A proof of the Palamodov's total instability conjecture

TL;DR

This paper provides the first detailed proof of Palamodov's total instability conjecture in Lagrangian dynamics, confirming related instability conjectures and analyzing charged rigid bodies in electrostatic fields.

Contribution

It offers the first rigorous proof of Palamodov's conjecture, linking it to classical instability problems and extending to charged rigid bodies in electrostatics.

Findings

Proves Palamodov's total instability conjecture in Lagrangian dynamics.

Confirms Lyapunov and Arnold's related instability conjectures.

Shows instability of charged rigid bodies in electrostatic fields.

Abstract

We give for the first time a detailed proof of the Palamodov's 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.