On a method for integrating the equations of rigid body motion in three homogeneous force fields (Q6640638)

Consider a rigid body, rotating about a fixed point \(O\) under the action of three homogeneous constant force fields. Let \(\boldsymbol{\gamma}, \boldsymbol{\gamma}_1, \boldsymbol{\gamma}_2\) be fixed in the space unit vectors in the directions of the resultant forces \(\boldsymbol{P}\), \(\boldsymbol{P}_1\), \(\boldsymbol{P}_2\) of these fields, and let \(C, C_1, C_2\) be fixed in the body centers of reduction of these forces. Then equations of motion of the rigid body with a fixed point in three homogeneous force fields are as follows\N\[\N\begin{array}{c} \boldsymbol A \boldsymbol{\dot\omega} = \boldsymbol A \boldsymbol\omega \times \boldsymbol\omega +\boldsymbol s \times \boldsymbol\gamma +\boldsymbol r \times \boldsymbol\gamma +\boldsymbol p \times \boldsymbol\gamma, \\\N \boldsymbol{\dot\gamma} = \boldsymbol\gamma \times \boldsymbol\omega, \quad \boldsymbol{\dot\gamma}^{(1)} = \boldsymbol\gamma^{(1)} \times \boldsymbol\omega, \quad \boldsymbol{\dot\gamma}^{(2)} = \boldsymbol\gamma^{(2)} \times \boldsymbol\omega. \end{array}\N\]\NHere \(\boldsymbol{A}\) is the tensor of inertia of the body for the fixed point, \(\boldsymbol{\omega}\) is the angular velocity vector of the body, vectors \(\boldsymbol{s}\), \(\boldsymbol{r}\), \(\boldsymbol{p}\) are defined by the formulas \(\boldsymbol{s}= P \cdot \boldsymbol{OC}\), \(\boldsymbol{r}= P_1 \cdot \boldsymbol{OC}_1\), \(\boldsymbol{p}= P_2 \cdot\boldsymbol{OC}_2\).\N\NIn the article, the author considers the existence problem for precession motions of the body relative to the vector \(\boldsymbol{\gamma}\). These motions are characterized by the relation\N\[\N\boldsymbol{a}\cdot\boldsymbol{\gamma} = a_0 \quad (a_0=\cos\theta_0),\N\]\Nwhere \(\theta_0\) is the angle between the unit vectors \(\boldsymbol{a}\) and \(\boldsymbol{\gamma}\), fixed in the body and in the space accordingly. The velocity vector of the body in the case of its precession motion can be presented in the form\N\[\N\boldsymbol\omega = \dot\varphi \boldsymbol{a} + \dot\psi \boldsymbol\gamma.\N\]\NThe variables \(\varphi\), \(\psi\) and the constant \(\theta_0\) can be treated as Euler's angles.\N\NIt is assumed further that the vectors \(\boldsymbol{\gamma}\), \(\boldsymbol{\gamma}_1\), \(\boldsymbol{\gamma}_2\) are mutually otrhogonal.\N\NIt is established in the article that for the spherical and for the dynamically symmetric rigid body there exist precession motions, satisfying the condition \(\varphi=2 \psi\).\N\NThen the case of spherical body is considered, when this condition is not satisfied, but the condition \(\varphi=2 (\psi+\vartheta)\) is fulfilled, where \(\vartheta\) is a new variable. A closed forth order normal system of ordinary differential equations is obtained for the variables \(\vartheta\), \(\psi\), \(\dot\vartheta\), \(\dot\psi\). Under additional conditions, integrating of this system is reduced to three algebraic equations and one first order differential equation with separating variables.