On the Qualitative Analysis of the Equations of Motion of a Nonholonomic Mechanical System

Abstract

The problem on the rotation of a dynamically asymmetric rigid body around a fixed point is considered. The body is fixed inside a spherical shell, which a ball and a disk adjoin to. The equations of motion of the mechanical system in the case of absence of external forces admit two additional first integrals and these are completely integrable. The nonintegrable case, when potential forces act upon the system, is also considered. The qualitative analysis of the equations of motion is done in the both cases: stationary sets are found and their Lyapunov stability is studied. A mechanical interpretation for the obtained solutions is given.

Appendix

$$\begin{aligned} \begin{array}{l} \beta _{11} = (4 e_2^2)^{-1} C \, \Big [ ((e_3^2 - 1) (\gamma _2^2 - 5) - e_2^2 (1 - \gamma _2^2 + z_1^2) + 2 e_2 \gamma _2 z_1 z_2) \, \lambda _0 \\ + C \, [(5 e_3^5 \gamma _2 \, (3 \gamma _3^2 - z_1^2) + e_2 e_3^4 \gamma _3 \, (43 \gamma _2^2 + 20 \gamma _3^2 - 25) + e_2 \gamma _3 \, (e_2^2 - 5) \\ \times (5 - 3 \gamma _2^2 - \gamma _3^2 + e_2^2 \, (4 \gamma _2^2 + \gamma _3^2 - 2)) + e_2 e_3^2 \gamma _3 (50 - 58 \gamma _2^2 - 25 \gamma _3^2 \\ + e_2^2 \, (59 \gamma _2^2 + 17 \gamma _3^2 - 27)) + e_3 \gamma _2 \, (e_2^2 (65 - 37 \gamma _2^2 - 46 \gamma _3^2) + 5 (\gamma _2^2 + 3 \gamma _3^2 - 5) \\ + e_2^4 \, (36 \gamma _2^2 + 23 \gamma _3^2 - 28)) + e_3^3 \gamma _2 \, (e_2^2 (37 \gamma _2^2 + 55 \gamma _3^2 - 33) - 5 (2 \gamma _2^2 + 7 \gamma _3^2 - 6))) \, z_1 \\ + (e_2^4 \gamma _2 \gamma _3 \, (4 (1 - \gamma _2^2) - 3 \gamma _3^2) + e_2^2 \gamma _2 \gamma _3 \, (21 \gamma _2^2 + 16 \gamma _3^2 - 25 + e_3^2 \, (53 - 57 \gamma _2^2 \\ - 45 \gamma _3^2)) - 5 \gamma _2 \gamma _3 \, (e_3^2 - 1) (5 - \gamma _2^2 - \gamma _3^2 + e_3^2 (3 \gamma _2^2 + 4 \gamma _3^2 - 3)) - e_2^3 e_3 \, (10 + 36 \gamma _2^4 \\ - 18 \gamma _3^2 + 7 \gamma _3^4 + \gamma _2^2 \, (41 \gamma _3^2 - 46)) + e_2 e_3 \, (25 - 60 \gamma _2^2 + 19 \gamma _2^4 + (44 \gamma _2^2 - 45) \gamma _3^2 \\ + 15 \gamma _3^4 - e_3^2 \, (10 - 29 \gamma _2^2 + 19 \gamma _2^4 + (53 \gamma _2^2 - 35) \gamma _3^2 + 20 \gamma _3^4))) \, z_2] \, \lambda _2 \Big ], \\ \beta _{22} = (4 e_2^2)^{-1} \, C \, \Big [(3 e_2^2 + 5 e_3^2 - (e_3 \gamma _2 - e_2 \gamma _3)^2) \, \lambda _0 + C \, [(3 e_2^5 \gamma _3 \, (4 \gamma _2^2 + \gamma _3^2 - 1) \\ + e_2^4 e_3 \gamma _2 \, (15 - 12 \gamma _2^2 + 19 \gamma _3^2) - 5 e_3^3 \gamma _2 \, (5 - \gamma _2^2 - 3 \gamma _3^2 + e_3^2 \, (\gamma _2^2 + 4 \gamma _3^2 - 1)) \\ + e_2^2 e_3 \gamma _2 \, (9 \gamma _2^2 - 13 \gamma _3^2 - 21 + e_3^2 \, (24 - 19 \gamma _2^2 + 5 \gamma _3^2)) + e_2 e_3^2 \gamma _3 \, (5 + 11 \gamma _2^2 - 15 \gamma _3^2 \\ + e_3^2 \, (15 - 21 \gamma _2^2 + 20 \gamma _3^2)) + e_2^3 \gamma _3 \, (3 (3 - 3 \gamma _2^2 - \gamma _3^2) + e_3^2 \, (8 - 3 \gamma _2^2 + 21 \gamma _3^2))) \, z_1 \\ + (3 e_2^4 \gamma _2 \gamma _3 \, (3 - 4 \gamma _2^2 - 3 \gamma _3^2) + e_2^2 \gamma _2 \gamma _3 \, (e_3^2 (9 \gamma _2^2 - 15 \gamma _3^2 - 14) + 3 \, (\gamma _2^2 + \gamma _3^2 - 3)) \\ + 5 e_3^2 \gamma _2 \gamma _3 \, (5 - \gamma _2^2 - \gamma _3^2 + e_3^2 \, (3 \gamma _2^2 + 4 \gamma _3^2 - 3)) + e_2^3 e_3 (6 + 12 \gamma _2^4 + 5 \gamma _3^2 - 11 \gamma _3^4 \\ - \gamma _2^2 \, (21 + 13 \gamma _3^2)) + e_2 e_3 (5 \gamma _3^2 \, (\gamma _3^2 - 3) + \gamma _2^2 (15 + 2 \gamma _3^2) - 3 \gamma _2^4 + e_3^2 \, (13 \gamma _2^4 \\ + \gamma _2^2 \, (11 \gamma _3^2 - 23) - 5 \, (\gamma _3^2 + 4 \gamma _3^4 - 2)))) \, z_2] \, \lambda _2 \Big ], \\ \beta _{12} = (4 e_2^2)^{-1} C \, \Big [2 \, (e_2 \, (e_2 \gamma _3 - e_3 \gamma _2) \, z_1 + (e_3 \, (\gamma _2^2 - 5) - e_2 \gamma _2 \gamma _3) \, z_2) \, \lambda _0 \\ + C \, [2 e_2^4 e_3 \gamma _2 \gamma _3 \, (14 \gamma _2^2 + 9 \gamma _3^2 - 15) + 10 e_3 \gamma _2 \gamma _3 \, (e_3^2 - 1) (5 - \gamma _2^2 - \gamma _3^2 \\ + e_3^2 \, (3 \gamma _2^2 + 4 \gamma _3^2 - 3)) + 2 e_2^5 \, (3 + 12 \gamma _2^4 + \gamma _3^2 (\gamma _3^2 - 4 ) + \gamma _2^2 (11 \gamma _3^2 - 15)) \\ + 2 e_2^2 e_3 \gamma _2 \gamma _3 \, (29 - 20 \gamma _2^2 - 7 \gamma _3^2 + e_3^2 \, (39 \gamma _2^2 + 23 \gamma _3^2 - 40)) + e_2^3 \, ( \gamma _3^2 (10 + 3 \gamma _3^2) \\ - 24 \gamma _2^4 - 15 + \gamma _2^2 \, (51 - 13 \gamma _3^2) + e_3^2 \, (62 \gamma _2^4 + 2 \gamma _3^2 \, (2 \gamma _3^2 - 19 ) + 2 \gamma _2^2 \, (31 \gamma _3^2 - 46) \\ + 26)) + e_2 \, (3 \gamma _2^2 (\gamma _2^2 - 5) + (15 - 2 \gamma _2^2) \gamma _3^2 - 5 \gamma _3^4 + 4 e_3^4 \, (8 \gamma _2^2 - 5) (\gamma _2^2 + 2 \gamma _3^2 - 1) \\ + e_3^2 \, (\gamma _2^2 (98 - 53 \gamma _3^2) + 5 (\gamma _3^2 \, (2 \gamma _3^2 + 7) - 7) - 35 \gamma _2^4)) + 2 (2 e_2^3 e_3 \gamma _3 (7 \gamma _2^2 + \gamma _3^2 - 4) \\ + e_2^4 \gamma _2 \, (12 \gamma _2^2 + 5 \gamma _3^2 - 9) + e_2 e_3 \gamma _3 \, (10 - 13 \gamma _2^2 + 4 e_3^2 (8 \gamma _2^2 - 5) + 5 \gamma _3^2) \\ + 5 e_3^2 \gamma _2 \, (5 - \gamma _2^2 - 3 \gamma _3^2 + e_3^2 \, (\gamma _2^2 + 4 \gamma _3^2 - 1)) + e_2^2 \gamma _2 \, (15 - 6 \gamma _2^2 + 2 \gamma _3^2 \end{array} \end{aligned}$$

$$\begin{aligned} \begin{array}{l} + e_3^2 \, (25 \gamma _2^2 + 13 \gamma _3^2 - 26))) \, z_1 z_2] \, \lambda _2 \Big ]. \end{array} \end{aligned}$$