We are now able to examine how different types of forces affect the energy. For any force that can be written solely as a function of \(x\), we are able to define a corresponding \(V\left(x\right)\). The difference between kinetic energy at two points \(x\ =\ a\) and \(x=b\) can thus be found as a difference of \(V\left(x\right)\). We call these forces conservative forces, as they conserve the energy in the system. Note therefore that only conservative forces can have potential energies, and the force is related to the gradient of the potential energy.
Conversely, forces that cannot be written as solely a function of \(x\) are known as non-conservative forces.
When thinking about force fields that are defined everywhere in space, such a field is conservative if and only if the curl of \(\vec{F}\) is zero everywhere. If so, a potential energy function is well-defined (up to a constant).