PETER HAGEDORN 8r JEAN MAWHIN Communicated by K. KIgC~G~SSNER 1. Introduction The Lagrange-Dirichlet stability theorem states that the equilibrium posi- tion of a discrete, conservative mechanical sytem is stable (in the sense of Liapunov) if the potential energy U assumes a strict local minimum at this equilibrium. We know from PAINZEV~'S example [22] (see also [37]) that this sufficient condition, stated by LAGRANOE [13] and proved by LEJEVNE DIRICH~ET [17], is not necessary for stability, and the history of the inversion of the Lagrange-Dirichlet theorem, i.e., of the search for necessary conditions for stability, is rich and interesting. We refer to [27] for a discussion and a bibliography, and to the more recent references [1-3, 6-12, 14-16, 18, 20; 21, 23-26, 28-32, 35, 36]. In 1971, HA6EDORN proved, for systems of class C 2, that the equilibrium position is unstable if the potential energy U has a strict maximum at this equilibrium [4, 5]. This result was extended in various directions by TA~L~- FER~O [30], BOmTIN & KOZLOV [2], Kozmv [10], LIUBUSHIN [18] and SOFER [29]. A common feature of all those proofs, initiated in [4], is the application of some method of the calculus of variations, namely, the theory of geodesics on Finsler manifolds, to Jacobi's form of the principle of least action. This leads to technical difficulties, which are overcome in the above-mentioned papers through a lot of effort and ingenuity. The aim of this short paper is to show that the replacement of Jacobi's principle of least action by a variant of another variational principle, recently introduced by VAN GROESEN [33, 34] in the study of normal-mode vibrations of conservative nonlinear systems, allows a simple and elementary treatment, in the setting of a Hilbert space instead of a Finsler manifold, of the instabili- ty theorem stated in [4], with the minimal regularity assumptions of [18, 29, 30]. In this way the problem is reduced to proving the existence of a minimum for a coercive functional defined on a weakly closed subset of a Hilbert space, a problem solved at the beginning of any monograph on the direct methods of the calculus of variations (see, e.g., [19]). The reduction