Mechanical equilibrium

In classical mechanics, a particle is in mechanical equilibrium if the net force on that particle is zero. By extension, a physical system made up of many parts is in mechanical equilibrium if the net force on each of its individual parts is zero.

In addition to defining mechanical equilibrium in terms of force, there are many alternative definitions for mechanical equilibrium which are all mathematically equivalent.

More generally in conservative systems, equilibrium is established at a point in configuration space where the gradient of the potential energy with respect to the generalized coordinates is zero.

If a particle in equilibrium has zero velocity, that particle is in static equilibrium. Since all particles in equilibrium have constant velocity, it is always possible to find an inertial reference frame in which the particle is stationary with respect to the frame.

An important property of systems at mechanical equilibrium is their stability.

In a function which describes the system's potential energy, the system's equilibria can be determined using calculus. A system is in mechanical equilibrium at the critical points of the function describing the system's potential energy. These points can be located using the fact that the derivative of the function is zero at these points. To determine whether or not the system is stable or unstable, the second derivative test is applied. With ${\displaystyle V}$ denoting the static equation of motion of a system with a single degree of freedom the following calculations can be performed:

When considering 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. Generally an equilibrium is only referred to as stable if it is stable in all directions.

Sometimes the equilibrium equations – force and moment equilibrium conditions – are insufficient to determine the forces and reactions. Such a situation is described as statically indeterminate.