A proof of the Palamodovs total instability conjecture

We give for the first time a detailed proof of the Palamodovs total instability conjecture in Lagrangian dynamics. This proves an older related Lyapunov instability conjecture posed by Lyapunov and Arnold and reduces the Lagrange-Dirichlet converse problem in the class of real analytic potentials to the Lyapunov instability of non strict minimum critical points. It also proves the instability of charged rigid bodies under the presence of an external electrostatic field.