# "External" potential energies strike back

Goldstein, the oracle of classical mechanics, says: But Morin also reliably tells me that
[The total energy of the system] may come in the form of (1) overall kinetic energy, (2) internal potential energy, or (3) internal kinetic energy

These two definitions contradict in their treatment of external conservative forces. Morin only counts the action of internal conservative forces in the definition of the potential energy of the system, whilst Goldstein also includes those of external conservative forces. Both are mathematically sound, but I think Morin's definition is better. This is because the work done by a single external conservative force in a pair cannot always be written as the negative of a change in a potential energy function on its own (e.g. if the position of the external body exerting the force is changing).

I wonder then what forces contribute to ##\sum V_i##. The only examples I can think of that would be perfectly fine to include in that term are the potential energies of fictitious/inertial forces, for which there are no other source bodies that we would otherwise need to include in our system. I wondered whether anyone could clarify what other forces contribute to that first term? Thank you.

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This is because the work done by a single external conservative force in a pair cannot always be written as the negative of a change in a potential energy function on its own (e.g. if the position of the external body exerting the force is changing).
If the external force can change on its own then it is not appropriately conservative.

Of course, if the external forces change their behavior only slowly then we may still be able to usefully model them as being conservative. If, for instance, the Earth slowly gains mass from meteoric bombardment, we can still accurately size the motors on our elevators using the concept of gravitational potential energy.

• etotheipi
Of course, if the external forces change their behavior only slowly then we may still be able to usefully model them as being conservative. If, for instance, the Earth slowly gains mass from meteoric bombardment, we can still accurately size the motors on our elevators using the concept of gravitational potential energy.

Sure, but the choice here is between just extending the system to include the Earth, in which case we can exactly consider the potential energy, or treating the work done by the Earth as an external force before algebraically introducing a "sort of" potential energy.

It just seems clearest to me to let all potential energies be internal to a chosen system, and let all external works be external works. The choice doesn't even change our equations, it's just a case of which is clearer conceptually.

If the external force can change on its own then it is not appropriately conservative.

This is a good point. A single external gravitational force would not be appropriately conservative if the other body could move.

Often in mechanics texts, we define arbitrary external forces which can be derived from potential energy functions. But are there any real forces that act in this manner? It appears to me that the only types of external forces that are appropriately conservative are some fictitious forces.

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Often in mechanics texts, we define arbitrary forces which can be derived from potential energy functions. But are there any real forces that act in this manner? It appears to me that the only types of external forces that are appropriately conservative are fictitious forces.
Much may depend on how the notion of "conservative" was introduced in our respective schooling.

For me, the notion of something being "conservative" was introduced in the context of mathematics. Vector calculus, in particular. You begin with the notion of a "scalar field". This is an assignment of a numeric value to every point in a space. For instance, altitude over a two dimensional map. Or mass density over a three dimensional fluid volume.

One intuition is that a scalar field amounts to a real-valued function of two or three real-valued arguments. Or a real-valued function of one vector-valued argument.

Then one can introduce the notion of a "partial derivatives". How rapidly does the altitude change with changes in the east-west coordinate value? How rapidly with north-south coordinate. This naturally leads to the notion of a "gradient". You put the x rate of change together with the y rate of change and you get a (x,y) vector -- the gradient.

The intuition is that the gradient is a vector whose direction gives the direction up the slope and whose magnitude gives the rate of increase (rise over run) in that direction.

Another intuition is that taking the gradient of a function of multiple variables is analogous to taking the first derivative of a regular function of one variable.

By this time one has also been introduced to the notion of a "vector field". This is an assignment of a vector value to every point in a space. So instead of altitude at every point on a two dimensional map we might have something like wind velocity or gravitational force.

One can see that by taking the gradient of a scalar field at every point in a space, one can obtain a corresponding vector field over that same space. The "gradient" of a scalar field is a vector field.

One can introduce the notion of a "path integral". If you consider a path through the underlying space and you integrate the dot product of incremental distance and local vector value you get a scalar result.

Now you are in a position to introduce the notion of a "conservative vector field" in several equivalent ways:

1. A conservative vector field is a vector field that is the gradient of a corresponding scalar field. The scalar field can be called the "potential" of the original vector field. It will typically be unique up to addition of an arbitrary constant.

2a. A conservative vector field is a field in which the path integral between two endpoints is independent of the path.

2b. A conservative vector field is a field in which the path integral around any loop is zero.

Having done all this work you eventually realize that the notion of time has not entered in. If you want to stick with this simplistic view, you pretty well have to just assume that all of the fields are static. To me, the value of this nice simple view is high enough that I am willing to ignore the pesky fact that the Earth's gravitational field is not static. Maybe that is because I am a mathematician -- willing to tolerate falsehood in the name of beauty. Or maybe that is because I am an engineer -- willing to tolerate minor falsehood in the name of simplicity.

Or maybe that is because I am a lawyer -- willing to tolerate falsehood in the name of winning an argument.

• • etotheipi and vanhees71
Having done all this work you eventually realize that the notion of time has not entered in. If you want to stick with this simplistic view, you pretty well have to just assume that all of the fields are static. To me, the value of this nice simple view is high enough that I am willing to ignore the pesky fact that the Earth's gravitational field is not static. Maybe that is because I am a mathematician -- willing to tolerate falsehood in the name of beauty. Or maybe that is because I am an engineer -- willing to tolerate minor falsehood in the name of simplicity.

Thanks for the really nice answer!

If the source is static, either due to being fixed in position or approximated as static due to e.g. having lots of mass, then it is definitely valid to treat it in the way Goldstein did. I still prefer the approach of Morin, but accept also that it is fine mathematically to describe certain conservative (or approximately conservative) external forces due to static fields as contributing to ##\sum V_i##. Even if it doesn’t sit right conceptually.

I don’t think there’s much else I can add... that’s a good enough resolution for me . Thanks!

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I would extend @jbriggs444's ideas and ask the following question: A force that is mathematically derivable from a potential function, may be deemed conservative, but what about the entity that is responsible for setting up that potential function? When learning about conservative forces, it is common practice to be given an exercise where one starts from a scalar function, e.g. ##\varphi(x,y)=x^2y-xy^2##, and is asked to calculate all sorts of goodies such as the force derived from it, the curl of the force a line integral or two and perhaps a verification of Stokes' theorem. In doing so, one is not concerned with whether the scalar function describes something physical; the concern is to ensure that the student has acquired a basic understanding of conservative forces and the scalar functions from which they are derived. Whether these are time dependent is also of secondary importance to the main task.

My point in asking the question is that when we do physics, we can certainly define a system and consider the forces acting on it. A system can only have external and internal forces acting on it. Internal forces are between pairs of system components can be derived from internal scalar functions if conservative resulting in an internal potential energy term. External forces, by definition, result from interaction between entities one of which is part of the system and one which is external to it. Replacing these forces with scalar potentials is a way to put the external entities out of sight-out of mind for purposes of doing the appropriate calculations, but this doesn't mean that these entities do not exist. They do and are outside the system. In that sense I disagree with Goldstein and argue that potential energy can only be internal and to a system that must have at least two components.

• etotheipi
I think you hit the nail on the head with what was bothering me, and put it a lot more eloquently than I could have! A potential energy function like ##\varphi(x,y)=x^2y-xy^2## could only occur physically for a static (or approximated as static) arrangement of source particles, and to define an actual potential energy we'd need to include the source particles in the system. That said, it's not really physical, like you say.

I think perhaps the best way to think about an external potential energy might be just be that they can exist for convenience of mathematics, so long as the field is static (or also for fictitious forces in non-inertial frames).

But these must not be confused with cases in which the external bodies are not static, in which case assigning the potential energy to one particle is wrong (since we would be ignoring the work done on the other body, which will also affect ##V_{ij}##)!