Conservative and non-conservative forces

Not every force lets us define a potential energy in the first place -- only conservative forces do. A force is called conservative if the work done in moving a body between two points depends only on those two points (the initial and final positions), and not at all on the particular path taken between them.

Gravity is the textbook example: whether an object is carried straight up, along a zig-zag path, or up a ramp, the work done against gravity between the same two heights is identical (this is what makes it sensible to write $U=mgh$ using only the height, regardless of the path used to get there). Equivalently, a conservative force is the negative gradient of a potential energy function: in one dimension, $F_x = -dU/dx$. Elastic spring force, electrostatic force, magnetic force and gravitational force are all conservative.

A force is non-conservative if the work done between two points does depend on the path -- friction is the classic example, since the work done against friction grows with the length of the path travelled, not just the net displacement; viscous drag and air resistance are non-conservative for the same reason (they depend on speed and path, not just endpoints). …