Stability and instability. Part One: Use of integrals of motion in the stability problem

The concept of equilibria stability in a dynamical system is clarified, def ining both linear (spectral and formal) and nonlinear stability. Cases for which there are equilibria that are linearly stable but nonlinearly unstabl e and viceversa are analyzed. The integrals of motion of the system are use d to study the evolution of states initially close to an equilibrium; it is shown how to derive a priori bounds for the distance to that point. The ex amples used here correspond to Hamiltonian systems both in the canonical an d singular sense. The process of state space reduction and the concept of r estricted stability are exemplified.