Conservative and Non-Conservative Forces and Concept of Potential Energy

A force is called CONSERVATIVE if the work done by (or against) it, in moving an object between two points, is independent of the actual path taken -- it depends only on the initial and final positions. Gravitational force is the standard example: lifting an object from the ground onto a table (ignoring air resistance) always takes the same amount of work regardless of the path followed, and the same amount of work is recovered when it is lowered back down.

For a conservative force, the small work done $dW=\vec{F}\cdot d\vec{x}$ in an infinitesimal displacement is defined to equal $-dU$, i.e. $dU=-\vec{F}\cdot d\vec{x}$, where U is the POTENTIAL ENERGY -- the energy a body possesses on account of its position. During motion under a purely conservative force, the total mechanical energy (kinetic + potential) is conserved: work done AGAINST the conservative force appears as a corresponding INCREASE in potential energy (e.g. lifting the object), and work done BY the force appears as a corresponding DECREASE (e.g. lowering it) -- either way the change in potential energy is path-independent, $U=-\int \vec{F}\cdot d\vec{x}$, depending only on the endpoints. (This idea is developed further, for gravitation specifically, in the next chapter.) …