Potential Energy

Potential energy is the energy a body possesses on account of its position or configuration relative to its surroundings -- it exists precisely because the forces acting on the body (gravity, a spring's restoring force, the Coulomb force between charges) themselves depend on position.

Formally, the potential energy of an object at a point $P$ is defined as the work done by an external force in moving the object, at constant velocity, from a chosen reference point $O$ (where potential energy is taken to be zero) to the point $P$:

Two design choices in this definition deserve attention. First, the reference point $O$ is arbitrary -- only changes in potential energy have direct physical meaning, so $O$ is simply chosen for convenience (the ground, the spring's natural length, etc.). Second, the object must move at constant velocity: if it did not, the work-kinetic energy theorem tells us the external force would also be imparting some extra kinetic energy, and we specifically want to isolate the energy that goes into changing the object's position, not its speed -- this requires another force (e.g. gravity itself) to exactly cancel the external force so that the net force, and hence the acceleration, is zero. …