Keywords

1 Introduction

The problem considered in this paper goes back to the Chaplygin work [1] of rolling a dynamically asymmetric balanced ball along a horizontal plane without slipping. The integrability of the system was revealed by Chaplygin with the help of its explicit reduction to quadratures. A sufficient number of works are devoted to the Chaplygin problem and its integrable generalizations (see, e.g., [2]). One of them is investigated in the paper. In [3] the generalization of system [2] is given. The motion of a dynamically asymmetric rigid body around fixed point O is considered (see Fig. 1). The body is rigidly enclosed in a spherical shell, the geometrical center of which coincides with the fixed point of the body. One ball and one disk adjoin to the spherical shell. It is supposed that slipping at a contact point of the ball with the shell is absent. The disk – nonholonomic hinge – concerns the external surface of the spherical shell. The centers of the balls and the axis of the disk are fixed in space. The study of dynamics of such systems is of interest, e.g., for robotics in the problems of the design and control of mobile spherical robots (see., e.g., [4]). The motion of the mechanical system is described by the differential equations [3]

Fig. 1.
figure 1

The rigid body enclosed in a spherical shell, which a ball and a disk adjoin to.

Full size image
$$\begin{aligned}{} & {} \textbf{I} \boldsymbol{\dot{\omega }} = \textbf{I} \boldsymbol{\omega } \times \boldsymbol{\omega } + R \boldsymbol{\gamma } \times \textbf{N} + \mu \textbf{E} + \mathbf{M_Q}, \ D_1 \boldsymbol{\dot{\omega }_1} = D_1 \boldsymbol{\omega _1} \times \boldsymbol{\omega } + R_1 \boldsymbol{\gamma } \times \textbf{N}, \nonumber \\{} & {} \boldsymbol{\dot{\gamma }} = \boldsymbol{\gamma } \times \boldsymbol{\omega }, \ \mathbf{\dot{E}} = \textbf{E} \times \boldsymbol{\omega }, \end{aligned}$$
(1)

and the equations of constraints

$$\begin{aligned} R \boldsymbol{\omega } \times \boldsymbol{\gamma }+R_1 \boldsymbol{\omega _1 } \times \boldsymbol{\gamma }=0, \ (\boldsymbol{\omega }, \textbf{E})=0. \end{aligned}$$
(2)

Here \(\boldsymbol{\omega }\) = \((\omega _1, \omega _2, \omega _3)\), R is the angular velocity of the body and the radius of the spherical shell, \(\boldsymbol{\omega _1}\) = \((\omega _{1_1}, \omega _{1_2}, \omega _{1_3})\), \(R_1\) is the angular velocity and the radius of the adjoint ball, \(\boldsymbol{\gamma } = (\gamma _1, \gamma _2, \gamma _3) \) is the unit vector of the axis connecting the fixed point with the center of the adjoint ball, \(\textbf{E} = (e_1, e_2, e_3)\) is the vector of the normal to the plane containing the fixed point and the axis of the disk, \(\textbf{I} = \text{ diag }(A, B, C)\) is the inertia tensor of the body, \(D_1\) is the inertia tensor of the adjoint ball, \(\textbf{N}\)= \((N_1, N_2,\) \(N_3)\), \(\mu \) are indefinite factors related to the reactions of constraints (2), \(\mathbf{M_Q}\) is the moment of external forces. One supposes that the position of the vectors E and \(\boldsymbol{\gamma }\) with respect to each other is arbitrary.

By means of the equations of constraints (2) the differential Eqs. (1) are reduced to the form [3]:

$$\begin{aligned}{} & {} \textbf{I} \boldsymbol{\dot{\omega }} + D \boldsymbol{\gamma } \times \boldsymbol{(\dot{\omega }\times \gamma )} = \textbf{I} \boldsymbol{\omega } \times \boldsymbol{\omega } + \mu \textbf{E} + \mathbf{M_Q}, \ \boldsymbol{\dot{\gamma }} = \boldsymbol{\gamma } \times \boldsymbol{\omega }, \ \mathbf{\dot{E}} = \textbf{E} \times \boldsymbol{\omega }, \end{aligned}$$
(3)

where \( D = \frac{R^2}{R_1^2} D_1\).

The indefinite factor \(\mu \) is found from the condition that the derivative of the 2nd relation (2) in virtue of differential Eqs. (3) is equal to zero.

If the body is subject to external forces, e.g., potential ones

$$\begin{aligned} \mathbf{M_Q} = \boldsymbol{\gamma } \times \frac{\partial U}{\partial \boldsymbol{\gamma }} + \textbf{E} \times \frac{\partial U}{\partial \textbf{E}}, \end{aligned}$$

where \(U = U(\boldsymbol{\gamma }, \textbf{E})\) is the potential energy of external forces, Eqs. (3) admit the following first integrals

$$\begin{aligned}{} & {} 2H=(\mathbf{I_Q} \, \boldsymbol{\omega }, \boldsymbol{\omega })+2U(\boldsymbol{\gamma }, \textbf{E})= 2\,h, \ V_1 = (\boldsymbol{\gamma }, \boldsymbol{\gamma }) = 1, \ V_2 = (\textbf{E}, \textbf{E}) = 1, \nonumber \\{} & {} V_3 =(\boldsymbol{\gamma }, \textbf{E})= c_1, \ V_4 = (\boldsymbol{\omega }, \textbf{E})=0 \end{aligned}$$
(4)

and are nonintegrable in the general case. Here \(\mathbf{I_Q} = \textbf{I} + D - D \boldsymbol{\gamma } \otimes \boldsymbol{\gamma }\), \(\boldsymbol{\gamma } \otimes \boldsymbol{\gamma } =[c_{ij}], \, c_{11}= \gamma _1^{2}, \, c_{12}= \gamma _1 \gamma _2,\ldots \)

In the case of the absence of external forces \((U = 0)\) and \((\textbf{E} \times \boldsymbol{\gamma }) \ne 0\), Eqs. (3) have two additional first integrals

$$\begin{aligned} F_1=( \textbf{K}, \textbf{E} \times \boldsymbol{\gamma }), \ F_2=(\textbf{K}, \textbf{E} \times (\textbf{E} \times \boldsymbol{\gamma })), \end{aligned}$$

where \(\textbf{K}= \mathbf{I_Q} \, \boldsymbol{\omega } -(\mathbf{I_Q} \, \boldsymbol{\omega },\textbf{E}) \textbf{E}\), and then system (3) is completely integrable.

2 Problem Statement

The qualitative analysis of the problem under consideration was not conducted so far. In the present work, the qualitative analysis of the equations of motion (3) on the invariant set defined by the relation \(V_4 = 0\) (4) is done. We find invariant sets of various dimension from the necessary conditions of extremum of the first integrals of the problem (or their combinations) and study their Lyapunov stability. The sets found in this way are called stationary ones. The stationary sets of zero dimension are known as stationary solutions, while the positive dimension sets are called stationary invariant manifolds (IMs).

We use the Routh–Lyapunov method [5] and some its generalizations [6] for the study of the problem. The computer analysis of the problem is mainly done symbolically. Computer algebra system (CAS) Mathematica and the software package [7] written in the language of this system are applied to solve computational problems. With the help of the package, the stability of the found solutions is investigated.

The paper is organized as follows. In Sect. 2 and 3, we describe finding stationary sets both in the case of absence of external forces and when potential forces act upon the mechanical system. Solutions obtained in these sections correspond to equilibria of the mechanical system. In Sect. 4, solutions corresponding to pendulum-type motions are presented. In Sect. 5, the stability of the found solutions is analyzed. In Sect. 6, we give some conclusions.

3 On Stationary Sets in the Case of Absence of External Forces

The equations of motion (3) in an explicit form on the invariant set \(V_4 = 0\) when \(U=0\) are written as

$$\begin{aligned}{} & {} \dot{\omega }_1 = -\frac{1}{\sigma _1} \Big [ D ((A - B) (B + D) \gamma _3 \bar{\omega }_2 - (A - C) (C + D) \gamma _2 \omega _3) \, \gamma _1 \omega _1 \nonumber \\{} & {} + (B - C) ((C + D) (B + D - D \gamma _2^2) - D (B + D) \gamma _3^2) \, \bar{\omega }_2 \omega _3 + \mu \, [(C + D) \nonumber \\{} & {} \times ((B + D) e_1 + D \gamma _2 (e_2 \gamma _1 - e_1 \gamma _2)) + D (B + D) \gamma _3 (e_3 \gamma _1 - e_1 \gamma _3)] \Big ], \nonumber \\{} & {} \dot{\omega }_3 = -\frac{1}{\sigma _1} \Big [ (A - B) ((B + D) (A + D - D \gamma _1^2) - D (A + D) \, \gamma _2^2) \, \omega _1 \bar{\omega }_2 \nonumber \\{} & {} - D ((A - C) (A + D) \gamma _2 \omega _1 - (B - C) (B + D) \gamma _1 \bar{\omega }_2) \, \gamma _3 \omega _3 + \mu \, [D (A + D) \nonumber \\{} & {} \times \gamma _2 \, (e_2 \gamma _3 - e_3 \gamma _2) + (B + D) (e_3 (A + D - D \gamma _1^2) + D e_1 \gamma _1 \gamma _3)] \Big ], \nonumber \\{} & {} \dot{\gamma }_1 = -\gamma _3 \bar{\omega }_2 + \gamma _2 \omega _3, \ \dot{\gamma }_2 = \gamma _3 \omega _1 - \gamma _1 \omega _3, \ \dot{\gamma }_3 = -\gamma _2 \omega _1 + \gamma _1 \bar{\omega }_2, \nonumber \\{} & {} \dot{e}_1 = -e_3 \bar{\omega }_2 + e_2 \omega _3, \ \dot{e}_2 = e_3 \omega _1 - e_1 \omega _3, \ \dot{e}_3 = -e_2 \omega _1 + e_1 \bar{\omega }_2, \end{aligned}$$
(5)

where \( \bar{\omega }_2 = -\frac{e_1 \omega _1 + e_3 \omega _3}{e_2}\),

$$\begin{aligned}{} & {} \mu = - \frac{1}{\sigma _2} \Big [(A \! - \! B) ((B + D) (A + D) \, e_3 + D (B + D) \gamma _1 (\gamma _3 e_1 \! - \! e_3 \gamma _1) \\{} & {} + D (A + D) \gamma _2 (\gamma _3 e_2 - e_3 \gamma _2)) \, \omega _1 \bar{\omega }_2 \\{} & {} - (A - C) (e_2 (A + D) (C + D) + D (C + D) \gamma _1 (e_1 \gamma _2 - e_2 \gamma _1) \\{} & {} + D (A + D) \gamma _3 (e_3 \gamma _2 - e_2 \gamma _3)) \, \omega _1 \omega _3 + (B - C) ( (B + D) (C + D) e_1 \\{} & {} + D (C + D) \gamma _2 (e_2 \gamma _1 - e_1 \gamma _2) + D (B + D) \gamma _3 (e_3 \gamma _1 - e_1 \gamma _3)) \, \bar{\omega }_2 \omega _3 \Big ], \\{} & {} \sigma _1 = D ((B \! + \! D) (C \! + \! D) \, \gamma _1^2 + (A \! + \! D) (C + D) \, \gamma _2^2 + (A \! + \! D) (B \! + \! D) \, \gamma _3^2) \\{} & {} - (A + D) (B + D) (C + D), \\{} & {} \sigma _2 = (B + D) (C + D) \, e_1^2 + (A + D) (C + D) \, e_2^2 + (A + D) (B + D) \, e_3^2 \\{} & {} - D [(C + D) (e_2 \gamma _1 - e_1 \gamma _2)^2 + (B + D) (e_3 \gamma _1 - e_1 \gamma _3)^2 \\{} & {} + (A + D) (e_3 \gamma _2 - e_2 \gamma _3)^2], \end{aligned}$$

Equations (5) admit the following first integrals:

$$\begin{aligned}{} & {} 2\,H = (A + D - D \gamma _1^2) \, \omega _1^2 + (B + D - D \gamma _2^2) \, \bar{\omega }_2^2 + (C + D - D \gamma _3^2) \, \omega _3^2 \nonumber \\{} & {} - 2 D (\gamma _1 \gamma _2 \omega _1 \bar{\omega }_2 + \gamma _1 \gamma _3 \omega _1 \omega _3 + \gamma _2 \gamma _3 \bar{\omega }_2 \omega _3) = 2\,h, \nonumber \\{} & {} V_1 = \gamma _1^2 + \gamma _2^2 + \gamma _3^2 = 1, \ V_2 = e_1^2 + e_2^2 + e_3^2 = 1, \nonumber \\{} & {} V_3 = e_1 \gamma _1 + e_2 \gamma _2 + e_3 \gamma _3 = c_1, \nonumber \\{} & {} F_1 = -(A + D) (e_3 \gamma _2 - e_2 \gamma _3) \, \omega _1 + (B + D) (e_3 \gamma _1 - e_1 \gamma _3) \, \bar{\omega }_2 \nonumber \\{} & {} - (C + D) (e_2 \gamma _1 - e_1 \gamma _2) \, \omega _3 = c_2, \nonumber \\{} & {} F_2 = [e_1 \, (A + D - 2 D \gamma _1^2) (e_2 \gamma _2 + e_3 \gamma _3) - \gamma _1 (A \, (e_2^2 + e_3^2) \nonumber \\{} & {} - D ((e_2^2 + e_3^2) ( \gamma _1^2 - 1) + (e_3 \gamma _2 - e_2 \gamma _3)^2 + e_1^2 (\gamma _2^2 + \gamma _3^2)))] \, \omega _1 \nonumber \\{} & {} + [e_2 \, (B + D - 2 D \gamma _2^2) (e_1 \gamma _1 + e_3 \gamma _3) - \gamma _2 (B \, (e_1^2 + e_3^2) \nonumber \\{} & {} - D ((e_1^2 + e_3^2) ( \gamma _2^2 - 1) + (e_3 \gamma _1 - e_1 \gamma _3)^2 + e_2^2 (\gamma _1^2 + \gamma _3^2)))] \, \bar{\omega }_2 \nonumber \\{} & {} + [e_3 \, (e_1 \gamma _1 + e_2 \gamma _2) (C + D - 2 D \gamma _3^2) - \gamma _3 (C \, (e_1^2 + e_2^2) \nonumber \\{} & {} - D ((e_2 \gamma _1 - e_1 \gamma _2)^2 + e_3^2 (\gamma _1^2 + \gamma _2^2) + (e_1^2 + e_2^2) ( \gamma _3^2 - 1)))] \, \omega _3 = c_3. \end{aligned}$$
(6)

Here \(F_1, F_2\) are the additional integrals of the 3rd and 5th degrees, respectively.

As was remarked above, the stationary conditions for the first integrals of the problem (or their combinations) are used to obtain solutions of interest for us. In the problem under consideration, because of rather high degrees of the first integrals, another approach [8] turned out to be more effective for seeking the desired solutions: first, obtain the desired solutions from the equations of motion, and, then, find the conditions on the parameters of the problem under which these solutions satisfy the stationary equations for the first integrals.

Obviously, Eqs. (5) have the solution \(\omega _1 = \omega _3 = 0\). These relations together with the integrals \(V_1 = 1, V_2 = 1\) define the invariant manifold (IM) of codimension 4 for the equations of motion (5). It is easy to verify by direct calculation according to the IM definition. The equations of the IM are written as:

$$\begin{aligned} \omega _1 = \omega _3 = 0, \, e_1^2 + e_2^2 + e_3^2 = 1, \, \gamma _1^2 + \gamma _2^2 + \gamma _3^2 = 1. \end{aligned}$$
(7)

With the help of maps on IM (7)

$$\begin{aligned} \omega _1 = \omega _3 = 0, \, \gamma _1 = \pm \sqrt{1 - \gamma _2^2 - \gamma _3^2}, \, e_1 = \pm \sqrt{1 - e_2^2 - e_3^2}, \end{aligned}$$
(8)

we find that the integral \(V_3\) takes the form

$$\begin{aligned} e_2 \gamma _2 + e_3 \gamma _3 \pm \sqrt{1 - \gamma _2^2 - \gamma _3^2} \sqrt{1 - e_2^2 - e_3^2} = c_1 \end{aligned}$$

on this IM. Thus, IM (7) exists for any angles between the vectors \(\textbf{E}\) and \(\boldsymbol{\gamma }\), i.e., it is the family of IMs.

The differential equations \(\dot{\gamma }_2 = 0, \, \dot{\gamma }_3 = 0, \, \dot{e}_2 = 0, \, \dot{e}_3 = 0\) on IM (7) have the family of solutions:

$$\begin{aligned} \gamma _2 = \gamma _2^0 = \text{ const }, \, \gamma _3 = \gamma _3^0 = \text{ const }, \, e_2 = e_2^0 = \text{ const }, \, e_3 = e_3^0 = \text{ const }. \end{aligned}$$
(9)

The latter relations together with the IM equations determine four families of solutions for the equations of motion (5)

$$\begin{aligned}{} & {} \omega _1 = \omega _3 = 0, \, e_1 = \pm \sqrt{1 - e{_2^0}^2 - e{_2^0}^2}, \, e_2 = e{_2^0}, \, e_3 = e{_3^0}, \, \gamma _1 = \sqrt{1 - \gamma {_2^0}^2 - \gamma {_2^0}^2}, \nonumber \\{} & {} \gamma _2 = \gamma {_2^0}, \, \gamma _3 = \gamma {_3^0}; \nonumber \\{} & {} \omega _1 = \omega _3 = 0, \, e_1 = \pm \sqrt{1 - e{_2^0}^2\! - e{_2^0}^2}, \, e_2 = e{_2^0}, \, e_3 = e{_3^0}, \, \gamma _1 = -\sqrt{1 - \gamma {_2^0}^2 - \gamma {_2^0}^2}, \nonumber \\{} & {} \gamma _2 = \gamma {_2^0}, \, \gamma _3 = \gamma {_3^0} \end{aligned}$$
(10)

that can be verified by substituting the solutions into these equations. Here \(e{_2^0}, e{_3^0},\) \(\gamma {_2^0}, \gamma {_3^0}\) are the parameters of the families. Evidently, the solutions belong to IM (7).

From a mechanical point of view, the elements of the families of solutions (10) correspond to equilibria of the mechanical system under study.

Using the stationary equations

$$\begin{aligned} {\partial K_1}/{\partial \omega _1} = 0, \, {\partial K_1}/{\partial \omega _3} = 0, \, {\partial K_1}/{\partial \gamma _j} = 0, \, {\partial K_1}/{\partial e_j} = 0 \ (j=1,2,3) \end{aligned}$$

for the integral \(2 K_1 = 2 \lambda _0 H - \lambda _1 (V_1 - V_2)^2 - \lambda _2 F_1 F_2 \ (\lambda _i = \text{ const})\), it is not difficult to show that this integral takes a stationary value both on IM (7) and solutions (10). For this purpose, it is sufficient to substitute expressions (8) (or (10)) into the above equations. These become identity.

Directly, from differential Eqs. (5), it is also easy to obtain the following their solutions:

$$\begin{aligned} \omega _1 = \omega _3 = 0, \, e_1 = \pm \gamma _1, \, e_2 = \pm \gamma _2, \, e_3 = \pm \gamma _3. \end{aligned}$$
(11)

Relations (11) together with the integral \(V_1 = 0\) define two IMs of codimension 6 of differential Eqs. (5) that is verified by direct computation according to the IM definition. The equations of these IMs have the form:

$$\begin{aligned} \omega _1 = \omega _3 = 0, \, e_1 \mp \gamma _1 = 0, \, e_2 \mp \gamma _2 = 0, \, e_3 \mp \gamma _3 = 0, \, \gamma _1^2 + \gamma _2^2 + \gamma _3^2 = 1. \end{aligned}$$
(12)

On substituting expressions (12) into the stationary conditions for the integral

$$\begin{aligned} 2 K_2 = 2 \lambda _0 H - \lambda _1 V_1 - \lambda _2 V_2 - 2 \lambda _3 V_3 - 2 \lambda _4 F_1 - 2 \lambda _5 F_2 \ (\lambda _i = \text{ const}) \end{aligned}$$

we find the values \(\lambda _2 = \lambda _1, \, \lambda _3 = \mp \lambda _1\) under which the integral \(K_2\) assumes a stationary value on IMs (12).

The integrals \(K_1\) and \(K_2\) (under the corresponding values of \(\lambda _2, \lambda _3\)) are used for obtaining the sufficient conditions of stability of the above solutions.

The differential equations \(\dot{\gamma }_2 = 0, \, \dot{\gamma }_3 = 0\) on each IMs (12) have the following family of solutions: \(\gamma _2 = \gamma _2^0 = \text{ const }, \, \gamma _3 = \gamma _3^0 = \text{ const }\). Thus, geometrically, in space \(R^8\), two-dimensional surface corresponds to each of IMs (12), each point of which is a fixed point of the phase space.

The integral \(V_3\) takes the values \(\pm 1\) on IMs (12). Thus, IMs (12) correspond to the cases when the vectors \(\textbf{E}\) and \(\boldsymbol{\gamma }\) are parallel or opposite in direction.

4 On Stationary Sets in the Case of the Presence of External Forces

Let the mechanical system under study be under the influence of external potential forces with the potential energy \(U = (\textbf{a}, \boldsymbol{\gamma }) + (\textbf{b}, \textbf{E})\), where \(\textbf{a} = (a_1, a_2, a_2)\), \(\textbf{b} = (b_1, b_2, b_2)\) are the indefinite factors. In this case, the equations of motion (3) on the invariant set \(V_4 = 0\) are written as:

$$\begin{aligned}{} & {} \dot{\omega }_1 = -\frac{1}{\sigma _1} \Big [ D ((A - B) (B + D) \gamma _3 \bar{\omega }_2 - (A - C) (C + D) \gamma _2 \omega _3) \, \gamma _1 \omega _1 \nonumber \\{} & {} + (B - C) ((C + D) (B + D - D \gamma _2^2) - D (B + D) \gamma _3^2) \, \bar{\omega }_2 \omega _3 + \mu \, [(C + D) \nonumber \\{} & {} \times ((B + D) e_1 + D \gamma _2 (e_2 \gamma _1 - e_1 \gamma _2)) + D (B + D) \gamma _3 (e_3 \gamma _1 - e_1 \gamma _3)] \nonumber \\{} & {} + ((C + D) (B + D - D \gamma _2^2) - D (B + D) \gamma _3^2) M_{Q_1} + D (C + D) \gamma _1 \gamma _2 M_{Q_2} \nonumber \\{} & {} + D (B + D) \gamma _1 \gamma _3 M_{Q_3} \Big ], \nonumber \\{} & {} \dot{\omega }_3 = -\frac{1}{\sigma _1} \Big [ (A - B) ((B + D) (A + D - D \gamma _1^2) - D (A + D) \, \gamma _2^2) \, \omega _1 \bar{\omega }_2 \nonumber \\{} & {} - D ((A - C) (A + D) \gamma _2 \omega _1 - (B - C) (B + D) \gamma _1 \bar{\omega }_2) \, \gamma _3 \omega _3 + \mu \, [D (A + D) \nonumber \\{} & {} \times \gamma _2 \, (e_2 \gamma _3 - e_3 \gamma _2) + (B + D) (e_3 (A + D - D \gamma _1^2) + D e_1 \gamma _1 \gamma _3)] \nonumber \\{} & {} + D \gamma _3 ((B + D) \gamma _1 M_{Q_1} + (A + D) \gamma _2 M_{Q_2}) + ((B + D) (A \! + \! D \! - \! D \gamma _1^2) \nonumber \\{} & {} - D (A + D) \gamma _2^2) M_{Q_3} \Big ], \nonumber \\{} & {} \dot{\gamma }_1 = -\gamma _3 \bar{\omega }_2 + \gamma _2 \omega _3, \ \dot{\gamma }_2 = \gamma _3 \omega _1 - \gamma _1 \omega _3, \ \dot{\gamma }_3 = -\gamma _2 \omega _1 + \gamma _1 \bar{\omega }_2, \nonumber \\{} & {} \dot{e}_1 = -e_3 \bar{\omega }_2 + e_2 \omega _3, \ \dot{e}_2 = e_3 \omega _1 - e_1 \omega _3, \ \dot{e}_3 = -e_2 \omega _1 + e_1 \bar{\omega }_2, \end{aligned}$$
(13)
$$\begin{aligned}{} & {} \text{ where } \ \mu = - \frac{1}{\sigma _2} \Big [(A \! - \! B) ((B + D) (A + D) \, e_3 + D (B + D) \, \gamma _1 \\{} & {} \times (\gamma _3 e_1 - e_3 \gamma _1) + D (A + D) \gamma _2 (\gamma _3 e_2 - e_3 \gamma _2)) \, \omega _1 \bar{\omega }_2 \\{} & {} - (A - C) (e_2 (A + D) (C + D) + D (C + D) \gamma _1 (e_1 \gamma _2 - e_2 \gamma _1) \\{} & {} + D (A + D) \gamma _3 (e_3 \gamma _2 - e_2 \gamma _3)) \, \omega _1 \omega _3 + (B - C) ( (B \! + \! D) (C \! + \! D) \, e_1 \\{} & {} + D (C + D) \gamma _2 (e_2 \gamma _1 - e_1 \gamma _2) + D (B + D) \gamma _3 (e_3 \gamma _1 - e_1 \gamma _3)) \, \bar{\omega }_2 \omega _3 \\{} & {} -((C + D) ((B + D) e_1 + D \gamma _2 (e_2 \gamma _1 - e_1 \gamma _2)) \\{} & {} + D (B + D) \gamma _3 (e_3 \gamma _1 - e_1 \gamma _3)) M_{Q_1} - ((C + D) ((A + D) e_2 \\{} & {} + D \gamma _1 (e_1 \gamma _2 - e_2 \gamma _1)) + D (A + D) \gamma _3 (e_3 \gamma _2 - e_2 \gamma _3)) M_{Q_2} \\{} & {} - (D (A + D) \gamma _2 (e_2 \gamma _3 - e_3 \gamma _2) + (B + D) ((A + D) e_3 \\{} & {} + D \gamma _1 (e_1 \gamma _3 - e_3 \gamma _1))) M_{Q_3} \Big ]. \\{} & {} M_{Q_1} = b_3 e_2 \! - \! b_2 e_3 + a_3 \gamma _2 \! - \! a_2 \gamma _3, \ M_{Q_2} = -b_3 e_1 + b_1 e_3 \! - \! a_3 \gamma _1 \! + \! a_1 \gamma _3, \\{} & {} M_{Q_3} = b_2 e_1 - b_1 e_2 + a_2 \gamma _1 - a_1 \gamma _2. \end{aligned}$$

Here \(\bar{\omega }_2, \sigma _1, \sigma _2\) have the same values as in Sect. 2.

The first integrals of Eqs. (13):

$$\begin{aligned}{} & {} 2\,H = (A + D - D \gamma _1^2) \, \omega _1^2 + (B + D - D \gamma _2^2) \, \bar{\omega }_2^2 + (C + D - D \gamma _3^2) \, \omega _3^2 \nonumber \\{} & {} - 2 D (\gamma _1 \gamma _2 \omega _1 \bar{\omega }_2 + \gamma _1 \gamma _3 \omega _1 \omega _3 + \gamma _2 \gamma _3 \bar{\omega }_2 \omega _3) + a_1 \gamma _1 + a_2 \gamma _2 + a_3 \gamma _3 \nonumber \\{} & {} + b_1 e_1 + b_2 e_2 + b_3 e_3 = 2\,h, \nonumber \\{} & {} V_1 = \gamma _1^2 + \gamma _2^2 + \gamma _3^2 = 1, \ V_2 = e_1^2 + e_2^2 + e_3^2 = 1, \nonumber \\{} & {} V_3 = e_1 \gamma _1 + e_2 \gamma _2 + e_3 \gamma _3 = c_1. \end{aligned}$$
(14)

We shall seek solutions of differential Eqs. (13) of the following type:

$$\begin{aligned} \omega _1 = \omega _3 = 0, \, e_1 = e_1^0, \, e_2 = e_2^0, \, e_3 = e_3^0, \, \gamma _1 = \gamma _1^0, \, \gamma _2 = \gamma _2^0, \, \gamma _3 = \gamma _3^0, \end{aligned}$$
(15)

where \(e_2^0, \, e_3^0, \, \gamma _2^0, \, \gamma _3^0\) are some constants, and \(e_1^0 = \pm \sqrt{1 - e_2^{0^2} - e_3^{0^2}}\),

\(\gamma _1^0 = \pm \sqrt{1 - \gamma _2^{0^2} - \gamma _3^{0^2}}\).

On substituting (15) into Eqs. (13) these take the form:

$$\begin{aligned}{} & {} \bar{\mu }\, [(C + D) ((B + D) \, e_1^0 + D \gamma _2^0 (e_2^0 \gamma _1^0 - e_1^0 \gamma _2^0)) + D (B + D) \gamma _3^0 (e_3^0 \gamma _1^0 - e_1^0 \gamma _3^0)] \nonumber \\{} & {} + ((C + D) (B + D - D \gamma _2^{0^2}) - D (B + D) \gamma _3^{0^2}) \bar{M}_{Q_1} + D (C + D) \gamma _1^0 \gamma _2^0 \bar{M}_{Q_2} \nonumber \\{} & {} + D (B + D) \gamma _1^0 \gamma _3^0 \bar{M}_{Q_3} = 0, \nonumber \\{} & {} \bar{\mu }\, [D (A + D) \gamma _2^0 \, (e_2^0 \gamma _3^0 - e_3^0 \gamma _2^0) + (B + D) (e_3^0 (A + D - D \gamma _1^{0^2}) + D e_1^0 \gamma _1^0 \gamma _3^0)] \nonumber \\{} & {} + D \gamma _3^0 ((B + D) \gamma _1^0 \bar{M}_{Q_1} + (A + D) \gamma _2^0 \bar{M}_{Q_2}) + ((B + D) (A + D - D \gamma _1^{0^2}) \nonumber \\{} & {} - D (A + D) \gamma _2^{0^2}) \bar{M}_{Q_3} = 0. \end{aligned}$$
(16)
$$\begin{aligned}{} & {} \text{ Here } \ \bar{\mu }= \frac{1}{\bar{\sigma }_2} \Big [ [(C + D) ((B + D) \, e_1^0 + D \gamma _2^0 (e_2^0 \gamma _1^0 - e_1^0 \gamma _2^0)) + D (B + D) \\{} & {} \times \gamma _3^0 (e_3^0 \gamma _1^0 - e_1^0 \gamma _3^0)] \, \bar{M}_{Q_1} + [(C \! + \! D) ((A + D) \, e_2^0 + D \gamma _1^0 (e_1^0 \gamma _2^0 \! - \! e_2^0 \gamma _1^0)) \\{} & {} + D (A + D) \gamma _3^0 (e_3^0 \gamma _2^0 - e_2^0 \gamma _3^0)] \, \bar{M}_{Q_2} + [D (A + D) \gamma _2^0 (e_2^0 \gamma _3^0 - e_3^0 \gamma _2^0) \\{} & {} + (B + D) ((A + D) e_3^0 + D \gamma _1^0 (e_1^0 \gamma _3^0 - e_3^0 \gamma _1^0))] \, \bar{M}_{Q_3} \Big ], \\{} & {} \bar{\sigma }_2 = (B + D) (C + D) \, e_1^{0^2} + (A + D) (C + D) \, e_2^{0^2} + (A + D) (B + D) \, e_3^{0^2} \\{} & {} - D [(C + D) (e_2^0 \gamma _1^0 - e_1^0 \gamma _2^0)^2 + (B + D) (e_3^0 \gamma _1^0 - e_1^0 \gamma _3^0)^2 \\{} & {} + (A + D) (e_3^0 \gamma _2^0 - e_2^0 \gamma _3^0)^2], \ \bar{M}_{Q_1} = b_3 e_2^0 - b_2 e_3^0 + a_3 \gamma _2^0 - a_2 \gamma _3^0, \\{} & {} \bar{M}_{Q_2} = -b_3 e_1^0 + b_1 e_3^0 - a_3 \gamma _1^0 + a_1 \gamma _3^0, \ \bar{M}_{Q_3} = b_2 e_1^0 - b_1 e_2^0 + a_2 \gamma _1^0 - a_1 \gamma _2^0. \end{aligned}$$

Equations (16) are linear with respect to \(a_i, b_i \, (i=1,2,3)\). Considering them as unknowns, we find, e.g., \(b_2, b_3\), as the expressions of \(a_1, a_2, a_3, b_1\), \(e_i^0, \gamma _i^0\):

$$\begin{aligned}{} & {} b_2 = \frac{1}{ e_1^0 (e_1^{0^2} + e_2^{0^2} + e_3^{0^2})} \,(b_1 e_2^0 (e_1^{0^2} + e_2^{0^2} + e_3^{0^2}) + a_3 ( e_1^0 e_3^0 \gamma _2^0 -e_2^0 e_3^0 \gamma _1^0) \nonumber \\{} & {} - a_2 ((e_1^{0^2} + e_2^{0^2}) \gamma _1^0 + e_1^0 e_3^0 \gamma _3^0) + a_1 ((e_1^{0^2} + e_2^{0^2}) \gamma _2^0 + e_2^0 e_3^0 \gamma _3^0)), \nonumber \\{} & {} b_3 = \frac{1}{e_1^0 (e_1^{0^2} + e_2^{0^2} + e_3^{0^2})} \,(b_1 e_3^0 (e_1^{0^2} + e_2^{0^2} + e_3^{0^2}) - a_3 ((e_1^{0^2} + e_3^{0^2}) \gamma _1^0 + e_1^0 e_2^0 \gamma _2^0) \nonumber \\{} & {} + a_2 e_2^0 (e_1^0 \gamma _3^0 -e_3^0 \gamma _1^0) + a_1 (e_2^0 e_3^0 \gamma _2^0 + (e_1^{0^2} + e_3^{0^2}) \gamma _3^0)). \end{aligned}$$
(17)

Assuming \(e_3^0 = e_2^0, \, \gamma _3^0 = \gamma _2^0\) and \(a_2 = a_3 = 0\), we obtain \( \gamma _2^0 = -(b_1 e_2^0 \pm b_2 \sqrt{1 - 2 e_2^{0^2}})/a_1\) from the 1st relation (17). The 2nd relation (17) under the above value of \(\gamma _2^0\) takes the form \(b_3 = b_2\). So, when \(a_2 = a_3 = 0, \, b_3 = b_2\), we have 4 families of solutions of differential Eqs. (13):

$$\begin{aligned}{} & {} \omega _1 = \omega _3 = 0, \, e_1 = -\sqrt{1 - 2 e{_2^0}^2}, \, e_2 = e_3 = e{_2^0}, \, \gamma _1 = \mp \frac{\sqrt{a_1^2 - 2 z_1^2}}{a_1} , \nonumber \\{} & {} \gamma _2 = -\frac{z_1}{a_1}, \, \gamma _3 = -\frac{z_1}{a_1}; \nonumber \\{} & {} \omega _1 = \omega _3 = 0, \, e_1 = \sqrt{1 - 2 e{_2^0}^2}, \, e_2 = e_3 = e{_2^0}, \, \gamma _1 = \pm \frac{\sqrt{a_1^2 - 2 z_2^2}}{a_1} , \nonumber \\{} & {} \gamma _2 = -\frac{z_2}{a_1}, \, \gamma _3 = -\frac{z_2}{a_1}. \end{aligned}$$
(18)

Here \(z_1 = b_1 e{_2^0} + b_2 \sqrt{1 - 2 e{_2^0}^2}\), \(z_2 = b_1 e{_2^0} - b_2 \sqrt{1 - 2 e{_2^0}^2}\), and \(e{_2^0}\) is the parameter of the families.

The integral \(V_3\) takes the form \( -(2 e_2^0 z_1 \pm \sqrt{1 - 2 e_2^{0^2}} \sqrt{a_1^2 - 2 z_1^2})/a_1 = c_1\) on the first two families of solutions (18), and on the last two families, it is \( -(2 e_2^0 z_2 \mp \sqrt{1 - 2 e_2^{0^2}}\) \(\sqrt{a_1^2 - 2 z_2^2})/a_1 = c_1\). Thus, solutions (18) exist under any angles between vectors \(\textbf{E}\) and \(\boldsymbol{\gamma }\).

From a mechanical point of view, the elements of the families of solutions (18) correspond to the equilibria of the mechanical system under study.

From the stationary conditions

$$\begin{aligned} {\partial \varPhi }/{\partial \omega _1} = 0, \, {\partial \varPhi }/{\partial \omega _3} = 0, \, {\partial \varPhi }/{\partial \gamma _j} = 0, \, {\partial \varPhi }/{\partial e_j} = 0 \ (j=1,2,3) \end{aligned}$$

of the integral \(2 \varPhi = 2 \lambda _0 H - \lambda _1 V_1 - \lambda _2 V_2 - 2 \lambda _3 V_3\) we find the constraints on \(\lambda _i\), under which the first two families of solutions (18) satisfy these conditions:

$$\begin{aligned} \lambda _0 =-\frac{e_2^0 \sqrt{a_1^2 - 2 z_1^2} \pm \sqrt{1 - 2 e_2^{0^2}} z_1}{a_1^2 e_2^0}, \ \lambda _2 = \frac{b_1 z_1 \mp b_2 \sqrt{a_1^2 - 2 z_1^2}}{a_1^2 e_2^0}, \ \lambda _3 = \frac{z_1}{a_1 e_2^0}. \end{aligned}$$

Having substituted the latter expressions into the integral \(\varPhi \), we have:

$$\begin{aligned}{} & {} 2 \varPhi _{1,2} = \mp \frac{2(e_2^0 \sqrt{a_1^2 - 2 z_1^2} \pm \sqrt{1 - 2 e_2^{0^2}} z_1)}{a_1^2 e_2^0} \, H -V_1 - \frac{b_1 z_1 \mp b_2 \sqrt{a_1^2 - 2 z_1^2}}{a_1^2 e_2^0} \, V_2 \nonumber \\{} & {} - \frac{2 z_1}{a_1 e_2^0} \, V_3. \end{aligned}$$
(19)

By the same way, we find the integrals taking a stationary value on the elements of the last two families of solutions (18):

$$\begin{aligned}{} & {} 2 \varPhi _{3,4} = \pm \frac{2(e_2^0 \sqrt{a_1^2 - 2 z_2^2} \pm \sqrt{1 - 2 e_2^{0^2}} z_2)}{a_1^2 e_2^0} \, H - V_1 - \frac{b_1 z_2 \pm b_2 \sqrt{a_1^2 - 2 z_2^2}}{a_1^2 e_2^0} \, V_2 \nonumber \\{} & {} - \frac{2 z_2}{a_1 e_2^0} \, V_3. \end{aligned}$$

5 On Pendulum-Like Motions

In the problem under consideration, we could not obtain solutions corresponding to permanent rotations of the mechanical system. These motions are typical of rigid body dynamics. Basing on the analysis of the equations of motion (5) and (13), one can suppose that there are no such solutions. However, under the action of external potential forces the mechanical system can perform pendulum-like oscillations.

When \(a_2 = a_3 = b_1 = 0\), the relations

$$\begin{aligned} \omega _3 = 0, \, \gamma _1 = \pm 1, \, \gamma _2 = \gamma _3 = e_1 = 0 \end{aligned}$$
(20)

define two IMs of codimension 5 of the equations of motion (13).

The differential equations on these IMs are written as

$$\begin{aligned} \dot{\omega }_1 = \frac{b_3 e_2 - b_2 e_3}{A}, \, \dot{e}_2 = e_3 \omega _1, \, \dot{e}_3 = -e_2 \omega _1 \end{aligned}$$

and describe the pendulum-like oscillations of the body with a fixed point relative to the axis Ox in the frame rigidly attached to the body.

The integral \(V_3\) on IMs (20) is equal to zero identically that corresponds to the case of orthogonal vectors \(\boldsymbol{\gamma }\), \(\textbf{E}\). The integral \(\varPsi = (V_1 - 1) V_3\) assumes a stationary value on IMs (20).

Let us consider another similar solution for equations (13). It is the IM of codimension 3:

$$\begin{aligned} \omega _1 = \gamma _3 = e_3 = 0. \end{aligned}$$
(21)

This solution exists for \(a_3 = b_3 = 0\).

The differential equations on IM (21)

$$\begin{aligned}{} & {} \dot{\omega }_3 = \frac{b_2 e_1 - b_1 e_2 + a_2 \gamma _1 - a_1 \gamma _2}{C + D}, \\{} & {} \dot{\gamma }_1 = \gamma _2 \omega _3, \, \dot{\gamma }_2 = -\gamma _1 \omega _3, \, \dot{e}_1 = e_2 \omega _3, \, \dot{e}_2 = -e_1 \omega _3 \end{aligned}$$

describe the pendulum-like oscillations of the body relative to the axis Oz. The motions exist under any angle between the vectors \(\boldsymbol{\gamma }\), \(\textbf{E}\), because the integral \(V_3\) on IM (21) takes the form: \(e_1 \gamma _1 + e_2 \gamma _2 = c_1\). So, it is the family of IMs.

6 On the Stability of Stationary Sets

In this Section, we investigate the stability of the above found solutions on the base of the Lyapunov theorems on the stability of motion. To solve the problems, which often arise in the process of the analysis, the software package [7] written in Mathematica language is applied. In particular, the package gives a possibility to obtain the equations of the first approximation and their characteristic polynomial, using the equations of motion and the solution under study as input data, and then, to conduct the analysis of the polynomial roots, basing on the criteria of asymptotic stability of linear systems. When the problem of stability is solved by the Routh–Lyapunov method, the package, using the solution under study and the first integrals of the problem as input data, constructs a quadratic form and the conditions of its sign-definiteness in the form of the Sylvester inequalities. Their analysis is performed by means of Mathematica built-in functions, e.g., Reduce, RegionPlot3D.

6.1 The Case of Absence of External Forces

Let us investigate the stability of one of IMs (12), e.g.,

$$\begin{aligned} \omega _1 = \omega _3 = 0, \, e_1 - \gamma _1 = 0, \, e_2 - \gamma _2 = 0, \, e_3 - \gamma _3 = 0, \, \gamma _1^2 + \gamma _2^2 + \gamma _3^2 = 1, \end{aligned}$$

using the integral \(2 K_{2_1} = 2 \lambda _0 H - \lambda _1 (V_1 + V_2 - 2 V_3) - 2 \lambda _4 F_1 - 2 \lambda _5 F_2\) for obtaining its sufficient conditions.

We use the maps

$$\begin{aligned} \omega _1 = 0, \, \omega _3 = 0, \, e_1 = \pm z, \, e_2 = \gamma _2, \, e_3 = \gamma _3, \, \gamma _1 = \pm z \end{aligned}$$

on this IM. From now on, \(z = \sqrt{1 - \gamma _2^2 - \gamma _3^2}\).

Introduce the deviations:

$$\begin{aligned}{} & {} y_{1}= \omega _1, \ y_{2}= \omega _3, \ y_{3} = e_1 - z, \ y_{4}= e_2 - \gamma _2, \, y_{5}= e_3 - \gamma _3, \ y_6 = \gamma _1 - z. \end{aligned}$$

The 2nd variation of the integral \(K_{2_1}\) on the set defined by the first variations of the conditional integrals

$$\begin{aligned}{} & {} \delta V_1 = \pm 2 z \, y_6 = 0, \ \delta V_2 = 2 (\gamma _2 y_4 + \gamma _3 y_5 \pm z \, y_3) = 0, \\{} & {} \delta V_3 = \gamma _2 y_4 + \gamma _3 y_5 \pm z \, (y_3 + y_6) = 0, \end{aligned}$$

is written as:

$$\begin{aligned}{} & {} 2 \delta ^{2} K_{2_1} = \alpha _{11} y_1^2 + \alpha _{12} y_1 y_2 + \alpha _{22} y_2^2 + \alpha _{33} y_3^2 + \alpha _{34} y_3 y_4 + \alpha _{24} y_2 y_4 + \alpha _{13} y_1 y_3 \\{} & {} + \alpha _{23} y_2 y_3 + \alpha _{14} y_1 y_4 + \alpha _{44} y_4^2, \end{aligned}$$

where

$$\begin{aligned}{} & {} \alpha _{11} = \frac{((A - B) \, \gamma _2^2 + (B + D)(1 - \gamma _3^2)) \, \lambda _0}{2 \gamma _2^2}, \, \alpha _{12} = \pm \frac{ (B + D) \, \gamma _3 z \lambda _0}{\gamma _2^2}, \\{} & {} \alpha _{22} = \frac{((C + D) \, \gamma _2^2 + (B + D) \, \gamma _3^2) \, \lambda _0}{2 \gamma _2^2}, \, \alpha _{33} = \frac{(\gamma _2^2 - 1) \, \lambda _1}{2 \gamma _3^2}, \, \alpha _{34} = \mp \frac{\gamma _2 \, \lambda _1 z}{\gamma _3^2}, \\{} & {} \alpha _{24} = \Big (\frac{(C + D) \, \gamma _2}{\gamma _3} + \frac{(B + D) \, \gamma _3}{\gamma _2} \Big ) \lambda _6 \mp (B - C) \, z \lambda _5, \, \alpha _{44} = -\frac{(\gamma _2^2 + \gamma _3^2) \, \lambda _1}{2 \gamma _3^2}, \\{} & {} \alpha _{13} = \mp \frac{((A - B) \, \gamma _2^2 + B + D) \, z \lambda _5}{\gamma _2 \gamma _3} - (A + D) \lambda _6, \\{} & {} \alpha _{23} = -\frac{1}{\gamma _2 \gamma _3} ( (B + D) \, \gamma _3 \lambda _5 \mp (C + D) \, \gamma _2 z \lambda _6) + (B - C) \, \gamma _2 \lambda _5, \\{} & {} \alpha _{14} = -\frac{1}{\gamma _2 \gamma _3} (((B + D) + (A - B) \, \gamma _2^2 \mp (B + D) \, \gamma _3 z \lambda _6) \, \gamma _2 \lambda _5) - (A - B) \, \gamma _3 \lambda _5. \end{aligned}$$

The conditions of sign-definiteness of the quadratic form \(\delta ^{2} K_{2_1}\)

$$\begin{aligned}{} & {} \varDelta _1 = \frac{(\gamma _2^2 - 1) \, \lambda _1}{\gamma _3^2}> 0, \ \varDelta _2 = \frac{\lambda _1^2}{\gamma _3^2}> 0, \nonumber \\{} & {} \varDelta _3 = \frac{ \lambda _1}{\gamma _2^2 \gamma _3^2} [((C + D) \, \gamma _2^2 + (B + D) \, \gamma _3^2) \, \lambda _0 \lambda _1 + ((C + D)^2 \, \gamma _2^2 + ((B + D)^2 \nonumber \\{} & {} - (B - C)^2 \, \gamma _2^2) \, \gamma _3^2) (\lambda _5^2 + \lambda _6^2)]> 0, \nonumber \\{} & {} \varDelta _4 = \frac{1}{\gamma _2^2 \gamma _3^2} ((C + D) (B + D + (A - B) \gamma _2^2) + (A - C) (B + D) \gamma _3^2) \nonumber \\{} & {} \times [\lambda _0^2 \lambda _1^2 + (B + C + 2 D + (A - B) \, \gamma _2^2 + (A - C) \, \gamma _3^2) \, \lambda _0 \lambda _1 (\lambda _5^2 + \lambda _6^2) \nonumber \\{} & {} + ((C \! + \! D) (B \! + \! D + \! (A \! - \! B) \, \gamma _2^2) \! + \! (A \! - \! C) (B \! + \! D) \, \gamma _3^2) (\lambda _5^2 \! + \! \lambda _6^2)^2] > 0. \end{aligned}$$
(22)

are sufficient for the stability of the IM under study.

The differential equations \(\dot{\gamma }_2 = 0, \, \dot{\gamma }_3 = 0\) on IMs (12) have the family of solutions:

$$\begin{aligned} \gamma _2 = \gamma _2^0 = \text{ const }, \, \gamma _3 = \gamma _3^0 = \text{ const }. \end{aligned}$$
(23)

Thus, each of IMs (12) can be considered as a family of IMs, where \(\gamma _2^0, \gamma _3^0\) are the parameters of the family.

Let \(\gamma _3^0 = \gamma _2^0\) and \(\lambda _5 = \lambda _6 = \lambda _1\). Taking into consideration (23) and the above constraints, inequalities (22) take the form:

$$\begin{aligned}{} & {} \frac{(\gamma _2^{0^2} - 1) \, \lambda _1}{\gamma _2^{0^2}}> 0, \ \frac{\lambda _1^2}{\gamma _2^{0^2}}> 0, \nonumber \\{} & {} \frac{\lambda _1^2}{\gamma _2^{0^2}} ((B + C + 2 D) \lambda _0 + 2 ((B + D)^2 + (C \! + \! D)^2 \! - \! (B \! - \! C)^2 \gamma _2^{0^2}) \, \lambda _1)> 0, \nonumber \\{} & {} \frac{\lambda _1^2}{\gamma _2^{0^4}} ((B + D) (C + D) + ((A - D) (B + C) + 2 (A D - B C)) \, \gamma _2^{0^2}) \nonumber \\{} & {} \times (\lambda _0^2 + 2 (B + C + 2 D - (B + C - 2\,A) \, \gamma _2^{0^2}) \lambda _0 \lambda _1 + 4 ((B + D) (C + D) \nonumber \\{} & {} + ((A - D) (B + C) + 2 (A D - B C)) \, \gamma _2^{0^2}) \lambda _1^2) > 0. \end{aligned}$$

With the help of the built-in function Reduce, we find the conditions of compatibility of the latter inequalities:

$$\begin{aligned}{} & {} A> B> C> 0 \ \text{ and } \ A< B + C, \, D> 0 \ \text{ and } \\{} & {} \Big [ \Big ( \Big (\lambda _0> 0 \ \text{ and } \ \Big (\sigma _1< \lambda _1< \sigma _2 - \frac{\sigma _3}{4} \ \text{ or } \ \sigma _2 + \frac{\sigma _3}{4}< \lambda _1< 0 \Big ) \ \text{ and } \\{} & {} \Big (-1< \gamma _2^0< -\frac{1}{\sqrt{2}} \ \text{ or } \ \frac{1}{\sqrt{2}}< \gamma _2^0< 1 \Big ) \Big ) \, \text{ or } \\{} & {} \Big ( \lambda _0 > 0 \ \text{ and } \ \sigma _2 + \frac{\sigma _3}{4}< \lambda _1< 0 \ \text{ and } \ \Big (-\frac{1}{\sqrt{2}} \le \gamma _2^0< 0 \ \text{ or } \ 0 < \gamma _2^0 \le \frac{1}{\sqrt{2}} \Big ) \Big ) \Big ]. \end{aligned}$$

Here

$$\begin{aligned}{} & {} \sigma _1 = \frac{(B + C + 2 D) \lambda _0}{2 ((B - C)^2 \gamma _2^{0^2} - (B^2 + C^2 + 2 B D + 2 D (C + D))) }, \\{} & {} \sigma _2 = \frac{((B + C + 2 D - (B + C-2\,A) \gamma _2^{0^2}) \lambda _0}{4 ( (2 B C + (B + C) D - A (B + C + 2 D)) \gamma _2^{0^2} - (B + D) (C + D))}, \\{} & {} \sigma _3 = \frac{\sqrt{(B - C)^2 - 2 (B - C)^2 \gamma _2^{0^2} + (B + C-2\,A)^2 \gamma _2^{0^4}} \lambda _0}{(B + D) (C + D) + (A (B + C + 2 D)-2 B C - (B + C) D) \gamma _2^{0^2}}. \end{aligned}$$

The constraints on the parameter \(\gamma _2^0\) give the sufficient conditions of stability for the elements of the family of IMs. The constraints imposed on the parameters \(\lambda _0, \lambda _1\) isolate a subfamily of the family of the integrals \(K_{2_1}\), which allows one to obtain these sufficient conditions. The analysis of stability of the 2nd IM of IMs (12) is done analogously.

Let us investigate the stability of IM (7), using the integral \(2 K_1 = 2 \lambda _0 H - \lambda _1 (V_1 - V_2)^2 - \lambda _2 F_1 F_2\) for obtaining sufficient conditions. The analysis is done in the map \(\omega _1 = 0\), \(\omega _3 = 0\), \(\gamma _1 = - z_1\), \(e_1 = - z_2\) on this IM. From now on, \(z_1 = \sqrt{1 - \gamma _2^2 - \gamma _3^2}\), \(z_2 = \sqrt{1 - e_2^2 - e_3^2}\).

In order to reduce the amount of computations we restrict our consideration by the case when the following restrictions are imposed on the geometry of mass of the mechanical system: \(A = 3\,C/2, \, B =2\,C, \, D = C/2\).

Introduce the deviations from the unperturbed solution:

$$\begin{aligned} y_1 = \omega _1, \ y_2 = \omega _2, \ y_3 = \gamma _1 + z_1, \ y_4 = e_1 + z_2. \end{aligned}$$

The 2nd variation of the integral \(K_1\) in the deviations on the set

$$\begin{aligned} \delta V_1 = -2 z_1 y_3 = 0, \ \delta V_2 = -2 z_2 y_4 = 0 \end{aligned}$$

has the form: \(2 \delta ^2 K_1 = \beta _{11} y_1^2 + \beta _{12} y_1 y_2 + \beta _{22} y_2^2\), where \( \beta _{11}, \beta _{12}, \beta _{22}\) are the expressions of \(C, \gamma _2, \gamma _3, e_2, e_3\). These are bulky enough and presented entirely in Appendix.

Taking into consideration that \( \gamma _2 = \gamma _2^0 = \text{ const }, \, \gamma _3 = \gamma _3^0 = \text{ const }, \, e_2 = e_2^0 = \text{ const }, \, e_3 = e_3^0 = \text{ const }\) (9) on IM (7), and introducing the restrictions on the parameters \(\gamma _3^0 = \gamma _2^0, \, e_3^0 = e_2^0\), we write the conditions of positive definiteness of the quadratic form \(2 \delta ^2 K_1\) (the Sylvester inequalities) as follows:

$$\begin{aligned}{} & {} \varDelta _1 = 2 [ \sqrt{1 - 2 e_2^{0^2}} (\gamma _2^{0^2} + e_2^{0^2} (1 - 4 \gamma _2^{0^2})) \nonumber \\{} & {} -2 e_2^0 \gamma _2^0 (1 - 2 e_2^{0^2}) \sqrt{1 - 2 \gamma _2^{0^2}} \, ] \, z + 1> 0, \nonumber \\{} & {} \varDelta _2 = - \frac{1}{e_2^{0^2}} \Big (8 \gamma _2^{0^2} + e_2^{0^2} (6 - 32 \gamma _2^{0^2}) - 15 - 16 e_2^0 \sqrt{1 - 2 e_2^{0^2}} \gamma _2^0 \sqrt{1 - 2 \gamma _2^{0^2}} \nonumber \\{} & {} + 2 \Big (2 e_2^0 \gamma _2^0 (1 - 2 e_2^{0^2}) (15 - 14 e_2^{0^2} -16 \gamma _2^{0^2} (1 - 4 e_2^{0^2})) \sqrt{1 - 2 \gamma _2^{0^2}} + \sqrt{1 - 2 e_2^{0^2}} \nonumber \\{} & {} \times (3 e_2^{0^2} (2 e_2^{0^2} -5) - ( 120 e_2^{0^4} - 106 e_2^{0^2} + 15) \, \gamma _2^{0^2}\nonumber \\{} & {} + 8 (32 e_2^{0^4} - 16 e_2^{0^2} + 1) \, \gamma _2^{0^4}) \Big ) z + \Big (\gamma _2^{0^4} (15 - 8\gamma _2^{0^2})^2 \nonumber \\{} & {} + 4 e_2^0 \sqrt{1 - 2 e_2^{0^2}} \gamma _2^0 \sqrt{1 - 2 \gamma _2^{0^2}} (15 \! - \! 14 e_2^{0^2} \! - \! 16 (1 - 4 e_2^{0^2}) \, \gamma _2^{0^2}) \nonumber \\{} & {} \times (3 e_2^{0^2} (2 e_2^{0^2} - 5) - (120 e_2^{0^4} -106 e_2^{0^2} + 15) \, \gamma _2^{0^2} + 8 (32 e_2^{0^4} - 16 e_2^{0^2} + 1) \, \gamma _2^{0^4}) \nonumber \\{} & {} + e_2^{0^2} (9 e_2^{0^2} (5 - 2 e_2^{0^2})^2 - 2 (1504 e_2^{0^6} - 4508 e_2^{0^4} + 3420 e_2^{0^2} - 675) \, \gamma _2^{0^2} \nonumber \\{} & {} + 4 ( 8736 e_2^{0^6} - 17264 e_2^{0^4} + 9761 e_2^{0^2} - 1785) \gamma _2^{0^4} - 32 ( 3840 e_2^{0^6} - 5312 e_2^{0^4} \nonumber \\{} & {} + 2300 e_2^{0^2} - 325 ) \, \gamma _2^{0^6} - 4096 (1 - 4 e_2^{0^2})^2 (1 - 2 e_2^{0^2}) \, \gamma _2^{0^8}) \Big ) z^2 \Big ) > 0. \end{aligned}$$
(24)

Here \(z = C \lambda _2, \, \lambda _0 = 1\).

The system of inequalities (24) has been solved graphically. The built-in function RegionPlot3D is used. The region, in which the inequalities have common values, is shown in Fig. 2 (dark region). Thus, when the values of the parameters \(z, e_2^0, \gamma _2^0\) lie in this region, the IM under study is stable.

Fig. 2.
figure 2

The region of stability of the IM for \(\gamma _2^0 \in [ -\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}]\),\(e_2^0 \in (0, \frac{1}{\sqrt{2}}]\), \(z \in (0, 3]\)

Full size image

6.2 The Case of the Presence of External Forces

In this Subsection, we analyze the stability of the elements of the families of solutions (18). Let us investigate one of the first two families, e.g.,

$$\begin{aligned}{} & {} \omega _1 = \omega _3 = 0, \, e_1 = -\sqrt{1 - 2 e{_2^0}^2}, \, e_2 = e_3 = e{_2^0}, \, \gamma _1 = - \frac{\sqrt{a_1^2 - 2 z^2}}{a_1}, \nonumber \\{} & {} \gamma _2 = -\frac{z}{a_1}, \, \gamma _3 = -\frac{z}{a_1}, \end{aligned}$$
(25)

where \(z = b_1 e{_2^0} + b_2 \sqrt{1 - 2 e{_2^0}^2}\).

The integral

$$\begin{aligned} 2 \varPhi _{1} \! = \! - \frac{2(e_2^0 \sqrt{a_1^2\! -\! 2 z^2}\! + \sqrt{1 - 2 e_2^{0^2}} z)}{a_1^2 e_2^0} \, H \! - \! V_1 \! - \! \frac{b_1 z - b_2 \sqrt{a_1^2\! - 2 z^2}}{a_1^2 e_2^0} \, V_2 - \frac{2 z}{a_1 e_2^0} \, V_3 \end{aligned}$$

is used for obtaining the sufficient conditions.

In the deviations

$$\begin{aligned}{} & {} y_1 = e_1 + \sqrt{1 - 2 e_2^{0^2}}, \, y_2 = e_2 - e_2^0, \, y_3 = e_3 - e_2^0, \, y_4 = \gamma _1 + \frac{\sqrt{a_1^2 - 2 z^2}}{a_1}, \\{} & {} y_5 = \gamma _2 + \frac{z}{a_1}, \, y_6 = \gamma _3 + \frac{z}{a_1}, \, y_7 = \omega _1, \, y_8 = \omega _2 \end{aligned}$$

on the linear manifold

$$\begin{aligned}{} & {} \delta H \! = \! b_1 y_1 + b_2 (y_2 + y_3) + a_1 y_4 \! = \! 0, \, \delta V_1 \! = \! -\frac{2}{a_1} \Big ( z (y_5 + y_6) \! + \! \sqrt{a_1^2 \! - \! 2 z^2} \, y_4 \Big ) \! = \! 0, \\{} & {} \delta V_2 = 2 (e_2^0 \, (y_2 + y_3) - \sqrt{1 - 2 e_2^{0^2}} y_1) = 0, \\{} & {} \delta V_3 = e_2^0 (y_5 + y_6) - \sqrt{1 - 2 e_2^{0^2}} y_4 - \frac{1}{a_1} \Big ( z (y_2 + y_3) + \sqrt{a_1^2 - 2 z^2} \, y_1 \Big ) = 0 \end{aligned}$$

the 2nd variation of the integral \(\varPhi _1\) has the form: \(\delta ^2 \varPhi _1 = Q_1 + Q_2\), where

$$\begin{aligned}{} & {} Q_1 = \frac{1}{2 a_1^2 e_2^{0^3}} \Big ((3 b_2 e_2^0 \sqrt{1 \! - \! 2 e_2^{0^2}} \! - \! b_1 (1 - 4 e_2^{0^2})) \, z + b_2 (1 - e_2^{0^2}) \sqrt{a_1^2 - 2 z^2} \\{} & {} \! - \! a_1^2 e_2^0 \Big ) \, y_1^2 \! + \! \frac{1}{a_1^2 e_2^{0^2}} \Big (\sqrt{1 - 2 e_2^{0^2}} (b_1 z \! - \! b_2 \sqrt{a_1^2 \! - \! 2 z^2}) \! - \! \sqrt{a_1^2 \! - \! 2 z^2} z \Big ) \, y_1 y_2 \\{} & {} \! + \! \frac{1}{a_1^2 e_2^0} \, \Big ( b_2 \sqrt{a_1^2 \! - \! 2 z^2} \! - \! b_1 z \Big ) y_2^2 \! + \! \frac{1}{a_1 e_2^{0^2}} \Big (e_2^0 \sqrt{a_1^2 \! - \! 2 z^2} \! - \! \sqrt{1 \! - \! 2 e_2^{0^2}} z \Big ) \, y_1 y_6 \\{} & {} + \frac{2 z}{a_1 e_2^0} \, y_2 y_6 - y_6^2, \\{} & {} Q_2 = -\frac{B + C + 2 D}{2 a_1^2 e_2^0} \Big (\sqrt{1 - 2 e_2^{0^2}} z + e_2^0 \sqrt{a_1^2 - 2 z^2} \Big ) \, y_8^2 \\{} & {} + \frac{(B + D) \sqrt{ 1 - 2 e_2^{0^2}}}{a_1^2 e_2^{0^2}} \Big (\sqrt{1 - 2 e_2^{0^2}} \, z + e_2^0 \sqrt{a_1^2 - 2 z^2} \Big ) \, y_7 y_8 \\{} & {} -\frac{1}{2 a_1^4 e_2^{0^3}} \Big (a_1^2 [(A e_2^{0^2} + B (1 - 2 e_2^{0^2})) (\sqrt{1 - 2 e_2^{0^2}} \, z + e_2^0 \sqrt{a_1^2 - 2 z^2}) \\{} & {} + D \sqrt{1 - 2 e_2^{0^2}} ((1 - 4 e_2^{0^2}) \, z + e_2^0 \sqrt{1 - 2 e_2^{0^2}} \sqrt{a_1^2 - 2 z^2})] \\{} & {} - D \, [(1 - 8 e_2^{0^2}) (b_1^3 e_2^{0^3} \sqrt{1 - 2 e_2^{0^2}} + b_2^3 (1 - 2 e_2^{0^2})^2 + 3 b_1 b_2 e_2^0 (1 - 2 e_2^{0^2}) \, z) \\{} & {} + e_2^0 (3 - 8 e_2^{0^2}) \sqrt{a_1^2 - 2 z^2} \, z^2] \Big ) \, y_7^2. \end{aligned}$$

The analysis of sign-definiteness of the quadratic forms \(Q_1\) and \(Q_2\) was done for the case when \(b_1 = 0\) and \(A = 3\,C/2, \, B = 2\,C, \, D = C/2\). Under these restrictions on the parameters, the conditions of negative definiteness of the quadratic forms \(Q_1\) and \(Q_2\) are respectively written as:

$$\begin{aligned}{} & {} \varDelta _1 = -1< 0, \ \varDelta _2 = - \frac{1}{a_1^2 e_2^{0^2}} \, \Big (b_2 (b_2 (1 - 2 e_2^{0^2}) + e_2^0 \sqrt{a_1^2 - 2 b_2^2 (1 - 2 e_2^{0^2})}) \Big ) > 0, \nonumber \\{} & {} \varDelta _3 = \frac{b_2}{ a_1^4 e_2^{0^5}} \Big ( 2 b_2 e_2^0 \, (a_1^2 (1 - 3 e_2^{0^2}) - b_2^2 \, (16 e_2^{0^4} - 14 e_2^{0^2} + 3)) \nonumber \\{} & {} + \sqrt{a_1^2 -2 b_2^2 (1 - 2 e_2^{0^2})} \, (a_1^2 e_2^{0^2} + b_2^2 (16 e_2^{0^4} - 10 e_2^{0^2} + 1)) \Big ) < 0 \end{aligned}$$
(26)

and

$$\begin{aligned}{} & {} \varDelta _1 = -\frac{2\,C}{a_1^2 e_2^0} \Big (b_2 \, (1 - 2 e_2^{0^2}) + e_2^0 \sqrt{a_1^2 - 2 b_2^2 (1 - 2 e_2^{0^2})} \Big ) < 0, \nonumber \\{} & {} \varDelta _2 = \frac{C^2}{a_1^6 e_2^{0^4}} \, \Big ( 3 a_1^4 e_2^{0^2} \, (5 - 2 e_2^{0^2}) - 8 b_2^4 \, (1 - 2 e_2^{0^2})^2 (32 e_2^{0^4} - 16 e_2^{0^2} + 1) \nonumber \\{} & {} + a_1^2 b_2^2 \, (15 - 4 e_2^{0^2} \, (60 e_2^{0^4} - 83 e_2^{0^2} + 34)) - 2 b_2 e_2^0 \, (1 - 2 e_2^{0^2}) \nonumber \\{} & {} \times (a_1^2 \, (14 e_2^{0^2} \! - \! 15) + 16 b_2^2 \, (8 e_2^{0^4} \! - \! 6 e_2^{0^2} \! + \! 1)) \, \sqrt{a_1^2 \! - \! 2 b_2^2 (1 \! - \! 2 e_2^{0^2})} \Big ) > 0. \end{aligned}$$
(27)

Taking into consideration the conditions for solutions (25) to be real

$$\begin{aligned} a_1 \ne 0 \ \text{ and } \ \Big (e_2^0 = \pm \frac{1}{\sqrt{2}} \ \text{ or } \ \Big (-\frac{1}{\sqrt{2}}< e_2^0 < \frac{1}{\sqrt{2}} \ \text{ and } \, -\sigma _1 \le b_2 \le \sigma _1 \Big ) \Big ) \end{aligned}$$
(28)

under the above restrictions on the parameters \(b_1, A, B, D\), inequalities (26) and (27) are compatible when the following conditions

$$\begin{aligned}{} & {} a_1 \ne 0, \, C> 0 \ \text{ and } \ \Big ( \Big (b_2< 0, \, \sigma _2< e_2^0 \le \frac{1}{\sqrt{2}} \Big ) \ \text{ or } \nonumber \\{} & {} \Big (b_2 > 0, \, -\frac{1}{\sqrt{2}} \le e_2^0 < -\sigma _2 \Big ) \Big ) \end{aligned}$$
(29)

hold.

$$\begin{aligned} \text{ Here } \ \ \sigma _1 = \sqrt{ \frac{a_1^2}{2 (1 - 2 e_2^{0^2})}}, \ \sigma _2 = \sqrt{\frac{b_2^2}{a_1^2 + 2 b_2^2}}. \end{aligned}$$

The latter conditions are sufficient for the stability of the elements of the family of solutions under study. Let us compare them with necessary ones which we shall obtain, using the Lyapunov theorem on stability in linear approximation [9].

The equations of the 1st approximation in the case considered are written as:

$$\begin{aligned}{} & {} \dot{y}_1 = 2 e_2^0 y_8 - \sqrt{z_1} \, y_7, \, \dot{y}_2 = e_2^0 y_7 + \sqrt{z_1} \, y_8, \, \dot{y}_3 = \Big (e_2^0 - \frac{1}{e_2^0} \Big ) \, y_7 + \sqrt{z_1} \, y_8, \nonumber \\{} & {} \dot{y}_4 = \frac{b_2}{a_1} \Big ( \frac{z_1}{e_2^0} \, y_7 - 2 \sqrt{z_1} \, y_8 \Big ), \, \dot{y}_5 = \frac{1}{a_1} \Big ( \sqrt{a_1^2 - 2 b_2^2 z_1} \, y_8 -b_2 \sqrt{z_1} \, y_7 \Big ), \nonumber \\{} & {} \dot{y}_6 = \frac{1}{a_1} \, \Big ( \frac{ \sqrt{a_1^2 - 2 b_2^2 z_1} \, (e_2^0 \, y_8 - \sqrt{z_1} \, y_7)}{e_2^0} + b_2 \sqrt{z_1} \, y_7 \Big ), \nonumber \\{} & {} \dot{y}_7 = \frac{1}{z_2} \Big (16 a_1^2 b_2 e_2^{0^2} (y_3 - y_2) + 2 a_1^2 e_2^0 \sqrt{z_1} \, (5 a_1 y_5 - 2 b_2 y_1 - 3 a_1 y_6) \Big ), \nonumber \\{} & {} \dot{y}_8 = \frac{1}{z_2} \, \Big (2 a_1^2 [b_2 \, (4 e_2^{0^2} - 5) \, y_1 + 5 a_1 y_5 + a_1 e_2^{0^2} (3 y_6 - 7 y_5)] \nonumber \\{} & {} + 10 a_1^2 b_2 e_2^0 \sqrt{z_1} \, (y_3 - y_2) + 2 b_2^2 z_1 \, (4 e_2^{0^2} - 1) (a_1 \, (y_5 + y_6) - 2 b_2 y_1) \nonumber \\{} & {} - 4 b_2 e_2^0 z_1 \sqrt{a_1^2 - 2 b_2^2 z_1} \, (a_1 \, (y_5 + y_6) -2 b_2 y_1) \Big ). \end{aligned}$$
(30)

Here \(z_1 = 1 - 2 e_2^{0^2}\), \(z_2 = C (3 a_1^2 (2 e_2^{0^2} - 5) - 8 b_2 z_1 \, (b_2 \, (4 e_2^{0^2} - 1) - 2 e_2^0 \sqrt{a1^2 - 2 b_2^2 z_1}))\).

The characteristic equation of system (30) has the form:

$$\begin{aligned} \lambda ^4 \, (\lambda ^4 + \alpha _1 \lambda ^2 + \alpha _2) = 0, \end{aligned}$$
(31)

where

$$\begin{aligned}{} & {} \alpha _1 = \frac{4\,C }{z_2^2} \, \Big (a_1^4 e_2^0 \, [ 2 b_2 \, (251 e_2^{0^2} - 122 e_2^{0^4} - 137) + 3 (10 e_2^{0^4} - 33 e_2^{0^2} + 20) \\{} & {} \sqrt{a_1^2 - 2 b_2^2 z_1} ] + 8 b_2^4 z_1^2 \, [ (64 e_2^{0^4} \! - \! 24 e_2^{0^2} + 1) \sqrt{a_1^2 - 2 b_2^2 z_1} \! - \! 2 b_2 e_2^0 \, (64 e_2^{0^4} \! - \! 40 e_2^{0^2} \\{} & {} + 5)] - a_1^2 b_2^2 z_1 \, [ (432 e_2^{0^4} - 518 e_2^{0^2} + 47) \sqrt{a_1^2 - 2 b_2^2 z_1} -2 b_2 e_2^0 \, (560 e_2^{0^4} - 706 e_2^{0^2} \\{} & {} + 173) ] \Big ), \\{} & {} \alpha _2 = \frac{1}{z_2^2} \, \Big ( (8 a_1^4 b_2^2 \, (240 e_2^{0^6} - 408 e_2^{0^4} + 206 e_2^{0^2} - 19) + 12 a_1^6 \, (4 e_2^{0^2} \, (e_2^{0^2} - 3) \\{} & {} + 5) - 64 a_1^2 b_2^4 \, (64 e_2^{0^6} - 80 e_2^{0^4} + 24 e_2^{0^2} - 1) \, z_1) - 8 a_1^2 b_2 e_2^0 \sqrt{a_1^2 - 2 b_2^2 z_1} \\{} & {} \times (a_1^2 (56 e_2^{0^4} - 110 e_2^{0^2} + 53) - 8 b_2^2 \, (32 e_2^{0^4} - 32 e_2^{0^2} + 5) \, z_1) \Big ). \end{aligned}$$

The roots of the bipolynomial in the round brackets are purely imaginary when the conditions

$$\begin{aligned} \alpha _1> 0, \, \alpha _2> 0, \, \alpha _1^2 - 4 \alpha _2 > 0 \end{aligned}$$

hold.

Taking into consideration (28), the latter inequalities are hold under the following constraints imposed on the parameters \(C, a_1, b_2, e_2^0\):

$$\begin{aligned}{} & {} C> 0 \ \text{ and } \ \Big [ a_1< 0 \ \text{ and } \ \Big ( \Big ( \Big ( b_2< \frac{3 a_1}{\sqrt{2}} \ \text{ or } \ \frac{3 a_1}{\sqrt{2}}< b_2< \frac{a_1}{\sqrt{2}} \Big ) \ \text{ and } \nonumber \\{} & {} \frac{\rho _1}{2} \le e_2^0 \le \frac{1}{\sqrt{2}} \Big ) \ \text{ or } \ \Big (b_2 = \frac{3 a_1}{\sqrt{2}} \ \text{ and } \ \frac{\rho _1}{2} \le e_2^0< \frac{1}{\sqrt{2}} \Big ) \ \text{ or } \ \big ( b_2 = \frac{ a_1}{\sqrt{2}} \ \text{ and } \nonumber \\{} & {} -\frac{\rho _1}{2}< e_2^0 \le \frac{1}{\sqrt{2}} \Big ) \ \text{ or } \Big ( \frac{a_1}{\sqrt{2}}< b_2< 0 \ \text{ and } \ \rho _2< e_2^0 \le \frac{1}{\sqrt{2}} \Big ) \Big ) \Big ] \ \text{ or } \nonumber \\{} & {} C> 0 \ \text{ and } \ \Big [ a_1> 0 \ \text{ and } \ \Big ( \Big ( 0< b_2< \frac{a_1}{\sqrt{2}} \ \text{ and } \ -\frac{1}{\sqrt{2}} \le e_2^0< -\rho _2 \Big ) \ \text{ or } \nonumber \\{} & {} \Big ( \Big ( \frac{a_1}{\sqrt{2}}< b_2< \frac{3 a_1}{\sqrt{2}} \ \text{ or } \ b_2 > \frac{3 a_1}{\sqrt{2}} \Big ) \ \text{ and } \ -\frac{1}{\sqrt{2}} \le e_2^0 \le -\frac{\rho _1}{2} \Big ) \ \text{ or } \nonumber \\{} & {} \Big ( b_2 = \frac{ a_1}{\sqrt{2}} \ \text{ and } \ -\frac{1}{\sqrt{2}} \le e_2^0< -\frac{\rho _1}{2} \Big ) \ \text{ or } \ \Big (b_2 = \frac{3 a_1}{\sqrt{2}} \ \text{ and } \nonumber \\{} & {} -\frac{1}{\sqrt{2}} < e_2^0 \le -\frac{\rho _1}{2} \Big ) \Big ) \Big ]. \end{aligned}$$
(32)
$$\begin{aligned} \text{ Here } \ \rho _1 = \sqrt{\frac{ 2 b_2^2 - a_1^2}{b_2^2}}, \ \rho _2 = \sqrt{\frac{ a_1^2 + b_2^2 - \sqrt{b_2^2 \, (2 a_1^2 + 5 b_2^2)}}{2 a_1^2 + 4 b_2^2}}. \end{aligned}$$

The analysis of zero roots of characteristic Eq. (31) was done by the technique applied in [10]. The analysis shown that the characteristic equation has zero roots with simple elementary divisors. Whence it follows, the elements of the family of solutions under study are stable in linear approximation when conditions (32) hold. Comparing them with (29), we conclude that the sufficient conditions are close to necessary ones. The analogous result has been obtained for the 2nd family of solutions. Instability was proved for the rest of the families of solutions.

7 Conclusion

The qualitative analysis of the differential equations describing the motion of the nonholonomic mechanical system has been done. The solutions of these equations, which correspond to the equilibria and pendulum-like motions of the mechanical system, have been found. The Lyapunov stability of the solutions has been investigated. In some cases, the obtained sufficient conditions were compared with necessary ones. The analysis was done nearly entirely in symbolic form. Computational difficulties were in the main caused by the problem of bulky expressions: the differential equations are rather bulky, and the first integrals of these equations are the polynomials of the 2nd–5th degrees. Computer algebra system Mathematica was applied to solve computational problems. The results presented in this work show the efficiency of the approach used for the analysis of the problem as well as computational tools.