On stability of mechanical systems under the action of position forces

Abstract

The stability problem is solved for the equilibrium position of a holonomic mechanical system subject to stationary geometric constraints and to potential and nonconservative position forces /1/. It is assumed that the characteristic equation of the linear approximation has pure imaginary roots among which there are none equal. The system being examined is invertible and is Birkhoff-stable/3/ when the system does not have an internal resonance in the sense of /2/: finite-order instability can be detected only under internal resonance. Necessary and sufficient stability conditions for a model system and sufficient Liapunov-instability conditions have been formulated for odd-ordered resonances. A fourth-order resonance, of greatest importance among even-ordered resonances for applications, has been investigated and for it necessary and sufficient stability conditions in the first nonlinear approximation have been obtained in the absence of degeneracy; it is shown that Liapunov-instability follows from third-order instability. An example is presented.