On the inversion of the lagrange-dirichlet theorem in the case of an analytic potential

Abstract

We examine the following conjecture concerning equilibrium instability for con-servative systems: if the potential is analytic and has no minimum at the origin, the vanishing solution is unstable. It is well-known that for a system with two degrees of freedom, this conjecture is true if the Hessian matrix of the potential is not zero. We prove that the conjecture is still true if the Hessian matrix is zero and the third-order terms of the potential do not constitute a perfect cube. Then, for an n-dimensional system, we extend a result of Koiter to the case of a potential for which the Hessian matrix has ( n−2) strictly positive eigenvalues and two zero eigenvalues.