E=MC^2 Mass and Energy, synonymous?

[Dale]

That is fine. I am glad to hear what you have to say, although I think that the idea that energy defies definition is patently false given the fact that it is defined.

 

[Dale]

I say go ahead and post your rebuttal. I am not into games on the Internet.

I suspect that your objection to the Lagrangian definition will be as substantive as your objection to the work definition. Which is to say, a mere matter of wordsmithing, blown out of proportion and presented as though it were some fundamental flaw.

 

[Popper]

E/c² is what Newton called mass (the factor between momentum and velocity). Rest energy is equivalent to rest mass and total energy is equivalent to the so called relativistic mass.

The expression E = mc² was unknown to Newton. But Newton did define mass essentially as the m in p = mv.

I myself view this in the same way that Mendel Sachs does in the article I mentioned above.

 

[Popper]

As I red Einstein's original article I know what it means.

Why would you automatically assume that was what the author has to say about it?

 

[Dale]

Those textbooks you mentioned attempt to define energy as the ability to do work. That's too vague to have meaning.

I never objected to the work definition.

"Too vague to have meaning" sure sounds like an objection to me.

Then I won't bother posting it if you've already made up you're mind. You seem to have decided already that what I have or have to say on this is nonsense so I won't bother.

That is certainly your prerogative. It is hardly an impressive rhetorical approach.

 

[Popper]

...although I think that the idea that energy defies definition is patently false given the fact that it is defined.

Let h = Jacobi's integral aka energy function, what you called the Lagrangian definition of energy. You argue against my view because you believe that there is a self-consistent logically definion of energy. That's what this thread is about, is it not? If I accepted that assertion then we wouldn't be talking about it. If you hold that the definition of energy is h = T + V kinetic energy + potential energy then that is an error. That is not the definition of energy (although many physicists erroneously call it that), its the definition of mechanical energy. There's a theorem, not a law of physics, that the total mechanical energy of a closed system is conserved. That's something that can be proved from first principles and as such it can't be called a law of conservation of energy.

Let me go back and make a correction to what I said earlier. Feynman wrote It is important to realize that in physics today, we have no knowledge of what energy is. I agree of course. And I shouldn't have said that energy can't be defined but rather said there is no logically consistent definition of energy to date just in case someone comes up with a definition I am unable to find fault with someday.

Back to your argument - You keep saying that energy is defined. I assume that you're speaking about either the "energy is defined as the ability to do work" definition or the "energy is defined as Jaconi's integral," also known as the energy function, which you referred to above as the Lagrangian-based definition of energy. Is that correct?

I used the term "Jacobi's integral" for good reason. The reason is given in Classical Mechanics = Third Edition[/i], by Goldstein, Safko and Poole in the last sentence on page 61. It's known more popularly as the energy function and labeled h (small case h is used when its expressed in terms of generalized coordinates whereas when it's expressed in terms of generalized coordinates and generalized, aka canonical, momentum). As I mentioned above, this is what you referred to as the Lagrangian-based definition of energy. I know that function quite. I've studied Goldstein et al quite carefully and in all its gory detail.

Let me now clarify my previous statements about that function. As I mentioned above there are two names given to it in Goldstein: energy function and Jacobi's integral. I explained above that its an equality rather than a definition. In response to that you responded

So what? All good scientific definitions are in the form of an equality. This certainly is irrelevant in whether or not something qualifies as a definition.

Don't worry. There's always a method to my madness. :) Let me explain, i.e. answer your (perhaps rhetorical) question "so what?"

Do you know why it's called the energy function? It's because there are circumstances in which h is the total energy of the system. Under those assumptions (stated in Goldstein - do you want me to post them or do you have Goldstein's text?) it can shown h = E_m where E_m = T + V where T = kinetic energy and V = potential energy. The sum is called the "total energy" and that's by h is called the energy function. But there are conditions under which h is not the total energy of the system and the reason we know what T + V is, is because its a quantity derived from mechanics and is a constant of motion of V is not a function of time. The problem here is that E_m is not energy itself but is actually mechanical energy, hence the m subscript.

Jacobi's integral only applies to classical mechanics. It does not apply to thermodynamics and statistical mechanics and it doesn't apply to quantum mechanics. That's obviously because there are other forms of energy other than mechanical energy.

Back to your comments

So what? All good scientific definitions are in the form of an equality. This certainly is irrelevant in whether or not something qualifies as a definition.

The only way that h can be associated with energy is to first determine what T + V is and then show that h = T + V, i.e. an equality. We knoww what T + V is from classical mechanics. In order to write the definition of the Lagrangian you must know, outside of Lagrangian dynamics, what both T and V are. And that only applies to classical mechanics. You can't use it to show what energy is iin high energy particle physics. I recall that you can't prove that R = mc^2 using Lagrangian dynmics, although I may be wrong. There are other forms of energy and forms of energy we havn't even discovered yet and when we do we postulate that the sum of all forms will be conserved.

 

[DrStupid]

The expression E = mc² was unknown to Newton. But Newton did define mass essentially as the m in p = mv.

Of course. As everybody else in his century he wasn't in doubt about Galilei transformation. In classical mechanics mass as defined by Newton must be frame independent. With Lorentz transformation the same definition leads to E=mc².

 

[Dale]

I apologize in advance, your post is very big and due to connectivity issues my screen is very small.

You argue against my view because you believe that there is a self-consistent logically definion of energy.

I believe that there are several, actually. Different theories and different formulations of the same theory lend themselves to different definitions of energy. Each of these definitions are used in their respective theories, and nothing in the quotes demonstrates the self-inconsistency you are claiming.

What there may not be is a theory-independent definition of energy which is completely general. As I have mentioned, I think this is what the Feynman and French quotes refer to, not your claim that energy defies definition.

Back to your argument - You keep saying that energy is defined. I assume that you're speaking about either the "energy is defined as the ability to do work" definition or the "energy is defined as Jaconi's integral," also known as the energy function, which you referred to above as the Lagrangian-based definition of energy. Is that correct?

Among others. Definitions that I can think of are "capacity to do work", "Noether current for time symmetry", "the energy operator", "time time component of stress energy tensor", "KE and anything that can be converted to KE", "ADM energy", "Komar energy", etc. For your position to be correct you need to show that each is a self-contradictory definition.

I used the term "Jacobi's integral" for good reason.
...
Jacobi's integral only applies to classical mechanics. It does not apply to thermodynamics and statistical mechanics and it doesn't apply to quantum mechanics.

I was not talking about the Jacobi integral. I was talking about Noether's theorem, which is more general.

The only way that h can be associated with energy is to first determine what T + V is and then show that h = T + V, i.e. an equality

You have this exactly backwards. If you have a definition for energy then whatever meets that definition IS energy, by definition. If h is defined as energy then the point of the equality h=T+V is to show that T+V is energy.

You have a long way to go to show that any of the commonly used definitions of energy are inconsistent, let alone that all of them are.

 

[Popper]

Each of these definitions are used in their respective theories, and nothing in the quotes demonstrates the self-inconsistency you are claiming.

Thank you for your opinion.

As I have mentioned, I think this is what the Feynman and French quotes refer to, not your claim that energy defies definition.

And thus we are going to end up simply disagreeing. It’s not as if Feynman is here to clarify..

Among others. Definitions that I can think of are "capacity to do work", "Noether current for time symmetry", "the energy operator", "time time component of stress energy tensor", "KE and anything that can be converted to KE", "ADM energy", "Komar energy", etc. For your position to be correct you need to show that each is a self-contradictory definition.

I disagree with all those. I don’t think it’s worth going on about it though. I think we both understand what the other’s view is at this point and when that happens its no use continuing.

I was not talking about the Jacobi integral. I was talking about Noether's theorem, which is more general.

Consider the example of a discreet system. In such case Noether’s theorem refers to the fact that when the Lagrangian is independent of time that Jacobi’s integral is an integral of motion. However it’s a well known fact that Jacobi’s integral is not necessarily the total energy of the system. There are specific conditions which the system must meet in order for Jacobi’s integral to be the energy. It’s quite possible for h to be constant but not the energy. This is stated quite clearly in the following places

Classical Mechanics – Third Edition, Goldstein, Safko and Poole, page 345

Classical Dynamics, Donald T. Greenwood, pages 73 and 167

Analytical Mechanics – Fifth Edition, Fowles and Cassidy, page 368

Analytical Mechanics with an Introduction to Dynamical Systems, Josef S. Torok page 124

Greenwood’s text uses the term Natural System to refer to those systems where the value of Jacobi’s integral is the total mechanical energy.

The theorem is proven based on the definition of mechanical energy being the sum of the total kinetic energy T and the sum of the total potential energy V. I.e. the proof of the theorem starts out by defining T and defining V and defining E as E = T + V and then proving that E = constant for a natural system. For continuous systems something similar holds. I.e. one defines the time-time component of the stress-energy-momentum tensor to be energy, not by defining energy to be the the time-time component of the stress-energy-momentum tensor.

You have this exactly backwards.

What I said is precisely correct. Simply turn to Goldstein et al and turn to section 2.7 page 62 and follow the derivation. You’ll see that for a natural system the authors state “…, so that h = T + V + E, and the energy function is indeed the total energy.”

If you have a definition for energy then whatever meets that definition IS energy, by definition. If h is defined as energy then the point of the equality h=T+V is to show that T+V is energy.

I, like Feynman and French, have stated numerous times above that there is a well defined definition of total mechanical energy for discrete systems and electromagnetic energy for electromagnetic systems etc. The forms of energy are very well defined. It’s the definition of energy itself that defines definition.

Please recall where I quoted Newtonian Mechanics by A.P. French, The MIT Introductory Physics Series. From page 376-368

The above remarks do not really define energy. No matter. It is worth recalling once more the opinion that H.A. Kramers expressed: "The most important thin and most fruitful concepts are those to which it is impossible to attach a well-defined meaning." The clue to the immense value of energy as a concept lies in its transformation. It is conserved - that is the point. Although we may not be able to define energy in general, that does not mean that is only a vague, qualitative idea.

Just so that we’re clear I’m going to post another expression of my position as given in An Introduction to Thermal Physics by Daniel V. Schroeder, page 17

To further clarify matters, I really should give you a precise definition of energy. Unfortunately, I can’t do this. Energy is the most fundamental dynamical concept in all of physics, and for this reason, I can’t tell you what it is in terms of something more fundamental. I can, however, list the various forms of energy – kinetic, electrostatic, gravitational, chemical, nuclear – and the statement that, while energy can often be converted from one form to another, the total amount of the energy in the universe never changes.

The author has a PDF file online about this entitled What is Energy at
http://physics.weber.edu/schroeder/eee/chapter1.pdf

Again the author writes

What is energy, in the scientific sense? I’m afraid I don’t really know. I sometimes visualize it as a substance, perhaps a fluid, that permeates all objects, endowing baseballs with their speed, corn flakes with their calories, and nuclear bombs with their megatons. But you can’t actually see the energy itself, or smell it or sense it in any direct way—all you can perceive are its effects. So perhaps energy is a fiction, a concept that we invent, because it turns out to be so useful.

One example of a failed definition of energy is The ability to do work. One might mistake that for being momentum. After all, anything that has momentum can do work since changes in momentum means there is a force and that force can do work. But that's tricky stuff. E.g. suppose that during the time period t1 to t2 a block is being pushed across a table at constant speed. The total work done is zero since there is no change in kinetic energy. The force pushing it is opposed by friction so that it doesn't accelerate. Each force cancels since the mome ntum of the block doesn't change.

So again, I hold this to be quite true. Nothing that has been posted up to this point has there been any definition that would include things like zero point energy or the energy in the universe when it reaches its eventual heat death.

For some reason I failed to get across that certain forms of energy are well defined and that’s the forms of energy that Noether’s theorem is able to address. At least the proofs that I’ve seen to date.

By the way; my position is not rooted in a lack of trying to find a suitable definition. I’ve searched extensively everywhere that I have access to in order to find a definition which I hold to be correct but never found one.

Just to save you a lot of trouble I'll state that I understand that you’ll disagree with my position even given what I’ve posted just now. After all, no sense of wasting a time when I pretty much understand your position. I just disagree with it. And not because I think it's a matter of opinion, but because I haven't seen a proper definition to date including what you've posted in this thread.

Now I'm not sure if I made the mistake of saying that it can't be defined. Only that today I know of no logically acceptable definition of it. At least in my opinion.

To really understand what Feynman meant in that quote you must read that section? It's in Volume 1 page 4-1 to 4-2. Otherwise you'll get the wrong idea from just looking at that quote. That's why I think you're misinterpreting that quote.

Thanks for the engaging converson DaleSpam. :)

 

[Dale]

I disagree with all those. I don’t think it’s worth going on about it though.

I think it is worth it. For your position to hold you must show that each of those definitions is self-contradictory. Something you have not shown for even one.

However it’s a well known fact that Jacobi’s integral is not necessarily the total energy of the system.

None of your comments regarding Jacobi's integral are relevant since afaik energy is not defined as Jacobi's integral. (I certainly didn't define it thus nor have I seen any other source do so. You are simply setting up a strawman to knock down.)

I, like Feynman and French, have stated numerous times above that there is a well defined definition of total mechanical energy for discrete systems and electromagnetic energy for electromagnetic systems etc. The forms of energy are very well defined. It’s the definition of energy itself that defines definition.

This comment makes me think that you are making a metaphysical assertion, rather than a scientific assertion. Whenever people talk about "X itself" or "what Y is" I find that they are talking philosophically, particularly if they use italics or bold or the word "really".

Scientifically, a thing is defined in a theory if we can calculate a definite value for it in terms of the framework of the theory and if you can make an experimental procedure for measuring it. Anything beyond this is philosophy, not science. Energy meets the scientific criteria for being well defined.

Scientists, including Feynman, are certainly allowed to make philosophical statements, but you shouldn't make the mistake of taking a philosophical statement as a scientific one simply because it was stated by a scientist.

One example of a failed definition of energy is The ability to do work.

This is a definition for standard Newtonian physics. So restricting ourselves to "Newtonian" systems, do you agree that a system with energy has the ability to do work? Do you agree that a system without energy would not have the ability to do work? Do you agree that the amount of work that a system has the ability to do is equal to its energy?

If you agree with all of that then the rest is mere objections over the "wordsmithing" of the definition.

One might mistake that for being momentum.

No, one would never make that mistake since an object with no momentum can still do work. Therefore by the definition momentum cannot be energy.

Your objection to this definition seems ill founded.

By the way; my position is not rooted in a lack of trying to find a suitable definition. I’ve searched extensively everywhere that I have access to in order to find a definition which I hold to be correct but never found one.

I suspect that is because you have some philosophical preconceptions to what the definition should be that are not necessary for a perfectly good scientific definition.

And not because I think it's a matter of opinion, but because I haven't seen a proper definition to date including what you've posted in this thread.

And you have not shown that any of the many definitions of energy is inconsistent, let alone all of them.

It is an undeniable fact that the various definitions exist. Therefore energy can be defined. You may have certain philosophical biases that lead you to dislike the definitions but your dislike is an entirely different thing from them not existing. The only objective way to invalidate a definition is to show that it is self-contradictory, which you have not done.

 

[Popper]

I think it is worth it. For your position to hold you must show that each of those definitions is self-contradictory. Something you have not shown for even one.

I don't need to prove that it's self-contradictory. I explained that your definitions are definitions of forms of energy and not energy itself and as such nobody is saying that the forms of energy are not properly defined. I'v stated that many times so far and I don't see the point in repeating it again after this post.

The reason I don't think it's worth my time is because I've already explained why each attempt at a definition that you gave is wrong and the only response you gave was to state that I made an error. So why shouuld I continut. I hold that what I said is correct and you hold that its not. There's nowhere to do when a discussion gets to that point.

I'll give you an example to illustrate my point. The assertion is that the term energy has no proper definition but that the various forms of energy are very well defined. Your response was to post an example of a form of energy and then assert that you proved me wrong because you posted a definition of energy. The problem being that you didn't post a definition of energy itself but merely a form of energy. E.g. mechanical energy or EM energy etc. I then explained that the energy function isn't even always the energy in general. You also asserted that energy is the ability to do work. I then explained that such a statement is far too vauge to properly define energy. In fact other authors use that exact definition to define potential energy. I gave you an example of how it being vauge leands you nowhere. E.g. suppose you assert that energy is the capacity to do work. That tells you nothing about how to quantify it.

You can go through the entire therad and see my responses to all your assertions. Just becaus you state that I was wrong in no way shape or form makes that true in my opinion.

Si that's why I'm going to agree to disagree with you and then not continue on in this thread. It's not as if I'm the only physicist who holds this opinion and we don't hold it for no good reason either. But I won't continue to debateit merely because you state that you're correct and I'm wrong. That's not how a pleasant discssion works.

Plus there are many other questions on the forum to address. Nobody in this thead has any unanswered quesions so I'm going to help in other threads.

Thanks for the conversation. I enjoyed it.

Ps - In many ways, what you've referred to above was mechanical energy and the conservation of mechanical energy theorem whereas I've been referring to energy and the law of conservation of energy. Of course in continuum mechanics a similar thing holds, i.e. EM energy and the zero covariant divergence of the stress-energy momentum tensor - i.e. conserved currents if memory servers (warning: my memory is like a sieve)

 

[Dale]

I don't need to prove that it's self-contradictory

Yes, you do. You started this whole discussion with a claim that energy defies definition. That claim is disproven by counter example, ie the example of several definitions. You may not like the definitions, but they nevertheless exist, disproving your claim. The only recourse you have is to invalidate the definitions. Definitions are not invalidated merely due to your dislike of the definition, but only due to self-contradiction. That is why, logically, for your original claim to hold you must prove that every definition of energy is self contradictory. If you do not (as you have not) then your claim is disproven by counterexample.

I explained that your definitions are definitions of forms of energy and not energy itself and as such nobody is saying that the forms of energy are not properly defined.

Take the Newtonian definition, energy is anything which gives a system the ability to do work. That doesn't limit the definition to any specific form or set of forms of energy. Anything which meets that definition is energy in Newtonian mechanics.

If you disagree with the above then perhaps I don't understand the distinction you are trying to draw between "forms of energy" and "energy itself." I.e. How would I distinguish a definition of "forms of energy" vs a definition of "energy itself" according to you?

None of the definitions mentioned in 43 seem to be limited to specific forms to me, and furthermore all of them claim to be definitions of energy rather than forms of energy. I suspect you may be making a philosophical distinction, not a scientific one. But either way it is up to you to justify and clarify that assertion.

 

[Popper]

Is Energy and Matter synonymous through the medium of light speed?

I responded to this post and demonstrated my point but I just realized that I didn't address this precise question yet. The answer is no. In the relation E₀ = mc² appears two quantities, E and m. These are logically different concepts: energy as being a constant and equaling the sum of all its various forms for a system and mass representing the inertia of a body, i.e. the property whereby it resists changes to momentum of the body (i.e.m = p/v). Since they are entirely two different concepts, energy cannot be said to be "equivalent to" mass. It should be pointed out that in his paper on the mass-energy relation Einstein did not say that mass is equivalent to energy but rather that the mass of a body is a measure of its energy content.

Dale - Sorry but as I said, I won't continue with that line of discussion since I've already proven my point in a manner which was satisfactory to me. I understand that you dsagree that it as satisfactory and that you claim that you proved me wrong by giving a definition. If I thought what you posted was correct I'd say so. I've already explained that you repeatedly keep missing the point over and over and over again and therefore you keep repeating your mistake. So I don't see a point in repeating myself yet again. I have no problem letting you believe whatever it is you wish to. But if you read Feynam like I suggested you might just understand, i.e. that the various forms of energy are well defined. It's energy itself that is not defined. But I won't beat this horse anymore. It's already dead as a doorknob. :)

 

[HomogenousCow]

Oh look, another thread where half the forum regulars come to state the same thing and then argue amongst each other.

Why don't we just agree that energy is just the scalar generated by time invariance of the lagrangian.

 

[HomogenousCow]

I don't need to prove that it's self-contradictory. I explained that your definitions are definitions of forms of energy and not energy itself and as such nobody is saying that the forms of energy are not properly defined. I'v stated that many times so far and I don't see the point in repeating it again after this post.

The reason I don't think it's worth my time is because I've already explained why each attempt at a definition that you gave is wrong and the only response you gave was to state that I made an error. So why shouuld I continut. I hold that what I said is correct and you hold that its not. There's nowhere to do when a discussion gets to that point.

I'll give you an example to illustrate my point. The assertion is that the term energy has no proper definition but that the various forms of energy are very well defined. Your response was to post an example of a form of energy and then assert that you proved me wrong because you posted a definition of energy. The problem being that you didn't post a definition of energy itself but merely a form of energy. E.g. mechanical energy or EM energy etc. I then explained that the energy function isn't even always the energy in general. You also asserted that energy is the ability to do work. I then explained that such a statement is far too vauge to properly define energy. In fact other authors use that exact definition to define potential energy. I gave you an example of how it being vauge leands you nowhere. E.g. suppose you assert that energy is the capacity to do work. That tells you nothing about how to quantify it.

You can go through the entire therad and see my responses to all your assertions. Just becaus you state that I was wrong in no way shape or form makes that true in my opinion.

Si that's why I'm going to agree to disagree with you and then not continue on in this thread. It's not as if I'm the only physicist who holds this opinion and we don't hold it for no good reason either. But I won't continue to debateit merely because you state that you're correct and I'm wrong. That's not how a pleasant discssion works.

Plus there are many other questions on the forum to address. Nobody in this thead has any unanswered quesions so I'm going to help in other threads.

Thanks for the conversation. I enjoyed it.

Ps - In many ways, what you've referred to above was mechanical energy and the conservation of mechanical energy theorem whereas I've been referring to energy and the law of conservation of energy. Of course in continuum mechanics a similar thing holds, i.e. EM energy and the zero covariant divergence of the stress-energy momentum tensor - i.e. conserved currents if memory servers (warning: my memory is like a sieve)

That line of reasoning is flawed, what is a number? What is a color?
I don't really like how energy is taught as some kind of physical currency traded for stuff to happen, statements like "the body uses food for energy' is patent nonsense.

 

[PeterDonis]

Why don't we just agree that energy is just the scalar generated by time invariance of the lagrangian.

Probably because (a) that's not the only possible definition of energy, and (b) that definition only applies if the Lagrangian (or the metric, which is more appropriate since we're talking about GR here) *is* in fact invariant under time translations. Plenty of metrics (and Lagrangians, for that matter) aren't.

 

[WannabeNewton]

Why don't we just agree that energy is just the scalar generated by time invariance of the lagrangian.

This is an extremely restrictive definition of energy. There are many types of energies that don't come out of this. Even total mechanical energy doesn't always come out of this so that is enough to show that taking the above as the definition would be too restrictive.

 

[Dale]

I've already proven my point in a manner which was satisfactory to me. I understand that you dsagree that it as satisfactory and that you claim that you proved me wrong by giving a definition.

Your satisfaction or dissatisfaction is an irrelevant emotion. Your position (that energy defies definition) is logically disproven, by counterexample.

the various forms of energy are well defined. It's energy itself that is not defined.

And yet even the "work" definition which is the brunt of your ridicule does not apparently limit itself to any particular forms of energy, contrary to your claim. You have not even been able to explain this claim, let alone justify it.

I certainly understand your desire to stop arguing in favor of an untenable position, but don't kid yourself about the outcome here.

 

[Popper]

Oh look, another thread where half the forum regulars come to state the same thing and then argue amongst each other.

I'm not a forum regular nor do I ever intend on being one.

Why don't we just agree that energy is just the scalar generated by time invariance of the lagrangian.

This was already explained above and rexplained when DaleSpam was unable to understanding it the second and third time.

Reasons why that attempt is flawed
1) Not all systems can be described by a Lagrangian
2) The quantity you speak of is known as, among many other names, Jacobi's integral and given the letter h. h does not always equal the energy. It may even happen that h is constant but not the energy
3) In those instances where h is the energy of the system then its only mechanical energy, which is a well defined quantity. Energy, on the other hand, comes if many other forms.

We know that energy comes in many different forms besides mechanical energy. Each form is well defined. However we cannot use h for those forms since they're not forms of mechanical energy. Dale has been unable to understand that point. For some reason, which remains a mystery to me, he seems to think that merely stating that I'm wrong is not by itself a proof that I'm wrong. Nope. Neither is it a reason to convince me to respond to him again in this thread.

That line of reasoning is flawed, what is a number? What is a color?

And you believe that merely making a claim that my reasoning is flawed is an actual logical line of reasoning to prove your point? The answer is, no, it's not a logical line of reasoning.

I don't really like how energy is taught as some kind of physical currency traded for stuff to happen, statements like "the body uses food for energy' is patent nonsense.

Then you don't understand the concept of energy that well.

Probably because (a) that's not the only possible definition of energy, and (b) that definition only applies if the Lagrangian (or the metric, which is more appropriate since we're talking about GR here) *is* in fact invariant under time translations. Plenty of metrics (and Lagrangians, for that matter) aren't.

Ah! Music to my hears! Bravo, sir. Bravo!

One might be tempted to define energy as the sum of all forms of energy but one would be making a serious error in doing so.

I suggest that you take a look in The Feynman Lectures on Physics, Vol I, Feynman, Leighton, and Sands, Addison Wesley, (1963)(1989). pages 4- to 4-2

It is important to realize that in physics today, we have no knowledge of what energy is. We do not have a picture that energy comes in little blobs of a definite amount. It is not that way. However, there are formulas for calculating some numerical quantity, and we add it all together it gives “28” - always the same number. It is an abstract thing in that it does not tell us the mechanism or the reasons for the various formulas.

DaleSpam - Please understand that I can't be insulted into responding to your attempts at an argument nor can I be coerced into posting just because you make a claim about how right you are. Perhaps a 12 y.o. might fall for that nonsense but not I.

 

[slickjunt]

Oh look, another thread where half the forum regulars come to state the same thing and then argue amongst each other.

It seems to me a majority of the 'intelectual' discussion on this thread has been centered around wordsmithing and arguments over defenitions.
Could a logical stance be taken on the matter. Say energy in any system is motion relative to another object. And the absence of energy would then simply be the absence of motion?

 

[WannabeNewton]

Could a logical stance be taken on the matter. Say energy in any system is motion relative to another object. And the absence of energy would then simply be the absence of motion?

This is again far too simplistic. You can have energy when there is no motion (the rest energy of a particle for example). Also, how would this definition even make sense when defining the energy of an entire asymptotically flat space-time?

 

[Popper]

It seems to me a majority of the 'intelectual' discussion on this thread has been centered around wordsmithing and arguments over defenitions.
Could a logical stance be taken on the matter. Say energy in any system is motion relative to another object. And the absence of energy would then simply be the absence of motion?

The answer has already been given in a numerous previous posts. I.e. take a look in The Feynman Lectures on Physics, Vol I, Feynman, Leighton, and Sands, Addison Wesley, (1963)(1989). pages 4- to 4-2

It is important to realize that in physics today, we have no knowledge of what energy is. We do not have a picture that energy comes in little blobs of a definite amount. It is not that way. However, there are formulas for calculating some numerical quantity, and we add it all together it gives “28” - always the same number. It is an abstract thing in that it does not tell us the mechanism or the reasons for the various formulas.

The same thing is given in A.P. French's text Newtonian Mechanics as well as in the thermal physics text An Introduction to Thermal Physics, by Daniel V. Schroeder.

Please ignore all the other noise in this thread to the contrary. It's wrong for the reasons stated in previous posts.

 

[HomogenousCow]

Huh what, I thought the jacobi integral was that thing in Newtonian three body problems.

I probably do not know as much as you do popper, but from the texts which I have read the hamiltonian definition of energy seems reasonable to me, at least in the theory of relativistic charged particles.

Also, how can one argue that something is not energy, if energy "defies definition"?
I always just thought energy was a time-invariant scalar generated by symmetry in the lagrangian.In what situations does this fail?

 

[WannabeNewton]

I always just thought energy was a time-invariant scalar generated by symmetry in the lagrangian.
In what situations does this fail?

As noted, this is a very specialized definition of energy. It will fail to give you even the total mechanical energy of a system in general. The simplest example is that of a hoop rotating about the z-axis with constant angular velocity and a bead sliding without friction on the hoop. The Lagrangian will be time translation invariant and will thus lead to a conserved Hamiltonian but the total mechanical energy is not conserved; in this system the Hamiltonian energy is not even equal to the total mechanical energy.

The point is that there are many different types of energies so you suggesting that the energy be unequivocally defined as the quantity that comes out of time invariant Lagrangians is nonsensical I'm afraid. Furthermore, energy makes sense for systems where a Lagrangian can't even be defined.

 

[HomogenousCow]

As noted, this is a very specialized definition of energy. It will fail to give you even the total mechanical energy of a system in general. The simplest example is that of a hoop rotating about the z-axis with constant angular velocity and a bead sliding without friction on the hoop. The Lagrangian will be time translation invariant and will thus lead to a conserved Hamiltonian but the total mechanical energy is not conserved; in this system the Hamiltonian energy is not even equal to the total mechanical energy.

The point is that there are many different types of energies so you suggesting that the energy be unequivocally defined as the quantity that comes out of time invariant Lagrangians is nonsensical I'm afraid. Furthermore, energy makes sense for systems where a Lagrangian can't even be defined.

Hmm I see.
However, in such a case the mechanism by which the bead is constrained is not specified, the lagrangian is missing the "other part". In SR when the particle lagrangian is coupled with the field lagrangian the hamiltonian does come out to be right, yes?

 

[WannabeNewton]

Do you have a reference as to whether the Lagrangian for a particle interacting with any classical field propagating on background flat Minkowski space-time always lends to a Hamiltonian that is the total energy? It is certainly true in the specific case of the electromagnetic field (Goldstein Edition 3 page 342). Regardless, I do not see how this justifies using this narrow definition of energy as an all encompassing definition of said quantity.

 

[Dale]

I'm not a forum regular nor do I ever intend on being one.

That's too bad.

This was already explained above and rexplained when DaleSpam was unable to understanding it the second and third time.

Reasons why that attempt is flawed
1) Not all systems can be described by a Lagrangian

I understood that, and explicitly mentioned that as a reason why the definition of energy was theory-specific and even formulation-specific. In turn, this was, I think, what Feynman was describing.

2) The quantity you speak of is known as, among many other names, Jacobi's integral and given the letter h. h does not always equal the energy. It may even happen that h is constant but not the energy

The Jacobi integral is much more limited in scope (three-body gravity) than the time-symmetry of the Lagrangian. Your arguments against the Jacobi integral are a straw man fallacy since I never listed that as even a potential definition.

3) In those instances where h is the energy of the system then its only mechanical energy, which is a well defined quantity. Energy, on the other hand, comes if many other forms.

The Lagrangian for an isolated system of EM is also time invariant, leading to a conserved EM energy also, not just a conserved mechanical energy. So while your criticism may be valid for your strawman Jacobi, it is not valid for the actual Noether definition of energy.

I suggest that you take a look in The Feynman Lectures on Physics, Vol I, Feynman, Leighton, and Sands, Addison Wesley, (1963)(1989). pages 4- to 4-2

Where he never makes your specific claim that "energy defies definition".

DaleSpam - Please understand that I can't be insulted into responding to your attempts at an argument nor can I be coerced into posting just because you make a claim about how right you are.

Please . I never insulted you. Your argument is disproven and for some reason you choose to mention how satisfied your responses made you feel. That is irrelevant and I merely pointed it out.

You repeatedly dismiss the "work" definition of energy, but the only actual argument you provided against it was that momentum could qualify, which I rebutted and apparently you agreed with the rebuttal since you didn't even attempt to refute it and didn't bring it up again.

Your only remaining argument is your claim that the definitions define "forms of energy" rather than "energy itself". That doesn't seem correct at first glance since none of the definitions of energy claim to be definitions of "forms of energy" nor do they seem limited to any specific set of known forms of energy, but you haven't been able to clarify your meaning well enough to tell if this final argument has any merit.

In any case, I have done far more than merely claim how right I am. I have disproven your primary position by counterexample and rebutted the bulk of your arguments on logical grounds. I do understand your unwillingness to proceed, but you have not been mistreated, nor insulted, nor ignored, only refuted.

 

[Dale]

The simplest example is that of a hoop rotating about the z-axis with constant angular velocity and a bead sliding without friction on the hoop. The Lagrangian will be time translation invariant and will thus lead to a conserved Hamiltonian but the total mechanical energy is not conserved; in this system the Hamiltonian energy is not even equal to the total mechanical energy.

I am interested in this example. So what is the conserved Noether charge, if not energy?

Without working it out it does seem that mechanical energy should be conserved since the hoop rotates at constant angular velocity and the bead experiences no friction so it should also rotate with constant angular velocity. At first glance there appears to be no mechanical potential energy and no change in kinetic energy, so what is not conserved?

The point is that there are many different types of energies so you suggesting that the energy be unequivocally defined as the quantity that comes out of time invariant Lagrangians is nonsensical I'm afraid. Furthermore, energy makes sense for systems where a Lagrangian can't even be defined.

Agreed.

 

[WannabeNewton]

Mornin' DaleSpam! Consider the setup again: we have a hoop of radius $R$ and mass $M$ rotating about the $z$-axis with prescribed constant angular velocity $\Omega$ and we have a bead of mass $m$ sliding without friction along the hoop. Fix the origin of the coordinate system to the center of the hoop and let $\varphi$ be the angle that the position vector to the bead (the vector from the origin to the bead) makes with the rotation axis of the hoop. The potential energy of the bead is then $U = -mgR\cos\varphi$ and the kinetic energy is $T = \frac{1}{2}mR^{2}\dot{\varphi}^{2} + \frac{1}{2}m(R\sin\varphi)^{2}\Omega^{2}$. The first term in $T$ is just the kinetic energy of the bead coming from its velocity tangential to the hoop. Since the hoop is itself rotating with angular velocity $\Omega\hat{z}$, the bead acquires an additional velocity tangential to the rotation of the hoop with magnitude $v = \left \| \mathbf{r} \times \mathbf{\Omega}\right \| = (R\sin\varphi) \Omega$ which gives rise to the second term in $T$.

Our Lagrangian is then $L = \frac{1}{2}mR^{2}\dot{\varphi}^{2} + \frac{1}{2}m(R\sin\varphi)^{2}\Omega^{2} + mgR\cos\varphi$. Notice that $\frac{\partial L}{\partial t} = 0$ therefore the Hamiltonian $H = \sum p_{i}\dot{q_{i}} - L = \text{const.}$ and is given by $H = \frac{1}{2}mR^{2}\dot{\varphi}^{2}- \frac{1}{2}m(R\sin\varphi)^{2}\Omega^{2} - mgR\cos\varphi$ whereas the total mechanical energy is given by $E = \frac{1}{2}mR^{2}\dot{\varphi}^{2}+ \frac{1}{2}m(R\sin\varphi)^{2}\Omega^{2} - mgR\cos\varphi$ so the total energy is not equal to the Hamiltonian in this case. The reason $E$ is not conserved is because whatever is keeping the hoop rotating at a prescribed constant speed must be doing work on the system.

If you however let the hoop rotate freely, so that its azimuthal position is no longer prescribed but rather a generalized coordinate, then you will find that $E$ for this new system is conserved and that it will be equal to the Hamiltonian $H$.

 

[Dale]

I will have to look into the math in detail tomorrow as I have a full day of work and an assignment due. However, I had a few notational quenstions.

Mornin' DaleSpam! Consider the setup again: we have a hoop of radius $R$ and mass $M$ rotating about the $z$-axis

Is the z axis parallel to gravity or perpendicular?

with prescribed constant angular velocity $\Omega$ and we have a bead of mass $m$ sliding without friction along the hoop. Fix the origin of the coordinate system to the center of the hoop and let $\varphi$ be the angle that the position vector to the bead (the vector from the origin to the bead) makes with the rotation axis of the hoop.

How can $\varphi$ ever be anything other than 90º if the bead is constrained to be along the hoop? I would think that you would want an angular variable specifying the angle around the hoop.

The potential energy of the bead is then $U = -mgR\cos\varphi$

OK, from your initial description I didn't realize that you were doing this in the presence of gravity. This makes more sense. This potential makes sense for a horizontal z and for $\varphi$ the angle from the horizontal perpendicular to the axis of rotation.

and the kinetic energy is $T = \frac{1}{2}mR^{2}\dot{\varphi}^{2} + \frac{1}{2}m(R\sin\varphi)^{2}\Omega^{2}$. The first term in $T$ is just the kinetic energy of the bead coming from its velocity tangential to the hoop. Since the hoop is itself rotating with angular velocity $\Omega\hat{z}$, the bead acquires an additional velocity tangential to the rotation of the hoop with magnitude $v = \left \| \mathbf{r} \times \mathbf{\Omega}\right \| = (R\sin\varphi) \Omega$ which gives rise to the second term in $T$.

I don't see how the rotation of the hoop gives any kinetic energy to the bead. I would think that the rotation of the hoop only gives a constant KE to the hoop itself.

 

[WannabeNewton]

Here is the picture of the system DaleSpam: http://s21.postimg.org/vz4nttwqu/IMG_0568.jpg

The extra kinetic energy term arises because the bead is also swirling around with the hoop, since it is constrained to stay on the hoop as the hoop rotates. So on top of the kinetic energy the bead has from sliding around on the hoop, which would be there even if the hoop wasn't rotating, it now also has an extra kinetic energy term due to the hoop actually swirling around. You are correct that the hoop itself also has a kinetic energy term but I abused the Lagrangian a little by excluding this since it will drop out of the equations of motion anyways. You can include the kinetic energy of the hoop itself if you want, the discrepancy between the Hamiltonian vs Total Energy won't change.

 

[VantagePoint72]

This is a neat example, WannabeNewton, but I don't think it supports your larger point that the Noether definition of energy is limited. I think I agree with the point, I just don't think this supports it.

As you pointed out, the reason for the failure of the Hamiltonian to agree with the mechanical energy in this problem is that the condition that the hoop rotates with constant angular frequency effectively "smuggles in" an outside force. Hence, all it really demonstrates is that if you're not very careful, it's possible to construct a system in which work is being done by/on the environment without that being immediately obvious. That the Hamiltonian-as-energy definition fails in such cases in not surprising.

If you construct a Lagrangian for the total isolated system—the bead, the hoop, and whatever is interacting with the hoop to make it rotate with constant angular velocity—then, as far as I can tell, the Hamiltonian should once again agree with the total energy. So, I take this example more as a cautionary tale about what system constraints can subtly imply for outside forces then a genuine refutation of the Hamiltonian definition of mechanical energy.

 

[Dale]

Here is the picture of the system DaleSpam: http://s21.postimg.org/vz4nttwqu/IMG_0568.jpg

D'oh! Thanks for the picture. I was totally misunderstanding your intended scenario. I thought that the hoop was rotating about its own axis. You mean the hoop rotating perpendicular to its own axis, so the normal to the plane of the hoop is rotating in a horizontal plane.

OK, tomorrow I will look at the details again.

 

[PeterDonis]

In what situations does this fail?

It fails for any system that is not time translation invariant. For example, the universe as a whole is not; it's expanding. So there's no time translation symmetry, hence no invariant scalar derived from it.

 

[WannabeNewton]

If you construct a Lagrangian for the total isolated system—the bead, the hoop, and whatever is interacting with the hoop to make it rotate with constant angular velocity—then, as far as I can tell, the Hamiltonian should once again agree with the total energy. So, I take this example more as a cautionary tale about what system constraints can subtly imply for outside forces then a genuine refutation of the Hamiltonian definition of mechanical energy.

But how would you take into account the potentially non-conservative external forces doing work on the system in the Lagrangian if you were to include them in the system? There are ways of taking into account non-conservative forces in the Euler-Lagrange equations themselves using virtual work but how would you incorporate them into the Lagrangian itself?

Personally, I would say the concept of energy is much better codified in its various forms within the field of thermodynamics.

Interestingly, and unrelated to my previous comment, one can define notions of momentum and energy at spatial infinity for an entire asymptotically flat space-time using the Hamiltonian formulation of general relativity. This is called the ADM energy-momentum.

EDIT: I found the newly revised version of the original paper by Arnowitt et al. yey! Here it is: http://arxiv.org/pdf/gr-qc/0405109v1.pdf