Investigation of motion of rigid body in non-principal axes of inertia by small parameter method (Q2761539)

Using the method of small parameter the author studies the motion of a rigid body round the fixed point in non-principal axes of inertia. The corresponding system of motion equations has the form NEWLINE\[NEWLINE\begin{matrix} A\dot p-F\dot q-E\dot r-(-Fp+Bq-Dr)r+(-Ep-Dq+Cr)q=Mg(y_0\gamma''-z_0\gamma'),\\ -F\dot p+B\dot q-D\dot r+(Ap-Fq-Er)r-(-Ep-Dq+Cr)p=Mg(z_0\gamma-x_0\gamma''),\\ -E\dot p-D\dot q+C\dot r-(Ap-Fq-Er)q+(-Fp+Bq-Dr)p=Mg(x_0\gamma'-y_0\gamma)\\ d\gamma/dt=r\gamma'-q\gamma'',\;d\gamma'/dt=p\gamma''-r\gamma,\;d\gamma''/dt=q\gamma-p\gamma'\end{matrix},NEWLINE\]NEWLINE where \(A,B,C\) are central moments of inertia; \(F,E,D\) are non-central moments of inertia; \(p,q,r\) are projections of the angular velocity vector; \(\gamma,\gamma',\gamma''\) are direction cosines of the stationary axis; \(M\) is a mass of a rigid body; \(g\) is acceleration of gravity; \(x_0,y_0z_0\) are coordinates of the mass center of a rigid body in the moving coordinates system. The following result is proved.NEWLINENEWLINE If \(A\neq B\neq C,\;AB-BC+D^2-F^2<0\), then the considered system of equations has single-valued integrals only under conditions: \(x_0\sqrt{AC-AB+F^2-E^2}+z_0\sqrt{BC-AC+E^2-D^2}=0,\;y_0=0\).