# Mechanical equilibrium

A standard definition of mechanical equilibrium is:

A system is in mechanical equilibrium when the sum of the forces on each particle of the system is zero.

However, this definition is of little use in continuum mechanics, for which the idea of a particle is foreign. In addition, this definition gives no information as to one of the most important and interesting aspects of equilibrium states – their stability.

An alternative definition of equilibrium that is more general and often more useful is

A system is in mechanical equilibrium if its position in configuration space is a point at which the gradient of the potential energy is zero.

Because of the fundamental relationship between force and energy, this definition is equivalent to the first definition. However, the definition involving energy can be readily extended to yield information about the stability of the equilibrium state.

For example, from elementary calculus, we know that a necessary condition for a local minimum or a maximum of a differentiable function is a vanishing first derivative (that is, the first derivative is becoming zero). To determine whether a point is a minimum or maximum, we must take the second derivative. The consequences to the stability of the equilibrium state are as follows:

• Second derivative < 0 : The potential energy is at a local maximum, which means that the system is in an unstable equilibrium state. If the system is displaced an arbitrarily small distance from the equilibrium state, the forces of the system cause it to move even farther away.
• Second derivative = 0 : This could be a region in which the energy does not vary, in which case the equilibrium is called neutral or indifferent or marginally stable. To lowest order, if the system is displaced a small amount, it will stay in the new state.
• Second derivative > 0 : The potential energy is at a local minimum. This is a stable equilibrium. The response to a small perturbation is forces that tend to restore the equilibrium. If more than one stable equilibrium state is possible for a system, any equilibria whose potential energy is higher than the absolute minimum represent metastable states.

In more than one dimension, it is possible to get different results in different directions, for example stability with respect to displacements in the x-direction but instability in the y-direction, a case known as a saddle point. Without further qualification, an equilibrium is stable only if it is stable in all directions.

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