# Meaning of potential energy in external fields

Generally potential energies are associated with a system of two bodies. If more than two bodies are involved the total can be determined by summing the contributions pairwise. It would appear as though in any system, the potential energies are all internal to the system. However in classical mechanics we have the Hamiltonian and the Lagrangian, which both (from what I've seen, anyway) treat ##U## as the potential energy of a single particle. We might have a ball being thrown in the air, and we'd denote ##U = mgx##.

Normally this wouldn't bother me, since, for instance, we could write two work energy theorems for the ball and the Earth:$$W_{\text{Earth|Ball}} = \Delta KE_{Earth}$$ $$W_{\text{Ball|Earth}} = \Delta KE_{Ball}$$ ##W_{\text{Ball|Earth}} + W_{\text{Earth|Ball}}## is the total work done by gravity on the system, i.e. ##-\Delta U_{\text{system}}##. We might then approximate that the Earth is so massive it doesn't move at all such that ##-\Delta U_{\text{system}} = W_{\text{Ball|Earth}}##. In effect, we can think of all the PE belonging to the ball with no problems.

However, this just seems like a useful trick for conservation of energy calculations. We sometimes call ##U## the potential energy in an external field. Why don't we care about the source body? Surely it only makes sense to speak of potential energies internal to some system?

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Why don't we care about the source body?
Why should we? Sometimes there is no source body, like for the potential in an accelerating frame of reference.

• etotheipi
Why should we? Sometimes there is no source body, like for the potential in an accelerating frame of reference.

Fair enough, in accelerating frames we can indeed define potential energy terms.

Nonetheless, in general terms if we're speaking of some arbitrary potential in space e.g. we might have a proton in a harmonic potential, the potential is necessarily caused by another source charge (I've no idea what it looks like in that case 😁) which has to share a chunk of the potential energy.

Even if we use the abstraction that all of the potential energy can be thought of as belonging to one thing, it's still a little bit iffy.

Homework Helper
Even if we use the abstraction that all of the potential energy can be thought of as belonging to one thing, it's still a little bit iffy.
One can dribble a ball down a basketball court without worrying about the Earth rebounding from each impact of ball on floor. Ignoring the finite mass of the Earth is not nearly as iffy as the tacit assumption of a rigid floor.

• vanhees71 and etotheipi
One can dribble a ball down a basketball court without worrying about the Earth rebounding from each impact of ball on floor. Ignoring the finite mass of the Earth is not nearly as iffy as the tacit assumption of a rigid floor.

Though if we let our total energy be ##E = KE_{basketball} + mgh##, then wouldn't E necessarily be the total energy of the Earth-basketball system, and not the basketball?

Homework Helper
Though if we let our total energy be ##E = KE_{basketball} + mgh##, then wouldn't E necessarily be the total energy of the Earth-basketball system, and not the basketball?
And how accurately are you measuring the energy of the basketball?

And how accurately are you measuring the energy of the basketball?

I'm not sure I follow 😅, I just thought to model the system we could write a work energy equation for the basketball and the Earth

##W_{grav, basketball} = \Delta KE_{basketball}## and ##W_{grav,Earth} = \Delta KE_{Earth}##. Then we sum the two equations such that ##W_{grav, basketball} + W_{grav,Earth} = -\Delta GPE##, and finally approximate ##\Delta KE_{Earth} = 0##.

That leaves us with ##\Delta KE_{basketball} + \Delta GPE = 0##, but ##GPE## is still a property of the system in this equation!

That is, I don't think the total energy of the basketball is actually well defined.

If we take the system to be just the basketball, then ##E = KE_{basketball}##, and gravity does some work causing this to change.

In order to start applying conservation of mechanical energy, surely we need to designate the Earth and the basketball as our system?

Homework Helper
That is, I don't think the total energy of the basketball is actually well defined.

If we take the system to be just the basketball, then ##E = KE_{basketball}##, and gravity does some work causing this to change.

In order to start applying conservation of mechanical energy, surely we need to designate the Earth and the basketball as our system?
$$PE=mgh$$
More accurate than your measuring instruments.

• Motore
If we take the system to be just the basketball, then ##E = KE_{basketball}##, and gravity does some work causing this to change.
In practical applications the result does not depend on whether you say "gravity does work on the ball by using the potential energy of the Earth-ball-system" or "the ball has potential energy and converts it into kinetic energy".

In order to start applying conservation of mechanical energy, surely we need to designate the Earth and the basketball as our system?
For specific applications we often don't use the most general version of a law, but a more specific relationship derived from the general law.

• etotheipi
In practical applications the result does not depend on whether you say "gravity does work on the ball by using the potential energy of the Earth-ball-system" or "the ball has potential energy and converts it into kinetic energy".

For specific applications we often don't use the most general version of a law, but a more specific relationship derived from the general law.

It's just slightly odd since Resnick/Halliday and Morin would write the total mechanical energy of the basketball system as ##E_{mech} = KE_{basketball}## and treat the gravitational force as doing external work.

Whilst Kleppner/Kolenkow and some others seem quite happy to forget the idea of a system and just consider the basketball as having potential and kinetic. In this interpretation, the potential energy in an external field also contributes to ##E_{mech}##.

But you would have thought that everyone has to agree on something as well-founded as mechanical energy?

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If you want to describe the closed system consisting of the base ball and the Earth. The energy conservation law reads
$$E=\frac{1}{2} m_{\text{B}} \vec{v}_{\text{B}}^2 + \frac{1}{2} m_{\text{E}} \vec{v}_{\text{E}}^2 - \frac{\gamma m_{\text{E}} m_{\text{B}}}{|\vec{r}_{\text{B}}-\vec{r}_{\text{E}}|}=\text{const}.$$

• etotheipi
But you would have thought that everyone has to agree on something as well-founded as mechanical energy?
There is no disagreement, just different levels of generality applied, as already explained.

• • vanhees71, jbriggs444 and etotheipi
There is no disagreement, just different levels of generality applied, as already explained.

Okay, I must say I'm still not too comfortable with it, but I can appreciate that ultimately it doesn't matter whether one counts the source as within the system or not even when we include its contribution as a potential energy term so long as the source particle is stationary.

It still just feels a little too much like cheating; a potential energy is an interaction energy, the negative of the work done on the system to establish the system. It seems it only makes sense to speak of an internal potential energy.

Either way, thanks for your help It still just feels a little too much like cheating
If using approximations is cheating, then all of physics is cheating.

• jbriggs444 and vanhees71
If using approximations is cheating, then all of physics is cheating.

Though I don't think this is an approximation, but rather bending the rules to make things simpler. The following is from Morin Once we've clearly defined the boundaries of our system, it's difficult to see how a potential energy term in an external field fits in. For instance, in the first "book" system, we don't consider any potential energy.

Of course, we could do an algebraic substitution of a potential energy term, but this seems to be slightly misleading.

...rather bending the rules to make things simpler.
Also known as: deriving new simpler rules for specific cases, from a more general rule.

• Motore and etotheipi
Also known as: deriving new simpler rules for specific cases, from a more general rule.

All potential energies are internal to a system. If we include a term for a potential energy in an external field, we are effectively describing the energy of a system including that external source, however we can ignore this technicality in practice since it doesn't affect the question.

And in the case that there are no source particles (e.g. centrifugal potential energies), this potential energy is internal to the system anyway.

So at the end of the day, we can still only have internal potential energies. But there is a little bit of wiggle-room for defining the system boundaries in general practice.

Gold Member
In practical applications the result does not depend on whether you say "gravity does work on the ball by using the potential energy of the Earth-ball-system" or "the ball has potential energy and converts it into kinetic energy".
I think the problem can be when to include 'another body' or when not to. The crossover from one to the other would surely just be when the amount of Energy transferred to the 'other body' is unmeasurable or negligible. The word 'cheating' was dealt with earlier on but Science is based on identifying and isolating individual entities and relationships in the presence of the rest of the world.

It's those danged 'rules' that we learn without learning about the basic idea of Science. Over-classification strikes again.

• etotheipi
It's a little like, suppose we had two equal sized asteroids falling toward each other. It would be completely wrong to write the mechanical energy of asteroid ##1## as ##-\frac{Gm_1m_2}{r} + \frac{1}{2}m_1v_1^2##. Instead, everyone would consider the system of both of them, and instead take the mechanical energy as ##-\frac{Gm_1m_2}{r} + \frac{1}{2}m_1v_1^2 + \frac{1}{2}m_2v_2^2##.

Now we replace asteroid 2 with the Earth. Of course, practically, it doesn't now matter which interpretation we choose in order to get the right answer. But conceptually, the scenario is no different to the first, and really it still makes no sense to designate the potential energy to one body.

I'd agree that learning rules is a bad way of learning Physics, however it helps to have a consistent theoretical framework with which you can attack any sort of problem, in classical mechanics anyway. Otherwise, when you do eventually generalise, problems start to pop up.

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