# Relation between potential energy and conservative force

Does potential energy only happen when the work done is by a conservative force? Or does work done by non-conservative forces also create potential energy?

• I don't have enough to add an answer, but I'd like some answers which address the fact that the electric field does not appear to be conservative in dynamic systems: $\vec{\nabla}\times\vec{E} = \frac{\partial\vec{B}}{\partial t} \neq 0$ (assuming the magnetic field changes over time--which it doesn't in electrostatics). Sep 9, 2021 at 6:16
• No, and potential energy doesn't "happen". PE remains after all the work done by every force, of any description, has been factored in. "Every force" does include "non-conservative forces". Sep 9, 2021 at 21:09
• The question could be more meaningful, if it defines potential energy and conservative force. Otherwise it si tempting to say that conservative force is the one that can be associated with a potential energy and vice versa - it is a possible deifnition. Sep 10, 2021 at 7:12

Forces can be conservative or non-conservative. But conservative forces do work where this work is equal to the change in potential energy. Conservative forces are also characterized by the fact that the work done by the force that moves an object from one point to another is independent of the path taken between these points (and the total work done will be zero when the path forms a closed loop).

However, a non-conservative force is one where the work done will indeed depend on the path. A good example of a non-conservative force is friction. The work done against a frictional force will depend on the length of the path between the two points, and due to this path dependence, there will be no potential energy we can associate with this force, and indeed the same is true for all non-conservative forces.

Non-conservative forces will either add or remove mechanical energy from a system. Friction, energy dissipation in the form of heat, removes energy from a system which cannot be fully converted back to work.

• See my answer here for the way free energy is like potential energy, and thus allows a potential energy-like approach for non-conservative forces in some circumstances. Sep 10, 2021 at 15:09
• It's a well written analysis Andrew, though probably too idealized - in that it assumes reversible processes $dS=0$ and we know there aren't many (if any) real examples of such things in (macroscopic) nature. All real processes are irreversible. There is always energy loss/transfer. In fact, that appears to be the basis of the original question. Sep 10, 2021 at 22:50

For a conservative force the following property is true $$\oint_C \vec F \cdot \vec {dl}=0$$ Using stokes theorom: $$\oint_C \vec V \cdot \vec{dl}=\iint_S (\vec \nabla \times \vec V)\cdot \hat ndA$$ where $$\hat n$$ is a normal vector which gives $$\vec \nabla \times \vec F=0$$ as a matter of fact if $$\vec F$$ can be written as a gradient its curl will always be zero(curl of a gradient field is always 0) .If a fields curl is 0 we can find a potential $$\phi$$ such that $$\vec V=\vec \nabla \phi$$.

For a conservative force such field is $$-V$$. Which means $$\vec F=- \vec \nabla V$$ and V can be recognized as potential energy of the force. Which means that any conservative force has a potential energy and vice versa.

By definition, a non-conservative force does not increase the potential energy of the object on which it does work.

The potential energy of another system might be increased incidentally or on purpose. For instance, if we run electric current through a rechargeable battery, the battery's electrical resistance does work on the electrons without increasing their potential energy, but the specific processes that give the battery resistance cause changes in the chemistry of the battery which increase the battery's chemical potential energy.

More generally, most (possibly all) non-conservative forces increase the heat of the system. Anywhere there is a separation of hot stuff from cooler stuff, there exists thermal potential energy.

In any case, with a non-conservative force, the potential energy of the object on which it does work is not increased, and the potential energy of other systems that is created incidentally is always less than the amount of work done by the non-conservative force, as a consequence of entropy.

My take: the energy picture is a complementary way of looking at things to Newton's forces. Forces give a very low-level description of what is going on, they can kind of have arbitrary complexity and detail. Energies are more coarse because they discard directional information.

# Victories of the force picture

Forces were kind of invented because they give you a direct window into the equations of motion of the system. So maybe the first thing I should say about forces is that they are visualizable and not particularly hard for computers to simulate, often there are several tricks in simulating them to grant more or less numerical stability to the answers... If you want to write a game where people have to create a bridge out of boards and linkages and those boards have some elastic and bending abilities but break if they bend too much, you would build this game probably using forces.

Constraint forces are almost free to simulate in this way, the force picture is great at constraints. So if I am constrained to not break the stairs while I am falling down them, the the forces at play are “whatever they need to be to stop me from falling through the stairs.” You can model this with potential energies, you set up the surfaces of the stairs in your model as places where the potential energy rapidly changes from 0 to infinity, but because those changes are energy gradients and you are trying to make those gradients sharp, you might find yourself with weird numerical artifacts when you run the simulation and you start producing these near-infinite forces.

Another place where forces shine is indeed where forces are non-conservative. We have a couple of tricks for this. For example, forces can be non-conservative due to what we call hysteresis. This means that the force is not a function of position by itself, but also how you got to that position. But sometimes we can deal with hysteresis by defining a larger configuration space and we can get path-Independence in the larger configuration space.

A worse non-conservative example for the energy picture is friction, the energy picture models friction by creating a very large number of energy reservoirs, for example harmonic oscillators, and coupling their motion to your particles’ motion to get an energy flow, and then taking an average so that you see this one-way dissipation effect on the energy. This approach works, but it's tedious.

# Victories of the energy picture

At some point we invented variational calculus, and this was the missing link to deriving equations of motion from the functional form of an energy expression. These are called Lagrangians and Hamiltonians if you want to look them up. So we achieved basic feature parity with the force picture!

And of course, inventing potential energies instead of forces to describe things, seems to come with this law of conservation of energy, which describes a particular invariant, something that holds true as the equations of motion do their thing.

Well, along the way one of the most important theorems was proved, Emmy Noether’s amazing result that if you say that the Lagrangian is the laws of physics for a system, then conserved quantities are symmetries of the laws of physics (statements that the laws of physics are the same after some transformation). So for example Newton's third law kind of comes out of nowhere, Newton just tells you that this is something that makes sense for forces and you believe him. And you later find out that it's something called conservation of momentum. Noether’s theorem restates this as saying: the laws of physics are the same if we move everything one centimeter to the left or right. Now that sounds plausible! Conservation of energy is restated as saying that the laws of physics are the same if we move everything one millisecond into the past or future. Gorgeous!

This also matters for practical simulation. I said above that simulating the equations of motion directly on a computer often leads to the noise in the calculation violating your conservation laws; the energy picture sometimes allows you to constrain your configuration space to enforce the conservation laws directly.

And remember that complicated simulation of friction? it has the very nice property that you can prove so-called fluctuation-dissipation theorems: energy can't just flow one way from the particle to the springs, but if there's random vibrations and jitters in the springs it's going to contribute noise to the particle. So actually in one of Einstein's anuus mirabilis papers where he went from being a nobody to a recognized name, Einstein derived one of these and used it to argue that physicists need to take atoms seriously (not all of them did at the time!) because a connection between drag coefficients and Brownian motion could actually measure the size of atoms indirectly. So these theorems are really amazing!

# Do potential energies only exist for conservative forces?

If by potential energy you mean a bucket where energy that is not visible in our system can be said to be stored, then conservation of energy by Noether’s theorem applies as long as the laws of physics remain constant over time, and this allows us to invent a bucket where the energy is going, plus a plausible argument why it is not coming back.

If you mean something more particular, like a function of position whose gradient corresponds to a force on a particle in your system, well the bucket from the last paragraph probably comprises a lot of potential energies and a lot of kinetic energies for particles that are not in your system. And then you're totally right that to even define such a potential energy, you must require the force to be conservative.

• I think this answer is confused (but maybe I'm confused--definitely possible). If you bring Noether's Theorem, then energy (and in particular potential energy) is extremely important because the force (direction you move) completely depends on which direction leads to the quickest release in energy. I'm not so sure this works with non-conservative forces...I'm also not convinced that non-conservative "forces" exist. Yeah, in macro, we have friction, but friction is also not well understood--and is a macro system. All theoretical physics depends on conservative forces. Sep 9, 2021 at 6:34
• And I'll add, "I'm confused"...probably because this the closest answer to mirroring modern physics. This is how "we" think now--it's literally the the basis for statistical mechanics and quantum mechanics. So I'm not saying this is wrong (at all!)--it's completely correct. Sep 9, 2021 at 6:40

There are a lot of answers at the time I write; I think this is an indication of some range of confusion out there because no one answer captured agreement sufficiently to stop other people adding their answers. This shows that potential energy is a slightly tricky concept, as my answer will show.

What is potential energy?

I begin here because I think it useful to take the point of view that whereas a force is a physical quantity, a potential energy is not an energy owned in any sense by a body (like the way the body has mass and position and velocity). Rather it is a convenient way to capture in mathematics notable aspects of some force which may be acting. (I'll say a little more about this in a moment).

Definition of the term "conservative force"

A force which can be expressed as the negative of the gradient of a single-valued scalar function is called "conservative" and the associated scalar function is called "potential energy".

The reason this is done is that if the force is the only one acting then in order to find the change in kinetic energy of a particle as it moves from one place to another under the action of the force, all you need to do is find the change in potential energy and apply a minus sign. This is often convenient and easy so it is a useful idea.

This is the main thing asked about in the question. The answer is that for these forces there is, by definition of the term "non-conservative", no function that can play the role of potential energy in general. However there can be cases where the force is non-conservative in general, but the physical behaviour under study has some constraint on it, and in the presence of this constraint you can introduce a function which acts quite like potential energy. There are many examples of this in thermodynamics.

One of the basic concepts in thermodynamics is "free energy". The word "free" here means, roughly speaking, "available". To illustrate, write down energy conservation for a simple mechanical system of pressure $$p$$ and volume $$V$$: $$dU = T dS - p dV$$ where $$U$$ is internal energy, $$T$$ is temperature, and $$S$$ is entropy. With this starting point we can write $$p = - \left. \frac{\partial U}{\partial V} \right|_S$$ This is saying that when the entropy of the system is not changing (e.g. because everything is reversible and there is no heat transfer) then the pressure can be related to the way internal energy changes with volume. For a piston moving in one dimension we have $$dV = A dx$$ where $$A$$ is the area of the piston, and $$p = f/A$$ where $$f$$ is the force, so $$f = - \left. \frac{\partial U}{\partial x} \right|_S .$$ If we now simply take it for granted that all the behaviour we are considering is at constant entropy then we might simply drop the mention of $$S$$ and write this as $$f = - \frac{d U}{d x}.$$ In this example, the internal energy $$U$$ is acting as a form of potential energy for movements of the piston.

Now why did I bring in thermodynamics here? It is because the original question was all about whether you can have potential energy for non-conservative forces. Well let's consider the following.

First introduce the Helmholtz function defined by $$F = U - TS$$. Note, this is not a force, it is a form of energy. We have $$dF = dU - T dS - S dT = - S dT - p dV$$ so $$p = - \left. \frac{\partial F}{\partial V} \right|_T.$$ So now the pressure is the partial derivative of the Helmholtz function at constant temperature. So, arguing as before for a piston moving in one dimension, $$f = - \left. \frac{\partial F}{\partial x} \right|_T$$ where $$f$$ is the force. The important point for our discussion is that this force would normally be termed "non-conservative" because when the conditions are isothermal then there is a flow of heat which means that the energy gained by the system is not just that provided by the work. And yet we have just identified a function---the Helmholtz function $$F$$---which is acting like a form of potential energy, as long as the conditions remain isothermal. So the answer to the original question is a qualified "yes, you can introduce a concept very much like potential energy when dealing with some kinds of non-conservative force," but you have to understand what you are doing, and therefore the best answer to offer at a preliminary level would be "no, don't try it, it doesn't work---but later on you will learn some methods which make it possible in some circumstances".

Concluding thought

But how can something we say is not possible then turn out to be possible? It is because potential energy is essentially a mathematical method. This is illustrated by the fact that if we add a constant to the potential energy everywhere then nothing changes: no physical effect will notice. You might think this isn't saying much, but it is still true even if the added constant fluctuates with time, which is a bit more striking (I found this quite remarkable when considering conditions in an ion trap, for example). (This is an example of a wider concept called gauge invariance). For another illustration of what it means to say that potential energy is a mathematical method rather than a physical quantity, consider when we do relativity and think about energy exchanges between charged particles and electromagnetic fields. In this case the concept of potential energy is simply not employed; it is not useful for gaining understanding so we don't bother with it. It was never a physical quantity waiting to be measured; it was always a mathematical tool available to be used where useful.

(To be clear, I am here talking about interpreting the combination $$q \phi$$ as an energy which can be assigned to a particle, where $$\phi$$ is electric potential. This does not deny the useful role of $$\phi$$ in other calculations such as deriving the fields from a given source.)

The term conservative force is a term used for forces that will do work when released (when the spring is let go, when a book is knocked off the shelf, when electrons are allowed to flow through a circuit etc.).

Before it does this work, we think of the work that it will do as stored. Stored energy just waiting to be released. This stored energy has a potential to be released as work - we call it potential energy.

So, yes, the term potential energy belongs to conservative forces only, since the term conservative force is used for forces that are "waiting" to be released and thus have energy stored.

Starting with Newton second law

$$\mathbf M\,\mathbf{\ddot{x}}=\mathbf F_a+\mathbf F_c\tag 1$$

and the constraint equations

$$\mathbf g(\mathbf x~,\mathbf{\dot{x}})=\mathbf 0$$

• $$\mathbf F_a~$$ external forces
• $$\mathbf F_c~$$ constraint forces
• $$\mathbf M~$$ diagonal mass matrix

multiply Eq. (1) with $$~\left(\frac{d\mathbf x}{dt}\right)^T$$ and integrating you obtain

$$\int d\left(\frac {\mathbf{\dot{x}^T}\,M\,\mathbf{\dot{x}}}{2}\right)=\int \mathbf F_a\,\cdot\,d\mathbf x+ \int \mathbf F_c\,\cdot\,d\mathbf x$$

the LHS is the kinetic energy , the work that done by the external force $$~\mathbf F_a~$$ is the potential energy , only if the force $$~\mathbf F_a=\mathbf F_a(\mathbf x)$$ or $$~\mathbf F_a=~$$constant. in this case is the force “conservative force”

• Surely you mean "only if the force is a function of $x$", rather than $F\propto x$? In 1D, any force which is an integrable function of position is conservative, though of course that does not generalize to higher dimensions. Sep 10, 2021 at 13:20

Potential energy only occurs when there is a conservative force field, it do not occur for non conservative forces.