# Equilibrium that is not stable

Consider a system which is in an equilibrium state. Now, a small perturbation causes it to start oscillating about the equilibrium state, but over time, the momentum with which it overshoots the equilibrium state keeps on increasing. This can be considered as an exact opposite of what can be expected of reaching a normal stable equilibrium state (frictional forces reducing excess momentum until equilibrium is reached).

What is such kind of an equilibrium state called? Is it still referred to as a stable equilibrium since we keep oscillating about the state?

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## 3 Answers

If there is oscillation, the equilibrium is stable in at least one dimension.

The situation where momentum keeps increasing could apply to a driven mass oscillating about a stable equilibrium.

It would also apply to a non-driven mass at a saddle point. Consider that the mass begins oscillating approximately, but not exactly, along the concave up curve in the plane of symmetry of the saddle (such as displaced slightly parallel to this plane of symmetry). The mass oscillates but deviation progressively increases, and eventually the mass falls away from the saddle point.

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In the dynamical systems jargon, this is the usual Lyapunov stability. If additionally the system reaches the equilibrium state, then the equilibrium is called asymptotically stable.

For the formal definition, see this link: http://www.scholarpedia.org/article/Stability#Definitions:_Stability_of_an_Equilibrium

It is also worth to remark that a equilibrium point can attract all the trajectories near to it and yet be unstable; An interesting example is given in p.58 of Bhatia, N. & Szegö, G. (1967). Dynamical systems stability theory and applications. Berlin New York: Springer-Verlag.

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I am not sure of what you mean but in mechanic it's called unstable equilibrium.

Check this http://en.wikipedia.org/wiki/Mechanical_equilibrium to get illustration on differents equilibrium states

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But in unstable equilibrium, you will never reach back the equilibrium state. –  Hashken Apr 5 '14 at 15:55