On the Instability of Stationary Solutions of a Lagrangian System with Gyroscopic Forces

This paper deals with the problem of the instability of an equilibrium, say (q = 0, = 0), of a lagrangian differential system, in the presence of "gyroscopic forces." More precisely, we examine the case in which the gyroscopic forces start with linear terms A(0) , A(0) being an invertible antisymmetric matrix, while the conservative forces arise from a potential function U(q), which starts with a homogeneous form U[k](q) of order k, k ≥ 3. We require that the lack of a local maximum of U(q) at q = 0 be recognizable from the inspection of U[k](q). Then, assuming that the Lagrangian function is 3(k − 1), we are able to give a criterion for the existence of a motion of the Lagrangian system which tends, either in the future or in the past, to the equilibrium (q = 0, q = 0). From this result we deduce, in particular, the instability of the equilibrium.