Meaning of potential energy in external fields

• etotheipi
In summary, the potential energies in a system of two bodies can be determined by summing the contributions pairwise, but in classical mechanics, the Hamiltonian and the Lagrangian treat potential energy as that of a single particle. This may lead to confusion when considering potential energies in an external field, where the source body is not explicitly stated. Some textbooks treat the total energy of the system as the sum of the kinetic energy of the particle and the potential energy in the external field, while others consider the potential energy as belonging to the particle itself. However, in practical applications, the specific relationship used often does not depend on the interpretation of the potential energy.
etotheipi
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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etotheipi said:
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
A.T. said:
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.

etotheipi said:
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
jbriggs444 said:
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?

etotheipi said:
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?

jbriggs444 said:
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?

etotheipi said:
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
etotheipi said:
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".

etotheipi said:
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
A.T. said:
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?

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
etotheipi said:
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
A.T. said:
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

etotheipi said:
It still just feels a little too much like cheating
If using approximations is cheating, then all of physics is cheating.

jbriggs444 and vanhees71
A.T. said:
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.

etotheipi said:
...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
A.T. said:
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.

A.T. said:
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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1. What is potential energy in external fields?

Potential energy in external fields refers to the energy that an object possesses due to its position or configuration in relation to an external force or field. This energy can be stored and converted into other forms of energy, such as kinetic energy, when the object moves or interacts with the external field.

2. How is potential energy related to external fields?

Potential energy is directly related to external fields because the strength and direction of the external field can affect the potential energy of an object. For example, an object placed in a gravitational field will have a greater potential energy if it is at a higher altitude, as it has more energy stored due to its position in the field.

3. Can potential energy in external fields be negative?

Yes, potential energy in external fields can be negative. This occurs when the external force or field is repulsive, causing the object to have a lower potential energy than if it were in a neutral or attractive field. An example of this is an object with an electric charge placed in an electric field with the same charge, resulting in a repulsive force and negative potential energy.

4. How is potential energy in external fields calculated?

The calculation of potential energy in external fields depends on the specific type of field and the properties of the object. For example, the potential energy of an object in a gravitational field can be calculated using the equation PE = mgh, where m is the mass of the object, g is the acceleration due to gravity, and h is the height of the object in the field.

5. What is the significance of potential energy in external fields?

Potential energy in external fields is significant because it allows us to understand and predict the behavior of objects in relation to external forces. It also plays a crucial role in many natural phenomena, such as the movement of planets in gravitational fields and the behavior of particles in electric and magnetic fields.

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