Non-conservative Forces:
Let the motion of a body along with a surface from A to B which offers frictional force that is proportional to the normal reaction exerted on it by the body in motion (refer Figure). For any position of the body at K at distance (s) measured along with its path if KK ′ = (dr ) is a vector displacement, the normal reaction (Ns) will based on the slope
of the tangent at s. The frictional force throughout a very-small displacement dr is Fs. Frictional force is opposite to dr and is given by (- μ N s ) .
WD in distance dr = Fs dr ≈ Fs (ds)
Total WD from A to B = ∫ Fs ds
= ∫- μ N s ds
Therefore, you shall notice that the WD by frictional force based on the path of motion from A to B.
The WD being negative, this is the work lost from the body whereby its total-energy is decreased.
In case of non-conservative field such as Frictional-Force-Field, it is hence, observed that principle of conservation of energy is not valid. Though, this principle may be modified in the form of following equation assuming path of the body of weight (W) along a curve from A to B as illustrated in Figure
