The title basically covers it. I've actually thought about this question for a while now, and I am still not sure if I have a definitive answer. Most potential energies seem to just be the work that it would take to move something with a given force (i.e. $mg$ in gravitational fields $\implies$ $mgh$, and $F = -kx$ $\implies$ $\frac{1}{2} kx^2$). So, I've come to think of potential energy to be a useful mathematical model for simplifying calculations, but I don't know if this is actually correct (This also leads me to ask questions like, "Is all of physics just useful mathematical models?"). But, I also know that a conservative force is literally defined as the gradient of a potential $F = -\nabla \phi$, so I don't even know what to think. Most of the arguments I've heard in favor of potential energy being a distinct "real" quantity go along the lines of "Where else would that kinetic/thermal energy have come from??? How about that one, buddy?"; these arguments are unconvincing to me. I've discussed this with a couple of my physics buddies, and they are split on it. I heard @knzhou had an explanation for this in one of his handouts, but I can't find it. Any discussion/info would be appreciated.
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1$\begingroup$ If you change your definition of the potential energy by adding a constant then the physics remains the same. It is not clear how you would interpret this in terms of whether or not this means the potential energy "actually exists." As such your question is vague and unclear. $\endgroup$– hftCommented Jul 5, 2023 at 19:30
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1$\begingroup$ possible duplicates: physics.stackexchange.com/q/138972/50583 and its linked questions $\endgroup$– ACuriousMind ♦Commented Jul 5, 2023 at 19:32
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$\begingroup$ Electrostatic potential energy can be understood in terms of an energy density of an electrostatic field. This energy density appears in the energy-momentum-stress tensor on the right side of the Einstein field equations and contributes to spacetime curvature. Both kinetic and potential terms of all types of fields warp spacetime. $\endgroup$– GhosterCommented Jul 5, 2023 at 20:05
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1$\begingroup$ As an aside, words and phrases like "actually" or "in fact" or "true" often don't really mean anything when written. Compare: "the pen is on the table" to "it's true that the pen is in fact on the table" or "the pen is actually on the table." When you use these words and phrases it can indicate that you do not actually know what you are asking about because these words and phrases are only used to provide some kind of emphasis (akin to banging on a table). $\endgroup$– hftCommented Jul 5, 2023 at 20:27
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1$\begingroup$ What experiment do you propose to answer your question? $\endgroup$– John DotyCommented Jul 5, 2023 at 21:34
6 Answers
My reply which may sound glib is "What is real?" If you lift a rock from the ground to a 100 m high platform, without question it has "something" which can be manifested as motion (if it falls). This new ability to cause motion in the future we call potential energy. The way it is quantified (in Joules) also allows us to accurately predict how much motion (e.g. maximum speed) it will reach. The ontological question is more difficult. You might also ask if photons are real, since no one has ever seen or observed one. We've only made observations which are well-explained by quantum theory which includes photons.
If potential energy does not exist, then energy stored in a battery does not exist. Work is required to organize electric charges within a battery such that energy can be drawn from the battery at a later time. This is described as electric potential energy. If the energy stored in a battery is real, then at least electric potential energy is real. Now consider a spring used to launch a ball. Work is done to compress the spring. Three days later, the spring launches a ball. Was energy stored in the molecular structure of the compressed spring? If energy can be stored in an organization of electric charges (rather than in the individual charges), then why not in an organization of molecules? The only way in science to measure whether something is always true is to prove through measurement that it cannot ever be false. We can easily show that something can be false under some circumstances, but we cannot measure all circumstances. The best we prove is that something is useful or convenient, that it works.
You are exactly right: @knzhou would know, because he basically knows everything ;-). Furthermore, it is also true that all of physics consists of handy models which are used to calculate answers to physics questions.
Here is a bit of motivation on the issue of potential energy.
Let's imagine we have the equations of motion for some real system that we have modeled, and we know how to fool around with calculus and vectors. We play around with the system, note its behavior under different circumstances, and compare the results to what the model says.
In so doing we discover some regularities that suggest the existence of useful nuggets buried (not too deeply) in our equations, and we then rewrite those equations to isolate the nuggets from the sand, which to our delight furnish easy ways to get solutions to completely different physical systems than the specific one we started out with. We give names to them: kinetic energy, potential energy, and momentum.
Then some really smart person comes along, takes a look at those things, and then using nothing more than a straightedge, protractor, and a copy of the King James version of the bible, derives each of them from what (s)he calls first principles.
Potential energy as i understand it at the moment describes systems and procedures in relation to time.
It is useful in some ways to think it like that.
It will be more useful to use Potential energy in relation to space since entropy exists but it is rather difficult to define a "unit" in this regard.
The change in potential energy is defined as the negative of the work done on a particle in a conservative force field, the change in kinetic energy as the positive of the work done. So if you drop a mass in a gravitational field the kinetic energy increases and the potential energy decreases by the same amount. The total energy change is therefore zero i.e. total energy is conserved. So potential energy exists by definition if you want to speak about energy and energy conservation in mechanical systems at all.
The basis of concepts like potential or kinetic energy comes from the fact that we have sometimes equilibrium situations that are not static. And, while the equilibrium is a kind of movement, we can isolate some constants in this movement.
In the case of a simple pendulum without friction for example, the quantity $$\frac {1}{2}v^2 + gh$$ is constant. Or in the case of a system mass-spring, the quantity: $mv^2 + kx^2$ is constant for a given $k$. We call them kinetic term and potential term.
So, what exists is some cases of dynamic equilibrium, with constants of movement, part of which is called potential energy.
