Mechanical equilibrium

A standard definitionmechanical equilibrium isstate ofmechanical systemwhichsum offorces on each particle ofsystemzero. However, this definition islittle usecontinuum mechanics,whichidea ofparticleforeign. In addition, this definition gives no information asone ofmost importantinteresting aspectsequilibrium states – their stability.

An alternative definitionequilibrium thatmore generaloften more useful is

A systemin mechanical equilibrium if its positionconfiguration space ispoint at whichgradient ofpotential energyzero.

Because offundamental relationship between forceenergy, this definitionequivalent tofirst definition. However,definition involving energy can be readily extendedyield information aboutstability ofequilibrium state.

For example, from elementary calculus, we know thatnecessary condition forlocal minimum ormaximum ofdifferentiable function isvanishing first derivative (that is,first derivativebecoming zero). To determine whetherpoint isminimum or maximum, we must takesecond derivative. The consequences tostability ofequilibrium stateas follows:

• Second derivative < 0 : The potential energyatlocal maximum, which means thatsystemin an unstable equilibrium state. Ifsystemperturbed by an arbitrarily small force,forces ofsystem do not cause itreturnequilibrium.

• Second derivative = 0 : This could beregionwhichenergy does not vary,which caseequilibriummarginally stable. Orregion could besaddle point,which caseequilibriumunstable.

• Second derivative > 0 : The potential energyatlocal minimum. This isstable equilibrium. A small perturbation does not causesystemleaveregion ofequilibrium point. If more than one stable equilibrium statepossible forsystem, any equilibria whose potential energyhigher thanabsolute minimum represent metastable states.