In some places, it is stated that the work done by a conservative forces is equal to the negative of the change in the potential energy (without any proof). Then it is derived that the work done by non-conservative forces is equal to the change in mechanical energy using work energy theorem. In some other places it is just the reverse; it stated that the work done by non-conservative forces is equal to the change in mechanical energy and using this the work done by conservative force is derived.
This seems circular to me and I don't understand where to start from. If I start with conservative forces, why is the work done by conservative force equal to the negative of the change in potential energy or if I start with non-conservative forces why is the work done by non-conservative forces equal to the change in mechanical energy?
Best Answer
The work-energy theorem can be proven directly from Newton's 2nd law, without any reference to conservative or nonconservative forces.
The relations between conservtive forces and their potential energy (and in fact, the existence of a scalar function satisfying these relations) is an entirely mathematical theorem. See here, here, and here.
After both of these results have been proven, one can separate the total work appearing in the work-energy theorem into the conservative and nonconservative parts. Since the conservative work is minus the difference in potential energy one can move it to the other side of the equation and get a positive difference in the total mechanical energy. In fact, the potential energy is defined so that the work equals the negative difference in the potential energy exactly because we want to get the difference in the mechanical energy (and not the difference in kinetic energy minus the difference in potential energy). That way, when the nonconservative forces do $0$ work we get conservation of energy.