What Is a State of Equilibrium

A ball rolls downhill until it reaches the bottom, and then stays there. The bottom is its equilibrium position. Equilibrium defines where a system tends to go and stay. The force f on the ball defines the strength of its tendency toward equilibrium  

The point of equilibrium is the point of zero net force. Therefore the state of equilibrium can either be defined as an extremum in the energy or as the point 

Zero net force can be achieved in different ways. The potential energy is a function of the degrees of freedom, V(x) in Example 2.1. Potential energy functions can take any shape. They may be perfectly flat, or smooth and curved, or rough and bumpy. An equilibrium state is defined as stable where a potential is at 

The force on the ball due to gravity is/(xr) = -dV/dx. Our sign convention specifies that as we move the ball in a direction to increase its potential energy (up), the force acts in the opposite direction (down) to reduce the energy. 

A stable equilibrium can be identified by tests of the first and second derivatives  

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