Stability of a Mechanical Equilibrium

Abstract

This chapter is devoted to stability questions concerning mechanical equilibria, or in other words to stability of critical points for differential equations having the Lagrangian or Hamiltonian form, with or without friction forces. The central theorem in this context was stated by J.L. Lagrange [1788]: roughly speaking, it asserts that a mechanical equilibrium of a conservative system is stable at each point where the potential function is strictly minimum. Lagrange himself (as well as S.D. Poisson [1838]) failed to give a proof for a potential function more general than a quadratic form. G. Lejeune-Dirichlet [1846] gave an elegant general proof which yielded the model for the entire direct method of Liapunov.