Asymptotic solutions of Lagrangian systems with gyroscopic forces

We consider Lagrangian systems in the presence of nondegenerate gyroscopic forces. The problem of stability of a degenerate equilibrium point O and the existence of asymptotic solutions is studied. In particular we show that nondegenerate gyroscopic forces in general have, at least formally, a stabilizing effect when O is a strict maximum point of the potential energy. It turns out that when we switch on arbitrary small nondegenerate gyroscopic forces, a bifurcation phenomenon arises: the instability properties of O are transferred to a compact invariant set which collapses at O when the gyroscopic forces are switched off. © 1995 Birkhäuser Verlag.