad \gamma _2 = \pm \frac {D_2}{\sqrt {2 D}}, \quad \gamma _3 = 0. $$

(31)

These solutions correspond to fixed points in the phase space of the system.

Obviously, the solutions (29) belong to the invariant manifold (27). It can readily be shown that the solutions (30) and (31) belong to the invariant manifold (28). To this end, we substitute relations (30) into Eqs. (28), which turn into identities. It follows that the solutions (30) belong to the invariant manifold (28). A similar result can also be obtained for the solutions (31).

It can be verified by a straightforward calculation that the expressions (30), together with Eqs. (11), determine solutions of the differential equations (4) coinciding with (21) in the original phase space. The solutions (31) in the original space are associated with the solutions (22).

Let us study the stability of the solutions (30) using the integral

$$ 2 \Phi = 2 \overline K_1 + \frac {B_{12}^2 - B_{11} (B_{22} + D)}{A_2^2} \overline V_{2}, $$

which takes a stationary value on these solutions, to obtain sufficient conditions.

In the deviations

$$ y_1 = \omega _1 \mp \frac {D_1 \big (2 B_{12}^2 - B_{22} (B_{11} - B_{22} - D)\big )}{2 \sqrt {2 D} A_2 B_{12}},\quad y_2 = \gamma _2 \mp \frac {D_1}{\sqrt {2 D}}, \quad y_3 = \gamma _3 $$

on the linear manifold

$$ \delta \overline V_2 = \pm \frac {\sqrt {2}}{B_{22} \sqrt {D} D_1} \Big (2 A_2 B_{12} \thinspace y_1 + \big (B_{22} (B_{11} - B_{22} + D) - 2 B_{12}^2\big ) y_2 \Big ) = 0, $$

the variation of the integral $\Phi$ is written as follows:

$$ \Delta \Phi = -\frac {2 B_{11} \big (4 B_{12}^2 + (B_{11} - B_{22})^2\big ) (B_{11} - B_{22} + D)}{\big (2 B_{12}^2 - B_{22} (B_{11} - B_{22} + D)\big )^2} y_1^2 - \frac {B_{11} (B_{11} + B_{22} + D)}{2A_2^2} y_3^2. $$

The quadratic form $\Delta \Phi$ will be positive definite under the following constraints on the parameters $B_{11}$, $B_{12}$, and $B_{22}$:

$$ \bigg ((B_{12} \neq 0 )\wedge \bigg ((B_{22} < 0) \wedge \bigg ( \frac {B_{12}^2}{B_{22}} < B_{11} < 0 \bigg )\bigg ) \vee \big ((B_{22} > 0)\wedge ( B_{11} < 0 )\big ). $$

(32)

Conditions (32) are sufficient for the stability of the solutions (30).

Let us obtain necessary conditions for the stability of the solutions (30) using the Lyapunov theorem on the stability by the first approximation [13].

In the case under consideration, the equations of the first approximation are written in the form

$$ \begin {aligned} \dot y_1 &= \frac {(B_{11} B_{22} - B_{12}^2) D_1}{\sqrt {2 D} A_2^2} y_3, \\ \dot y_2 &= \pm \frac {\big (B_{22} (B_{11} - B_{22} - D) - 2 B_{12}^2\big ) D_1} {2 \sqrt {2 D} A_2 B_{12}} y_3, \\ \dot y_3 &= \pm \bigg ( \frac {B_{22} \big (B_{11}^2 + B_{22} (B_{22} + D)\big ) + (2 B_{12}^2 - B_{11} B_{22}) (2 B_{22} + D)}{2 \sqrt {2 D} A_2 B_{12} B_{22}} y_2 - \frac {\sqrt {D}}{\sqrt {2} B_{22}} y_1 \bigg ) D_1. \end {aligned} $$

(33)

The characteristic equation

$$ \lambda \big (2 A_2^2 \lambda ^2 + (B_{11} - B_{22})^2 + (B_{11} + B_{22}) D + 4 B_{12}^2\big ) = 0 $$

of system (33) has only zero and pure imaginary roots under the conditions

$$ ( B_{12} \neq 0)\wedge \bigg ((B_{22} < 0)\wedge \;\bigg (\frac {B_{12}^2}{B_{22}} \le B_{11} < 0)\bigg ) \vee (B_{11} > 0) \bigg ) \vee \big ((B_{22} > 0)\wedge (B_{11}\neq 0 )\big ). $$

Comparing the last inequalities with (32), we conclude that the sufficient conditions are close to the necessary ones.

Thus, the solutions (30) that are stable on the manifold correspond to the solutions (21) that are stable in part of the variables in the original phase space. The same result has also been obtained in the case of the solutions (31).

Based on the results presented, we can state the following assertion.

Assertion.

The generalizations of the Routh–Lyapunov method used to analyze the problem under consideration have made it possible to isolate special sets of differential equations (1) in the special case of the existence of an additional quadratic integral $K_1$ (3) of these equations—the stationary invariant manifolds (11), (14), (19) and the stationary solutions (21), (22) as the points of intersection of these invariant manifolds—and investigate the solutions found for stability. With the help of linear and nonlinear combinations of the first integrals of the problem delivering stationary values to the solutions found, sufficient stability conditions (23) and (24) have been obtained for the invariant manifolds (11), (19) and the invariant manifolds (14), respectively, and stability in terms of the variables (25) is proved for the stationary solutions. The approach used has also allowed carrying out a similar study of the differential equations on the invariant manifold (11).

CONCLUSIONS

New additional quadratic integrals are indicated in the problem of the motion of a rigid body with a fixed point under the action of a magnetic field generated by the Barnett–London effect and potential forces. A qualitative analysis of a system of differential equations admitting one of these integrals is carried out. Stationary invariant manifolds of codimensions 3 and 4 are isolated, and sufficient conditions for their stability are obtained. It is shown that the intersections of the manifolds are fixed points in the phase space of the system corresponding to the permanent rotations of the body. The stability of these motions with respect to part of the variables is proved. A qualitative analysis of the differential equations has also been carried out on one of the invariant manifolds found. Stationary solutions and invariant manifolds of codimension 2 are isolated. Sufficient stability conditions are obtained for stationary solutions on a manifold and are compared with the necessary ones. We note that in the considered case of the existence of an additional quadratic integral, the parameters characterizing the influence of the potential forces do not explicitly appear in either the equations of motion of the body or the first integrals. The conditions for the stability of solutions are obtained in the form of constraints on the parameters characterizing the magnetic forces. Thus, it can be assumed that potential forces do not have a considerable effect on the motion of the body in this case.