Investigation of solid rotation evolution close to the Lagrange case using the averaging method (Q2761551)

The author considers the movement of a dynamical symmetric solid around a fixed point under the action of the regenerating moment depending on the angle of nutation \(\theta\) and the disturbing moment. The corresponding equations have the form NEWLINE\[NEWLINEA\dot p+(C-A)qr=k(\theta)\sin\theta\cos\phi+M_1,\quad A\dot q+(A-C)pr=-k(\theta)\sin\theta\sin\phi+M_2,\;C\dot r=M_3,NEWLINE\]NEWLINE NEWLINE\[NEWLINE M_{i}=M_{i}(p,q,r,\psi,\theta,\phi,t),\quad i=1,2,3,NEWLINE\]NEWLINE NEWLINE\[NEWLINE\dot\psi=p\sin\phi+q\cos\phi, \quad \dot\theta=p\cos\phi-q\sin\phi,\quad \dot\phi=r-(p\sin\phi+q\cos\phi){\text{ctg}} \theta.NEWLINE\]NEWLINE Here, \(p,q,r\) are projections of the angular velocity vector on the main inertia axes; \(M_{i}, i=1,2,3\), are projections of the disturbing moment vector on the same axes; \(A\) and \(C\) are the equatorial and axial moments respectively, \(A\neq C\). A small parameter \(\epsilon\ll 1\) is introduced such that NEWLINE\[NEWLINEp=\epsilon P,\quad q=\epsilon Q,\quad k(\theta)=\epsilon K(\theta),\quad M_{i}=\epsilon^2 M^{*}_{i}(P,Q,r,\psi,\theta,\phi,t),\quad i=1,2,NEWLINE\]NEWLINE NEWLINE\[NEWLINEM_3=\epsilon M_3^{*}(P,Q,r,\psi,\theta,\phi,t).NEWLINE\]NEWLINE Using the averaging method, the asymptotic behaviour of the considered system is studied.