If the work done is **path independent,
** the underlying force is
** conservative**.

Example: the force of gravity:
dW_{grav} = m**g** • d**r** = - mg dy

W_{grav} =
- mg(y_{f}-y_{i}) = - mg Δ y

Because the work done depends only on the endpoints and not on the path the underlying external force is conservative.

Examples of conservative force: gravity, springs.

Example of non-conservative force: friction.

For an arbitrary conservative force:
W_{cons} =
- (U_{f} - U_{i})

Start with our previous connection between kinetic energy and work:

W = Δ K → W_{cons} + W_{non-cons} = Δ K

U_{i} - U_{f}
+ W_{nc} = K_{f} - K_{i}

K_{i} + U_{i} + W_{nc} = K_{f} + U_{f}