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, q = 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) q, 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 greater-than-or-equal-to 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 C3(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. (C) 1995 Academic Press, Inc.