On the instability of the equilibrium for a Lagrangian system with gyroscopic forces

We consider a Lagrangian Differential System (L.D.S.) with Lagrangian function L(q, {Mathematical expression})=T(q, {Mathematical expression})+U(q), sufficiently smooth in a neighbourhood of the critical point q=0 of the potential function U(q). The kinetic function T(q, {Mathematical expression}) is a non homogeneous quadratic function of the {Mathematical expression}'s, i.e. the L.D.S. contains the so-called gyroscopic forces. The potential function U(q) starts with a degenerate (but non zero), semidefinite-negative, quadratic form. Moreover, q=0 is not a proper maximum of U, and this property has to be recognized in a suitable way. By analizing the problem of the existence of solutions of the L.D.S., which asymptotically tend to the equilibrium solution, (q, {Mathematical expression})=(0,0), we provide a sufficient criterium for its instability. © 1994 Birkhäuser Verlag.