On equilibrium instability for conservative and partially dissipative systems

Abstract

The investigation brings some contributions to the classical problem of inverting the Lagrange-Dirichlet stability theorem. First, an example is given of a conservative holonomic mechanical system with a stable equilibrium at the origin, although the potential function is strictly negative along some rays issuing from the origin. Then, one establishes a new instability result in the conservative case. Last, by means of a vector auxiliary function, one proves an instability theorem for holonomic systems with partial dissipation.