INSTABILITY IN THE PRESENCE OF AN INDEFINITE CONSERVATION LAW

Abstract

Consider a C1 vector field X on a finite-dimensional manifold M and xe an equilibrium point for the dynamics [Formula: see text]. We will prove that if I is a C2 constant of motion for X such that xe is also a critical point of I, then [Formula: see text] is a constant of motion for the linearized system [Formula: see text]. As a consequence we will give an instability result under the condition that [Formula: see text] is an indefinite quadratic form. If xe is not a critical point of I, then we obtain [Formula: see text] as a constant of motion for the linearized system.