# What is the rigorous quantitative definition of the concept of "Energy"?

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First of all I acknowledge you that I posted this Question on many other forums and Q&A Websites. So don't be surprised if you found my question somewhere else.

I bet when the experts saw the title, many of them said: "...again another dumb guy seeking answers to useless questions...". But believe me I have a point.
Let me say I'm not worried if our conversation lead beyond conventional physics and violates or disrupts our standard classical epistemological system of physical concepts. What I want to do is to mathematically and physically clarify the definition of an important concept in physics.
Let's get started:
"What is energy?"
A High school teacher: Huh it's simple: "The Ability of a system to do work on another system".
Cool. Then "What's work done by a gravitational field?"
Same teacher: It is called "Gravitational potential energy".
Then you mean "Energy" is defined by "Work" and "Work" is defined by "Energy". So it leads to a paradox of "Circular definition".
The teacher: Oo
Let us even go further and accept this definition. What about a system reached its maximum entropy(in terms of thermodynamics being in "Heat death" state). Can it still do work? The answer is of course no. But still the system contains energy.
So the above definition is already busted.
Another famous (and more acceptable) definition is "any quantity that is constant when laws of physics are invariant under time translations". That's right but this is a consequence of Noether's theorem which uses the concept of "Lagrangian" and "Hamiltonian" to do this. Two quantities that are already using the concept of energy in their definitions. So again it gets circular.
Beside we can also define "Momentum" as "any quantity that is constant when laws of physics are invariant under space translations". But we don't. First of all we quantitatively define momentum as $p := mv$, then we deduce its conservation as a natural result of Noether's theorem or even when the scope is outside analytical mechanics, we consider it a principle or axiom. In both cases we first "Rigorously" and "Quantitatively" defined a concept then made a proposition using this concept.
Now this is my point and this is what I'm seeking: "What is a quantitative definition of energy" that is both rigorous and comprehensive. I mean I will be satisfied If and only if someone says:
$E := something$
Yes, I want a "Defining equation" for Energy.
I hope I wasn't tiresome or stupid for you. But believe me I think it's very important. Because energy is one of the most significant concepts in physics but we haven't any rigorous definition of it yet. By the way, we can even define other forms and other types of energy using a universal general definition of it. I hope you understand the importance of this and give me a satisfying answer.
Again I repeat I don't fear to go further than our standard conceptual framework of physics. Maybe it's time to redesign our epistemological conventions.
P.S. Somewhere I saw someone said it can be defined as the "Negative time derivative of Action" which means:
$E := -\frac{dS}{dt}$
Where $S$ is the action and $t$ is time. However since action is a concept based on Lagrangian and is already dependent on the concept of energy, I think, again it won't help.
P.P.S Some people say consider energy as a "Primitive notion" or an "Undefinable Concept". But it's not a good idea too. First because it's not a "SI base quantity" from which they couldn't be defined by any previous well defined quantity and since Energy haven't a base dimension(the dimension is $[ML^{2}T^{-2}]$) so it couldn't be a primitive notion. Second we often assume a quantity primitive or undefinable, when it's very trivial that it's almost understandable to everyone. At least to me the concept of energy is too vague and misty that when I work with it, I don't know what I'm actually doing.
P.P.P.S And also please don't tell me "Energy is another form of mass". I assume we are talking about non-relativistic Newtonian mechanics and also don't forget the concept of energy has been used long before appearance of "Relativity theory".

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Energy is the conserved current from the time invariance of the Lagrangian.

One problem with your objection to the circular nature of the definition is that ultimately all definitions are circular if you have no postulates or axioms. You have to start somewhere, with some things as given.

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Energy is the conserved current from the time invariance of the Lagrangian.

One problem with your objection to the circular nature of the definition is that ultimately all definitions are circular if you have no postulates or axioms. You have to start somewhere, with some things as given.

So if I don't want to take "Energy" as a primitive concept, Is there any "quantitative definition" of it? like:

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The definition of energy is really a combination of things- largely defined in order to maintain "conservation of energy". The simplest definition of energy is that of "work" : force times distance. Of course, applying force to something may cause it to move so we add "kinetic energy"- 1/2 mass times velocity squared. Of course there is friction which will slow an object, causing heat, so we add heat energy. "Energy" is really a "book keeping" device!

Edited by Country Boy

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The definition of energy is really a combination of things- largely defined in order to maintain "conservation of energy". The simplest definition of energy is that of "work" : force times distance. Of course, applying force to something may cause it to move so we add "kinetic energy"- 1/2 mass times velocity squared. Of course there is friction which will slow an object, causing heat, so we add heat energy. "Energy" is really a "book keeping" device!

So really doesn't someone aggregated these definitions to make one simple comprehensive definition? It seems really bad to have such a vague concept in a so formal field such as Physics.

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One problem with your objection to the circular nature of the definition is that ultimately all definitions are circular if you have no postulates or axioms. You have to start somewhere, with some things as given.

Very true.

But once I was nearly killed for saying something like that...

Hamed.Begloo, a circular definition is not a paradox. Every definition has to be circular. Of course you can define something in terms that everybody thinks are inherently clear, and then it seems as if you have a non-circular definition. But if you want to build up a rigorous conceptual system, then, as Swansont says, you have to take some definitions as axioms. The only way to get out this circle is to refer to something you can do, e.g. an experiment, or an observation, and so leave the domain of language. With the empirical meaning of your basic concepts, you give empirical meaning to the whole conceptual building that is derived from these basic concepts.

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So really doesn't someone aggregated these definitions to make one simple comprehensive definition? It seems really bad to have such a vague concept in a so formal field such as Physics.

I'm not sure that it matters if the concept is vague, or even undefined. The formality is in the mathematics. The "thing" we call energy appears consistently in various equations as a value that is conserved. As far as physics is concerned, I'm not sure anything more than that is required.

Attempting to pin down what this "thing" actually is, is philosophy not physics.

Edited by Strange

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"Energy" is really a "book keeping" device!

Very much so. We notice that some value stays the same under a set of conditions, so it's a useful quantity to know, when we can do so. And then we try and expand to new kinds of systems, we make sure we apply the concept consistently so it can continue to be a useful quantity.

So really doesn't someone aggregated these definitions to make one simple comprehensive definition? It seems really bad to have such a vague concept in a so formal field such as Physics.

Energy as the capacity to do work is usually the first definition that is offered up, because we usually start with Newton's laws so we know about forces, and then learn about work, and see that Work shows up as KE. We learn about conservative forces and how we can apply the concept of work there, too, as potential energy.

I don't see it as a vague concept at all. I don't have the problem you see with the relationship between work and energy. You can define mechanical work in terms of forces and displacements without referring to energy at all. Equating it with energy is an additional tool one can use to understand and predict how these systems behave.

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Energy is an intrinsic property of all bodies regardless of their mass.

In other words, the energy that is within a body can be accessed or cannot be, in the form of work.

Suppose there is a system which contains potential energy because it is near to a gravitational source. This energy can be utilised to perform useful work, by changing its state of motion, i.e., by moving the body we can convert its potential energy into kinetic energy.

Mathematically energy can be defined as the force ( may be abstract ) times the displacement.

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Hamed,

Looking at your posts here, I am unsure whether you are asking what we think energy is or whether you want to tell us what you think energy refers to.

I assume you are referring to non relativistic, non quantum energy since you have posted this in classical physics.

Here a formal definition would be:

That physical quantity which possesses the dimensions of mass times the square of distance divided by the square of time or ML2T-2

That also covers subjects, not yet mentioned in this thread, such as virtual work and virtual energy.

Edited by studiot

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In quantum physics we have collisions, interactions, scattering of particles.

One particle hits other, giving its entire or part of energy, and accelerating second particle.

Photon is absorbed by first particle, particle is accelerated, it hits second particle, and new photon appears (this way we can learn collision has happened, because photon is then absorbed by our eye/detector)

Newly created photon can have smaller energy than originally absorbed one had initially.

In such case the rest of energy remain as property of first particle.

It's still accelerated, but to smaller velocity. And can hit, scatter with other particles.

From one gamma photon with f.e. 116 keV there could be created this way 50000 green photons with 2.32 eV each.

And then further to even more quantity of photons with even lower energies each, in infrared and then microwave range.

Suppose so we're accelerating electrons in electric field. U = 1000 Volts.

Maximum kinetic energy of single electron will be up to 1000 eV.

After hitting stationary matter, this kinetic energy will be lost by electron, it will be decelerated, and up to 1000 eV photon could be created.

While passing through cloud of electrons in this stationary matter,

electron will ionize matter: give away it's entire or part of kinetic energy, and accelerating these electrons, so they will be ejected from their atoms in the all directions.

Once they are attracted back by atom, they have to emit photons (spectral lines), and these photons are detected by our eye/detector, thus we see event.

Energy could be imagined/modeled as fluid, which is transferred from one quantum object (particle) to other quantum object, or their groups.

Particles are carriers of energy.

ps. Are you humanist.. ?

Edited by Sensei

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Hamed.Begloo, a circular definition is not a paradox. Every definition has to be circular. Of course you can define something in terms that everybody thinks are inherently clear, and then it seems as if you have a non-circular definition. But if you want to build up a rigorous conceptual system, then, as Swansont says, you have to take some definitions as axioms. The only way to get out this circle is to refer to something you can do, e.g. an experiment, or an observation, and so leave the domain of language. With the empirical meaning of your basic concepts, you give empirical meaning to the whole conceptual building that is derived from these basic concepts.

So what are those Axioms(or primitive concepts). Is energy itself primitive. I don't think so.

I'm not sure that it matters if the concept is vague, or even undefined. The formality is in the mathematics. The "thing" we call energy appears consistently in various equations as a value that is conserved. As far as physics is concerned, I'm not sure anything more than that is required.

Attempting to pin down what this "thing" actually is, is philosophy not physics.

There are many other "things" that are completely well defined like "Momentum", "Velocity", "Acceleration", "Force", etc. Even those which are undefined like "Mass", "Charge", etc could reasonably be imagined as what they really are. But energy... that is really hard to imagine.

Mathematically energy can be defined as the force ( may be abstract ) times the displacement.

Your definition to me seems more like the definition of "Work".

Energy as the capacity to do work is usually the first definition that is offered up, because we usually start with Newton's laws so we know about forces, and then learn about work, and see that Work shows up as KE. We learn about conservative forces and how we can apply the concept of work there, too, as potential energy.

I don't see it as a vague concept at all. I don't have the problem you see with the relationship between work and energy. You can define mechanical work in terms of forces and displacements without referring to energy at all. Equating it with energy is an additional tool one can use to understand and predict how these systems behave.

I know, but I think despite all of this energy must have a "quantitative definition"

Hamed,

Looking at your posts here, I am unsure whether you are asking what we think energy is or whether you want to tell us what you think energy refers to.

I assume you are referring to non relativistic, non quantum energy since you have posted this in classical physics.

Here a formal definition would be:

That physical quantity which possesses the dimensions of mass times the square of distance divided by the square of time or ML2T-2

That also covers subjects, not yet mentioned in this thread, such as virtual work and virtual energy.

I just want to clarify what energy is for myself(in Classical physics of course) and I think it's only possible by defining it quantitatively

By the way with your definition I can say with multiplications of many different quantities I can define energy.

Energy could be imagined/modeled as fluid, which is transferred from one quantum object (particle) to other quantum object, or their groups.

Particles are carriers of energy.

To my knowledge Energy is a quantity which can not be localized. It's like a wave and is distributed all over system. But you're considering a locality for it.

ps. Are you humanist.. ?

No, I'm mostly naturalist. and I also think (nearly) everything must be describable by nature.

Guys please see this definition(link below) offered by a person named "Marcel":

http://physics.stackexchange.com/a/288811/126696

I wanted something like this. I mean a definition which literally defines something. But still, any other suggestions would be appreciated.

Edited by Hamed.Begloo

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I know, but I think despite all of this energy must have a "quantitative definition"

We have kinetic energy (which can be translational or rotational), electrostatic and gravitational potential energy, vibrational energy, photon energy. Getting into thermodynamics we have all of the terms you find on the Gibbs free energy (U, pV and TS). What constitutes work in a thermodynamics situation might not be considered work in some other context. And more.

How are you going to quantify the definition of energy when how you do so depends on the system you are investigating?

Energy could be imagined/modeled as fluid, which is transferred from one quantum object (particle) to other quantum object, or their groups.

Particles are carriers of energy.

It could be (possibly), but that would be a new paradigm and not part of current physics. There is no substance that we've discovered that would act this way. As it stands, energy is a property of particles and systems.

Guys please see this definition(link below) offered by a person named "Marcel":

http://physics.stackexchange.com/a/288811/126696

I wanted something like this. I mean a definition which literally defines something. But still, any other suggestions would be appreciated.

Which is something I described in post #8. It's the work-Energy theorem. I'm surprised you had not yet run across that.

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There was another definition of energy which has fallen out of favour over the last 50 years or so.

'Everything that is not matter is energy'

Since it is now apparent that matter is at least overwhelmingly, if not entirely, a highly structured manifestation of energy itself, then that old definition simplifies to:

'Energy is Everything'

As axioms go, it's a fairly comprehensive one. Whether or not one agrees with it depends upon whether you can answer a slightly different question.

'What is NOT energy?'

Edited by sethoflagos

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There are many other "things" that are completely well defined like "Momentum", "Velocity", "Acceleration", "Force", etc. Even those which are undefined like "Mass", "Charge", etc could reasonably be imagined as what they really are. But energy... that is really hard to imagine.

So what? Why is what you find hard to imagine relevant?

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Energy could be imagined/modeled as fluid, which is transferred from one quantum object (particle) to other quantum object, or their groups.

Particles are carriers of energy.

To my knowledge Energy is a quantity which can not be localized. It's like a wave and is distributed all over system. But you're considering a locality for it.

Particles during their collisions and interactions are scattered in the all directions randomly.

You don't know where is particle until it hits you, being decelerated, and give away its energy.

After that it's no longer "excited". System under observation has changed.

The smaller system under observation (like one unstable isotope which is decaying to daughter isotope and alpha,beta,gamma f.e.), the more devastating result to system.

It's often called "collapse of wave-function" while analyze of what happens with electrons around nucleus.

From all random possibilities they can be, there is appearing one definite state.

Particle accelerators have vacuum chambers (spherical,cylindrical) with detectors of photons around it's inner surface.

There is interaction of particles in the center of sphere f.e. annihilation, or decay of neutral pion meson.

And two gamma photons go in two opposite directions of chamber. It has thousands of detectors on inner surface.

One photon hits at one spot at time t0, second photon hits at opposite spot at time t1.

From where they hit, with what delays between them, there is calculated, where used to be place of annihilation or decay in the center of sphere.

Edited by Sensei

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There was another definition of energy which has fallen out of favour over the last 50 years or so.

'Everything that is not matter is energy'

Since it is now apparent that matter is at least overwhelmingly, if not entirely, a highly structured manifestation of energy itself, then that old definition simplifies to:

'Energy is Everything'

As axioms go, it's a fairly comprehensive one. Whether or not one agrees with it depends upon whether you can answer a slightly different question.

'What is NOT energy?'

Spin is not energy. Height is not energy. Color is not energy. Basically any property that is not mass is not energy.

I imagine that it fell out of favor (if it was ever actually in favor) is that it's just wrong on a fundamental level. Energy is a property, while matter is a classification. Photons are not matter, but they aren't energy, either. They have energy.

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So what are those Axioms(or primitive concepts). Is energy itself primitive. I don't think so.

If definitions are circular, then none of them is primitive. Physicists take, I assume, those as basic axioms that are most easily and/or exact measurable in a laboratory. But if one is really only interested in the theoretical framework of physics, then the only thing one must make sure of is that from the axioms the whole can be built up. There is no a priori reason to take e.g. force as the most fundamental concept, instead of energy or mass (or even acceleration). The only thing that counts is that you can derive the other concepts from them.

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So what are those Axioms(or primitive concepts). Is energy itself primitive. I don't think so.

There are many other "things" that are completely well defined like "Momentum", "Velocity", "Acceleration", "Force", etc. Even those which are undefined like "Mass", "Charge", etc could reasonably be imagined as what they really are. But energy... that is really hard to imagine.

Your definition to me seems more like the definition of "Work".

I know, but I think despite all of this energy must have a "quantitative definition"

I just want to clarify what energy is for myself(in Classical physics of course) and I think it's only possible by defining it quantitatively

By the way with your definition I can say with multiplications of many different quantities I can define energy.

To my knowledge Energy is a quantity which can not be localized. It's like a wave and is distributed all over system. But you're considering a locality for it.

No, I'm mostly naturalist. and I also think (nearly) everything must be describable by nature.

Guys please see this definition(link below) offered by a person named "Marcel":

http://physics.stackexchange.com/a/288811/126696

I wanted something like this. I mean a definition which literally defines something. But still, any other suggestions would be appreciated.

I'm sorry but I get the impression you posted this because you just want to cherry pick phrases from everybody and then snipe at them.

You posted this in classical physics, but seem to want to discuss quantum mechanics.

Yes there is more than one way to skin a cat, but you posted this in classical physics where there is an established structure, based on certain fundamental quantities, from which all others are derived.

So whilst you could develop an alternative set of fundamental quantities and use those, that would loose the advantages already endowed in the conventional set.

These are, minimum set, minimum complexity, maximum coverage, maximum flexibility.

One of the issues with using Force as a fundamental is the relationship Force x Distance does not always give you energy.

Classically it is not true to say the 'energy is always distributed or non localised'.

If you are thinking of Heisenberg, this is quantum and the correct pairing is energy and time. For position, the pairing is with momentum.

I gave you a sound quantitative definition of energy in terms of the fundamental quantities of mechanics.

What more do you want?

Edited by studiot

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I think the sort of thing all physicists take as axiomatic are the consistency of mathematical results and reasoning, the existence of empirical evidence and the possibility of obtaining same, the objectivity (or at least inter-subjectivity in a weak sense) of experiment, and perhaps the theoretical repeatability of results (if not the actual). Within sub-disciplines more concrete things would be taken as axiomatic - the invariance of the speed of light, the cosmological principle, the use of statistics and probability, and the rules of conservation/symmetry. The more applied the area of research the more things would be taken as given

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Spin is not energy. Height is not energy. Color is not energy. Basically any property that is not mass is not energy.

I imagine that it fell out of favor (if it was ever actually in favor) is that it's just wrong on a fundamental level. Energy is a property, while matter is a classification. Photons are not matter, but they aren't energy, either. They have energy.

Not much left of a photon after an electron has absorbed its energy.

Forgive me. In the absence of a better explanation, my mind can cope with seeing elementary particles as 'structured energy' with various properties of symmetry, directionality, physical size etc. Perhaps I give energy an inflated significance, But then, my main field is thermodynamics so what do you expect?

At least you didn't come up with 'fields'. Or 'space-time'.

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Not much left of a photon after an electron has absorbed its energy.

.

An atom can't absorb a photon if the angular momentum doesn't match up. You can't go from an s state to a d state with one photon. So there's more to a photon than energy.

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Now that discussion took here, let me put it this way:

Assume Space($\vec{r}$), Time($t$), Mass($m$) and Charge($q$) as primitive notions in Classical Mechanics(I know you may say any concepts could be regarded as primitive but they are good reasons to take them as primitive: Space and Time are primitive in mathematics and Mass and Charge are localized simple properties we could assign to particles and/or bodies).
Now we define new concepts based on previous ones: Velocity(Rate of change of Spatial position $\vec{v}:=\frac{d\vec{r}}{dt}$), Momentum(Mass multiplied by velocity $\vec{p}:=m\vec{v}$), Force(Rate of change of momentum $\vec{F}:=\frac{d\vec{p}}{dt}$), Current Intensity(Rate of change of charge $I:=\frac{dq}{dt}$), Angular momentum(Moment of momentum $\vec{L}:=\vec{r}\times \vec{p}$), etc.
But look at Energy. It have no rigouros quantitavie definition.
What Finally I Want Is A Quantitative Definition Of Energy. I Mean Something Like: $E := something$
Edited by Hamed.Begloo

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Strictly speaking, E is not equal to SOMETHING.

And as I have stated, by classical definition you can define energy of a system is equal to force times displacement. Energy spent by a system is the work done by it. Mathematically work is defined as the dot product of the vectors of force and displacement.

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hamed.begloo

WHAT FINALLY I WANT IS A QUANTITATIVE DEFINITION OF ENERGY. I MEAN SOMETHING LIKE:

First and foremost stop shouting.

Second I have already given you a correct and complete definition.

Third this is an exercise in physics not in computer programming so please write the equals sign correctly, instead of using the computer assignment statement.

Fourthly definitions do not require an equals or assignment operator.

Fifthly should report your infringement of forum rules in not replying to me?

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