It is important to note also that potential energy is only a useful quantity when friction is not involved . We will see why in a moment. As an example of a system where potential energy is a useful quantity, consider gravity. As usual, assume that we are near the surface of the Earth and that g is a constant. So the potential energy of an object is increased when it it is lifted higher, and decreases when the object is lowered.
Friction causes an object to expend additinal energy to attain a certain position.
But the extra energy is required whether moving toward or away from that position. Hence friction is incompatible with our notion of potential energy, since potential energy should depend only on the position of the object and not on how it came to be in that position. As a time derivative of momentum, force is necessary to change an object's momentum. As a spatial derivative of potential energy, any object which has potential energy feels a force exerted on it. There are two other dynamic variables which we will use.