Regarding the idea of potential energy

[walking]

I am studying basic mechanics and have reached the chapter on potential energy. However I am a bit confused about the difference between potential energy and the formula for the potential energy due to work done by a conservative force. I am not sure which of the following interpretations is correct:

**Possibility 1.**

One possibility is that potential energy is a general idea which doesn't have anything to do with conservative forces. The formula for potential energy however would only be for conservative forces, because the textbook says that the work done by nonconservative forces depends on the path and not just the end points, hence a formula for them doesn't exist.

So in this possibility, it would seem as though potential energy is one thing, and the formula for it is another (more specific) thing which only exists for conservative forces. But then why are both referred to as "potential energy", and why does the textbook say that potential energy is *defined* by the formula (meaning that potential energy as a general concept is specifically only for conservative forces?).

**Possibility 2.**

Another possibility is that potential energy is only defined for conservative forces to begin with. In this case, there would be no such thing as potential energy for nonconservative forces.

But this confuses me a bit because the book defines potential energy without reference to conservative forces, as simply being the energy a system possesses due to its configuration. However, the reason why I think possibility 2 is correct is because all of the textbooks I have read say that potential energy is *defined* by the formula for potential energy due to a conservative force. This would mean that potential energy indeed only exists for conservative forces.

**Conclusion**

I am unable to decide which of the two possibilities is correct. There seems to be a contradiction here which I am not seeing how to avoid.

 

[Dale]

**Possibility 2.**

Another possibility is that potential energy is only defined for conservative forces to begin with. In this case, there would be no such thing as potential energy for nonconservative forces.

This is correct.

But this confuses me a bit because the book defines potential energy without reference to conservative forces, as simply being the energy a system possesses due to its configuration.

If the force governing a system is non-conservative then there is not a well defined energy for a given configuration. The path taken to arrive at the configuration would also matter, not just the configuration.

 

[Henryk]

formula for the potential energy due to work done by a conservative force

Actually, it is the work done AGAINS the conservative force. Think of gravity. A body above the surface of the Earth has some potential energy but to get it up, you have to lift it, i.e. do work against the gravity. Now, the energy gained (relative to a reference height) is equal to $m \cdot g \cdot \delta h$. where m is the mass of the body, g, free fall acceleration and $\delta h$ is the height the body is lifted to. Obviously, the potential energy depend only on the difference between the original and final position of that body and its mass.
An example of a non-conservative field is an induce electric field by a change of a magnetic field. This is used commonly in transformers. The key is that the induced voltage (energy is equal to voltage * charge) depends in the number of turns of the winding of the transformer. The initial and final points do not matter much.
So yes, the potential energy can only be defined for a conservative field.