Influence of dissipative forces on the stability behavior of the steady motions of Lagrangian mechanical systems with cyclic coordinates

Abstract

We are concerned with stability problems for a holonomic system $\mathcal {S}$ with $n$ degrees of freedom having $n-m$ cyclic coordinates, $m<n$. Let $x$ be the set of the acyclic coordinates and let $v$ be the set of the generalized velocities corresponding to the cyclic coordinates. For $\mathcal {S}$ conservative or subject to a dissipation restricted to the acyclic coordinates, we revisit classical stability results concerning the steady motions of the system and give some new contribution. When $\mathcal {S}$ is strictly dissipative with respect to all the coordinates, the integrals of momenta disappear and so do the steady motions. In this case, under suitable conditions there exist motions for which $x$ is constant and, consequently, $v \rightarrow 0$ as $t \rightarrow +\infty$ (pseudosteady motions). We analyze the stability properties with respect to $(x, \dot x)$ of these motions. Such properties define a stable or unstable behavior with respect to $(x, \dot x)$ of corresponding steady motions of the conservative system under the influence of strictly dissipative perturbing