# Why can't we define a potential energy for a non-conservative force? [closed]

We could define potential energies for non-conservative forces too and then we could conserve it with kinetic and potential energy as we know it. But no one does that. Why is this? Please explain. Any help would be great.

## closed as unclear what you're asking by ACuriousMind♦, Kyle Kanos, HDE 226868, Norbert Schuch, user36790 Dec 24 '15 at 17:35

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• "We could define potential energies for non-conservative forces." Give an example of a potential energy for a non-conservative force. – Bill N Dec 23 '15 at 18:01
• Related: physics.stackexchange.com/q/122345/2451 , physics.stackexchange.com/q/31672/2451 and links therein. – Qmechanic Dec 23 '15 at 19:22
• What is your definition of "non-conservative force" if not "force that is not derived from a potential"? Being the gradient of a potential is one of the equivalent definitions of a conservative force. – ACuriousMind Dec 23 '15 at 19:23

Conservative force does not mean that the energy is conserved; it means that the force is in any point the gradient of a scalar function. It thus follows that the work done by the field does not depend on the path in its domain and consequently the work done along a closed curve is zero.

The same just does not hold true anymore whenever the field cannot be written as the gradient of a scalar function in any point.

• Interestingly enough, in non-simply-connected domains even being a gradient of a (locally defined) scalar function does not imply that the force is conservative, as demonstrated in Aharonov-Bohm experiment. So a tiny clarification is needed: the ability to define potential energy for a force is equivalent to that force being conservative in simply connected domains. – Michael Dec 24 '15 at 17:42
• Oh yes, sure, topological defects aside :). – gented Dec 24 '15 at 17:48
• @Michael do you mean irrotational vector fields are not necessarily conservative in non simply connected domains? Gradient fields are. – Bananach Aug 26 '18 at 11:30
• @Bananach, see en.wikipedia.org/wiki/Aharonov%E2%80%93Bohm_effect – Michael Aug 27 '18 at 19:29
• @Michael that is way beyond my physical abilities. Are you sure there is no simple way of saying what you want to say? – Bananach Aug 27 '18 at 20:05

You could try to define a potential energy with respect to some position, but it would not be unique in the case of a non-conservative force. That is because the work done to move from one position to another would depend on the path taken.

I don't see how the concept of a non-unique potential energy would be helpful.

A system with friction is a simple example of a non-conservative force field. Let's assume that in this system, an object has potential energy E. Now let's make the object make a small excursion - up, left, down, right. It returns to the same point, but it had to do work in order to overcome the friction. If that work is W, then its potential energy must have been reduced by the same W, and is now E-W

The implication is that there is no unique potential energy associated with a state of the system (position of the object in the field). And that means that potential energy is a meaningless concept (because a particular state cannot have a unique potential energy associated with it). And that is why nobody does it...