# Circular reasoning involving conservation of energy and the definition of potential energy? [closed]

This might seem a dumb question but it is at the heart of mechanics.

We learn that in our universe the total energy of a closed physical system is conserved, never destroyed, never created, only transformed in different types of energy as time goes by.

Then we learn that potential energy is the kind of energy reservoir that gets filled by the kinetic energy lost in, for example, a gravitational field; When you throw a rock into the air, the rock has a lot of kinetic energy but as it goes higher and higher the speed decreases (because of the force of gravity). Decreasing speed means decreasing kinetic energy. If we believe that energy is conserved, then that energy must go to another category: potential energy, which is not observable in the motion of the object itself.

Then, potential energy (in the example of the rock, a dam, a pendulum, a spring etc...) is just kinetic energy waiting to be actualized. Is energy that is not manifesting until something happens and gets released and transformed into kinetic. We know it's there because we think energy is conserved, so the balance means that the new kinetic energy of a falling object can't be coming out of nowhere.

At least that's how we learn it in school.

Then when we ask "how we know that energy is conserved?" we are told that when we calculate the potential energy and the kinetic energy, and then we add them together, we get a total that is independent of time. In that way we prove that energy is conserved.

But I see this as circular reasoning: We defined potential energy as the result of a balance, that we suppose exist because we assume energy is conserved. And then we justify our assumption that energy is conserved by invoking this seemingly made-up concept of potential energy to compensate for the unbalances in the energy we observe.

How can we break this loop? Can potential energy be defined independently of the law of conservation of energy? And on a related note: Do we have some kind of experiment or observation that allow us to know that something has potential energy that doesn't rely on transforming it to kinetic first and checking the balance with the conservation law (this seems an indirect way of knowing something has potential energy)?

If I knew nothing about the conservation of energy, could I define what potential energy is? If I didn't knew what potential energy is, would I be able to verify that energy is indeed conserved?

Imagine a universe where the law of conservation of energy is not true. Is there any plausible scenario were the physicist in this universe would first define potential energy and then discover that energy is in fact not conserved as a whole? Or would they make-up another non-accesible kind of energy to compensate for the residual and establish by convention that energy is conserved after all? They could call this other kind of energy another name. Could we live in that universe and be defining potential energy just to make it fit with our believe that energy must be conserved, when in fact it is not?

I guess my last rephrasing of this question might be the clearest: I have the feeling that one of both (potential energy or the conservation of total energy) is a human convention, while the other is an observable fact inside the framework of that convention. You can choose potential energy to be a non-physical thing defined by humans and then with that in hand see that in nature energy is conserved when we consider potential energy. Alternatively we could establish, as a convention, that energy must be conserved, so then we would say that any time we observe something slowing down it means that kinetic energy is going to another invisible reservoir that only manifests itself indirectly when that reservoir releases its content in the form of kinetic energy that we can observe. But if these concepts are tied in this way then we could throw both concepts altogether and still describe nature without invoking either of them.

The question was closed because It lacks focus apparently. I know It is a long question, but I really highlighted the key parts in bold. Also I asked the question in several different ways precisely to make It clearer to everyone. Each rephrasing is equivalent to the others: That doesn't make the question more vague but more concrete. The fact that It worked as intended can be verified by the fact that all 3 answers (before the question was closed) got the same idea and are indeed answering my question. I don't really know how to make It less ambiguous.

• I think your fundamental mistake here is in assuming that the definition of potential energy is circular simply because it is well-motivated. Going back to physics 101, given a conservative force $\vec F$, we define an associated potential energy function $U$ such that $\vec F = -\nabla U$. If we do this, then as long as $\vec F$ is the only force doing any work, then the quantity $E = 1/2 mv^2 + U$ is constant. There's nothing circular about this. If the potential energy was defined as $E-1/2 mv^2$, then there would be circularity, but it isn't. May 1, 2023 at 4:15

In General we define energy to be the quantity that is preserved by the dynamical evolution of a system. In Newtonian mechanics, for example, it turns out that if the trajectory $$x(t)$$ of an object is governed by Newton's law of motion, then there is a quantity that can be written as

$$\frac{1}{2}m\dot x^2 + V(x)$$

that is preserved, namely it remains constant over time. We simply call the part of that quantity that depends on the object's velocity ($$\dot x$$) kinetic energy, and the part that doesn't potential energy.

Any physical theory in which there exist some quantity that is preserved over time can be said to have conservation of energy, just by matter of definition. Of course one can imagine different laws of physics where no such quantity exists, but in such a universe any physical theory that does have a conserved quantity will necessarily be in contradiction to experimental observations, just by our assumption that no such quantity exists.

How can we break this loop?

With Noether’s theorem.

Very often when you see something that seems circular it is just because the teacher has made a judgement call and decided that the actual derivation of the concept is not beneficial for the students at that time. Either because the requisite background material has not been covered or because a detailed proof would detract from the flow of the class.

Noether’s theorem says that if the action has some differential symmetry then there is a corresponding conserved quantity. Energy is the conserved quantity that is associated with time translation invariance of the Lagrangian.

So the fact that the laws of physics are the same today as they were yesterday implies that there is a conserved quantity. We call this specific quantity energy.

In many scenarios this conserved quantity has one term which is proportional to the velocity squared and another term which is a function of position. We call the first term “kinetic energy” and the second term “potential energy”.

• Isn't this circular in some way? To apply Noether's theorem you need the Euler-Lagrange equations, this you need tondefine the lagrangian L=K-U. Isn't that a way of looking for a definition of potential energy? May 1, 2023 at 4:32
• @Swike no, it is not circular at all. The Lagrangian is obtained experimentally. I.e. we use the Lagrangian that we do because it produces the experimentally observed equations of motion. Particularly as we went beyond Newtonian mechanics, experiment historically led the way and the Lagrangian followed. But even for Newtonian mechanics, we can now take that conceptual approach to avoid circularity.
– Dale
May 1, 2023 at 4:41

For an intuitive answer, consider the following:

In many cases we see obvious and immediate energy conservation. Such as when billard balls hit each other, where some are slowed down while others are sped up so that the total kinetic energy is constant.

In other cases we see, shall we say "delayed" energy conservation. When a ball is thrown upwards and falls back down, then we see its kinetic energy "disappearing" and a second later "reappearing".

Overall when looking at all every-conducted controlled experiments of these types, we see either the immediate energy conversion between known energy types where energy always is conserved, or we see this "delayed" energy conservation where "disappearing" energy eventually "reappears" again. Since the "disappearing" energy seems to always "reappear" eventually, we might think of it as "stored" in some abstract sense - and just waiting to be released later on, in the sense that it actually never fully disappeared.

We might call it potential energy since it has a potential to "reappear" (to manifest as known energy forms, such as kinetic). If we do choose to consider this "disappearing-and-reappearing" energy amount as energy that in fact never disappears but just is converted "to storage", then suddenly every single energy-related experiment we have ever conducted shows energy conservation! With the idea of potential energy invented in this way, the energy seems to always be conserved - thus it is very, very helpful to accept this potential-energy definition and simultaneously define a law of energy conservation (the first law of thermodynamics) in general.

Actually calculating the potential energies is now a simple questions of finding out how much energy that is lost from other types (such as how much kinetic energy that is lost at any moment during an upwards throw). Often times you can find that out (in particular for mechanical energy types) by simply considering the work that the force of gravity does, since work equals an energy change. Then you find the expressions/formulas that describe how different types of potential energies can be calculated.

There is no circular reasoning now. Potential energy has been invented as a key necessity or energy conservation to be a reality. If it wan't accepted, then energy conservation would not be accepted either.

• Thank you. "If it wasn't accepted, then energy conservation would not be accepted either.", so these two concepts are very strictly linked, you have both or none, right? So it really is a convenient convention to say that potential energy exists. Could we make a physical theory that instead considers that kinetic energy is destroyed and then created out of nowhere by certain conditions? In that way we wouldn't have energy conservation an no need to conceive potential energy. Still the description would fit experimental data. Why not doing this? Apr 30, 2023 at 19:36
• @Swike Exactly, you have both or none. Energy conservation has turned out to be a rather simple and enormously useful workaround for otherwise impossibly complicated computations, but it does require the definition of potential energies as non-disappearing energy quantities. Sure, you could refuse these definitions, but then you would not have the law of energy conservation at hand and would be severally held back in your engineering work. Unless you have good reasons (such as other methods that then would take over, which otherwise would not be applicable), then it would be counterproductive. Apr 30, 2023 at 20:24

Potential energy can be defined indpendantly for conservative forces like gravity, and is often defined as work done against force in bringing the object from infinity to the current location, against the acting force field. And for convenience we define $$U(\infty)=0$$ generally. This statement is mathematically: $$\frac{dU}{dr}=-F(r)$$

or $$dU=\vec{F(r)}\cdot\vec{dr}$$ $$U(r)-U(\infty)=\int_\infty^{r}\vec{F(r)}\cdot\vec{dr}$$

And Kinetic energy can be seperately defined as work done on system amounts to change in kinetic energy(Work energy Theorem).

Also note that all forces cant have potential energy defined, like for friction.

Also conservation of energy can be independently verified by Noether's theorem, which is result of time symmetry of our universe.

Ig this breaks your circular logic?