Conditions for rigidity steady motions of mechanical systems with cyclic coordinates (Q1584159)

The author studies the stationary motion of the mechanical system with \(n\) degrees of freedom under condition that one of the coordinates is cyclic. It is assumed that the kinetic energy is \[ T(\dot q,q) = T^*(\dot q_1,\dots,\dot q_{n-1},q_1,\dots,q_{n-1})+\alpha\dot q^2_n/2, \] where \(\alpha\) is the function of \(\,q_1,\dots,q_{n-1}\); \(\,V=V(q_1,\dots,q_{n-1})\,\) is the potential energy of the system. The Rause function \(\,R=T^*-V-\frac{c^2}{2\alpha}\,\) of the system depends on \(c\) as on the para meter. Under some assumptions established is the following. Let the fourth degree form \(W_4\) be positive definite in the decomposition \[ W_c=W_c(0)+W_4+\dots, \] where \(\,W_c=\frac{c^2}{2\alpha}\). Then the stationary motion \(\,q_1=q_1=\dots=q_{n-1}=0\,\) of the system is stiff as \(\,c\to\infty\). For the definition of stiffness, see \textit{T.~A.~Yakovleva} [Vestn. Mosk. Univ. Ser.~I 1983, No.~3, 98--100 (1993)].