  # Hamed.Begloo

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## Posts posted by Hamed.Begloo

1. ### Rigorous definition of "Differential"

It is, however, difficult to guess the level to answer at since you are discussing a very simple function of a single variable but some of your comments hint at a much higher level of mathematics. For example you mentioned differential forms and your answer about the definition of a function.

Actually I said I prefer that the answer not to be in the context of "Theory of differential forms" because I have no knowledge of the notion of "Differential forms". However as far as I know the approach used in the discipline is quite different from calculus: introducing new spaces, usage of different geometries, constructing some form of algebras for differentials,... it seems they are treating differentials as some different kind of mathematical objects rather than simply functions.
Anyway considering more lower levels for answer is better. If you wanna consider a level to answer me, I can say for example much of "Calculus and Analytic Geometry, Thomas-Finney" are understandable to me.

There are a lot of different notations about for calculus.

Further many different sorts of people use calculus for different purposes.

Some of the people are much more rigorous than others.

Many people mix up the notation and/or terminology.

None of this really matters for the calculus of a single variable and people muddle through.

I'm still not convinced why it doesn't matter for single variable calculus. If it's just a notation then I think it's really bad to use Leibniz's notation in calculus. We should reformulate calculus using Lagrange's notation for both derivative and antiderivative operations so we get rid of this nonsense differential. Leibniz introduces symbols like "$\int$" and "$\mathrm{d}$" which seems to be insisting they must have separate standalone meanings and definitions. But when you talk about them everyone says these symbols has no meaning for themselves alone and we end up that "$\int f(x) \mathrm{d} x$" and "$\frac{\mathrm{d} f(x)}{\mathrm{d} x}$" are just mere notations for antiderivative and derivative respectively. Anyway I thought that widespread usage of Leibniz's notation must have an important reason. Not that just this makes them algebraically more flexible so students manipulate them easily... but from a more pure mathematical perspective.

More rigor becomes important with multivariable calculus.

This is where the 'differential becomes important.

The differential is really defined as a transformation from Rn to R where n is the number of independent variables.

I don't know if your studies are ready for this or where to start for you so I am going to post two pages from different textbooks.

Please say if anything is familiar.

Or do I need to start further back?

I will be away for a few days now so see what you can make of these pages. differentials11.jpg differentials10.jpg

OK, let's see what I get:
Here $X$ and $Y$ seem not to be simple numerical variables but rather $n$-tuples consisting of numerical variables. So the arguments of $f$ are $n$-tuples and this is why the domain is $\mathbb{R}^{n}$. Since the $f$ is a Real-valued function the output must be a number and this also justifies why the range is $\mathbb{R}$. On the other hand $X_0$ seems to be a determined known point in $\mathbb{R}^{n}$.
And here's what I don't get:
Now this is where my inexperience and knowledgelessness ruins everything. Now I'm going to ask some very basic questions that pops into my mind right now:
1) I know what "Linear transformation" is: Any transformation that carries the properties of "Additivity" and "Homogeneity". But I always have a headache distincting notions such as "Transformation", "Mapping" and "Function". Could you first tell me what are their difference?
2) The definition says about "the differential of $f$ at $X_0$", but then says about the value of the linear transformation at $Y$. I'm not sure what is the argument of the function/transformation. $X_0$ or $Y$?
3)It somehow reminds me of this relation about the differential of a multivariate function:
$\mathrm{d} f(x_1,x_2,\cdots ,x_n)=\mathrm{d}x_1 \frac{\partial }{\partial x_1}f(x_1,x_2,\cdots ,x_n) + \mathrm{d}x_2 \frac{\partial }{\partial x_2}f(x_1,x_2,\cdots ,x_n)+ \cdots + \mathrm{d}x_n \frac{\partial }{\partial x_n}f(x_1,x_2,\cdots ,x_n)$
But the problem with this definition is again some other differentials($\mathrm{d}x_1$, $\mathrm{d}x_2$, $\cdots$ , $\mathrm{d}x_n$) exists in the definition. However yours are in the form of $y_1$, $y_2$ , $\cdots$ , $y_n$. So maybe it's different. Anyway are these related?
I stop here so my basic issues would be resolved.
Thank you for helping me. 2. ### Rigorous definition of "Differential"

Whether you realize it or not, you included both functions and the values of those functions in your question and mixed them up in your thinking.

Yes, and... honestly I still don't know how I included the values of the functions in my question. I just wanted a definition for "Differential function" which haven't any other differentials in its definition as the definition of "Derivative function" doesn't contain any other derivative either.

I have said this before and already indicated that there are two functions involved

The derived (not derivative) function and the original function.

(That by the way is how the name derivative came about)

Both of these are functions in their own right and therefore have values.

Sorry but I didn't find what you mean by "Derived function"(if it's not the derivative function). Again would you explain more about what you meant here?

You can only subtract values to obtain a Delta [something]

Now I guess I know what you mean. You're saying since "Difference of function" is tied to two values of the function, then necessarily "Differential of function" is defined by the values of the function(if I'm right). So does it mean there is nothing such as a "Differential function"? And it's only meaningful at some points?
P.S. I would appreciate if you answer more descriptively and again consider the fact that... you know... I'm not a native English speaker .
3. ### Rigorous definition of "Differential"

Do you actually know what a function is?

A function is a "Relation(which is a subset of a cartesian product of two sets)" where the second entries of the ordered pair elements of the resultant set aren't pairwisely equal.

You have not even acknowledged my comment about the difference between a function and the value of a function at some point.

Yes but I'm not talking about the value of the functions or their differential at a point. Sure it's important but it's not everything. For example we may show the derivative of function $f(x)$ at some point like $x_0$ as follows:

$\frac{\mathrm{d} f(x)}{\mathrm{d} x}|_{x=x_0}={f}'(x_0)$

But we define something else and name it "Derivative function" which renders derivative as a function -and not the value of ${f}'$ at $x_0$- which its outcome is a function(and not a value) and we show it like this::

$\frac{\mathrm{d} f(x)}{\mathrm{d} x}={f}'(x)$

And I'm actually talking about "Differential function" -not the value of $\mathrm{d} f(x)$ at $x_0$- and I think it should have a precise definition(as the Derivative function has a definition too).

There is no reason to expect an explanation or definition of how to form d (f(x)) in the definition of a differential any more than to expect a definition of how to form say x3+x2+x+3 or sin(x) in the definition of f(x).
In fact d(f(x)) is quite rightly not mentioned at all.
So why bring it up?

I didn't get what you meant here. Could you explain more?

4. ### Rigorous definition of "Differential"

$\Delta f\left( {{x_1}} \right) \approx f'\left( {{x_1}} \right)\Delta x$

Basically in words it states
The change in the value of the function of x, f(x) at the point x1 is approximately equal to the value of the derived function f'(x) at x1 multiplied by the change in the value of x.
There is nothing circular about that definition.
All you need to do is distinguish between functions and their values.

Actually in that statement "The change in the value of a function/variable" is previously defined:

$\Delta f(x)= f(x_2)-f(x_1)$ or $\Delta x= x_2-x_1$

But how we defined $\mathrm{d} f(x)$ or $\mathrm{d} x$ is not precisely elaborated. One may say it's the same as a change in the variable/function when the change is small enough but what "small enough" means here again isn't precisely elaborated. I actually attempted to think of a definition for differential as like it's simply the limit of a difference as the difference approaches zero:

$\mathrm{d}x= \lim_{\Delta x \to 0}\Delta x$

But because of the so called "Archimedean property" of the "Real number system", that makes a differential simply zero:

$\mathrm{d}x= 0$

and it's not logical.

Can we make this situation better?

5. ### Rigorous definition of "Differential"

First of all I want to clarify that I posted this question on many forums and Q&A websites so the chances of getting an answer will be increased. So don't be surprised if you saw my post somewhere else.
Now let's get started:

When it comes to definitions, I will be very strict. Most textbooks tend to define differential of a function/variable in a way like this:

--------------------------------------------------------------------------------

Let $f(x)$ be a differentiable function. By assuming that changes in $x$ are small, with a good approximation we can say:

$\Delta f(x)\approx {f}'(x)\Delta x$

Where $\Delta f(x)$ is the changes in the value of function. Now if we consider that changes in $f(x)$ are small enough then we define differential of $f(x)$ as follows:

$\mathrm{d}f(x):= {f}'(x)\mathrm{d} x$

Where $\mathrm{d} f(x)$ is the differential of $f(x)$ and $\mathrm{d} x$ is the differential of $x$.

--------------------------------------------------------------------------------

What bothers me is this definition is completely circular. I mean we are defining differential by differential itself. Although some say that here $\mathrm{d} x$ is another object independent of the meaning of differential but as we proceed it seems that's not the case:

First of all we define differential as $\mathrm{d} f(x)=f'(x)\mathrm{d} x$ then we deceive ourselves that $\mathrm{d} x$ is nothing but another representation of $\Delta x$ and then without clarifying the reason, we indeed treat $\mathrm{d} x$ as the differential of the variable $x$ and then we write the derivative of $f(x)$ as the ratio of $\mathrm{d} f(x)$ to $\mathrm{d} x$. So we literally (and also by stealthily screwing ourselves) defined "Differential" by another differential and it is circular.

Secondly (at least I think) it could be possible to define differential without having any knowledge of the notion of derivative. So we can define "Derivative" and "Differential" independently and then deduce that the relation $f'{(x)}=\frac{\mathrm{d} f(x)}{\mathrm{d} x}$ is just a natural result of their definitions (using possibly the notion of limits) and is not related to the definition itself.

Though I know many don't accept the concept of differential quotient($\frac{\mathrm{d} f(x)}{\mathrm{d} x}$) and treat this notation merely as a derivative operator($\frac{\mathrm{d} }{\mathrm{d} x}$) acting on the function($f(x)$) but I think that it should be true that a "Derivative" could be represented as a "Differential quotient" for many reasons. For example think of how we represent derivatives with the ratio of differentials to show how chain rule works by cancelling out identical differentials. Or how we broke a differential into another differential in the $u$-substitution method to solve integrals. And it's especially obvious when we want to solve differential equations where we freely take $\mathrm{d} x$ and $\mathrm{d} y$ from any side of a differential equation and move it to any other side to make a term in the form of $\frac{\mathrm{d} y}{\mathrm{d} x}$, then we call that term "Derivative of $y$". It seems we are actually treating differentials as something like algebraic expressions.

I know the relation $\mathrm{d} f(x)=f'(x)\mathrm{d} x$ always works and it will always give us a way to calculate differentials. But I (as an strictly axiomaticist person) couldn't accept it as a definition of Differential.

So my question is:

Can we define "Differential" more precisely and rigorously?

P.S. I prefer the answer to be in the context of "Calculus" or "Analysis" rather than the "Theory of Differential forms". And again I don't want a circular definition. I think it is possible to define "Differential" with the use of "Limits" in some way(though it's just a feeling).

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

I suggest you postpone extended discussion of turning effects to another thread, it does not belong here. Start another if you wish.

I'm afraid starting another thread for the same subject being count as another infringement of the forum rules. If you don't mind, I prefer to wait for answers here.

Thank you. 7. ### What is the rigorous quantitative definition of the concept of "Energy"?

Quite frankly now I'm very happy that you noticed you are talking with someone who have much less knowledge than yourself.

I then asked if you know of such a situation.

I know that the outcome of the product of "Force" and "Distance/Position/Length" depends highly on either quantities being "Vector" or "Scalar" and also on either the operator acting between them being "Dot product" "Cross product" or simply just a "Multiplication of a vector and a scalar".

I also asked what else you might obtain that is not energy.

I know the outcome of this product could have the dimension of [Energy] or [Torque] but the outcoming quantity could be either "Difference in Energy", "Work" or "Torque"

To save time I will tell you that moment (the americans like to call it torque) is defined as force times distance.

I hope you know the difference between moment and energy.

Let me tell you what I know about Moment and Torque then you correct me if I'm wrong.

I know many people use these words interchangeably. But in overall what I saw from most textbooks or on the web usually "Torque" is defined as "Cross product of Position vector and Force vector" while "Moment" usually defined as "Cross product of Position vector and any other quantity" Like:

Angular momentum(moment of momentum) $\vec{L}:=\vec{r}\times \vec{p}$

Torque(moment of force) $\vec{\tau}:=\vec{r}\times \vec{F}$

etc.

And this makes Torque a special case of Moment.

This is important because Force is (always) a vector and 'distance' in this case is also a vector.

Since there are two possible and different ways of multiplying two vectors together,

Multiplying force and 'distance' will get you two different answers and two different quantities.

As I mentioned earlier, yes I know there are two possibilities.

Energy is the scalar (or dot) product of the two vectors.

Moment is the vector product of the same two vectors.

I will stop here for the time being until you can confirm that you have understood what I have said so far. Now this is my main problem: I agree about Moment/Torque but I know the dot product of "Force" and "Distance" gives you "Work". I know "Work" and "Energy" have the same dimensions but I think they are different quantities. They are other situations where two different quantities have same dimensions. For example "Translational kinetic energy" and "Rotational kinetic energy" have same dimensions but they are not describing one thing. I think If two quantities have same dimensions then not necessarily they describe one thing. Yes "Work" may represent the difference in energy but does not talk about "Energy itself".

I hope I delivered my purpose.

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

I'm sorry but I really didn't get what you mean. Maybe this problems arise because of me not being a native English speaker and not understanding you( or who knows being a total stupid ).

You have yet to answer my first question there.

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.

.

You have not yet answered my question attached to that statement now thrice repeated.

Honestly, I still don't know what question you are talking about. Can you repeat it(by taking my stupidity into account)?
Also:
First you said:

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

Then you said:

(note srinam correctly offered force times displacement)

Now again you say:

How can it possibly be a definition of energy?

It just seems to me you are negating your own words.

It was clearly a reason to not use this definition but employ the one that I actually stated.

As I already mentioned my stupidity let me review your first post:

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.

I didn't mean this but I was in a hurry and answered everybody shortly. However again I accept it was my fault and you are right to make this interpretation.

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

I don't know What makes you think I want to discuss quantum mechanics.

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.

If this is what makes you think so, I can say I was told for example Potential energy of a particle is stored in field lines around that particle. Not in particle itself. Correct me if I'm wrong.

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.

And this is the main reason I'm here: Determining fundamental and derived concepts in classical mechanics which you said is an established structure.

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.

What I understood from this line is: We need minimum number of Primitive notions and Axioms therefore the theory is more simple(as all theories seek to be simple) while it covers a wide range of phenomena(which is a very good property for a theory despite its simplicity) and can be easily altered to give some ad-hoc explanations for the phenomena not in its range of validity(like describing Hydrogen with Bohr's atomic model or some non-relativistic cosmological phenomena). Again correct me if I'm wrong.

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

This is the only quantitative statement you introduced which you yourself don't count as a definition of energy.

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

What more do you want?

Again I didn't find the quantitative statement.

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

So you didn't understand and asked for clarification since it offered all that you asked for?

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.

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What exactly did you not understand about my definition?

What did you make of my comment against the 'force times distance' argument? (note srinam correctly offered force times displacement)

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

My Impression from the above sentence is that you said "Force times Distance" is not necessarily "Energy". So how could I take this as a definition?

Or did you just carry on demanding an answer to conform to your terms, telling others that they are wrong?

No. Infact The main problem is I have no way to define my terms.

And as I mentioned in my "Post #29" I just think every theory should be formulated in an axiomatic way. For example think of "Classical electromagnetism". With just one primitive notion(Electric Charge) and a set of field equations(Maxwell's Equations) as axioms plus a description of electromagnetic force(Lorentz Force) every phenomena in classical electromagnetism are described. But classical mechanics although dating back to older ages, still seems to be not fully organized and axiomatized.

Now that you have got into the habit of replying, perhaps we can carry on the conversation towards understanding?

I think now I know why you thought I did an infringement in the forums. But that post was not specifically written for you. I read all the posts before, thought they didn't get what I mean, summed up my words and then posted it for everybody.

Listen when it comes to science, I confess I'm the most illiterate one. I'm here to learn something from you all, not to question your knowledge. The challenges that I make are just to (as you said) carry on the conversation. I hope the turbidity is removed.

No. Not in science.

Until someone comes up with a complete and comprehensive description of reality, no one will really be sure which phenomena are primitive and which are derived. In fact, the further we progress, the more abstract everything seems to become.

Correct. I agree in science as a whole we couldn't define an axiomatic system. Because the fundamentality of notions are always changing. For example I know in quantum mechanics the notion of "Rotational angular momentum(Spin)" have not the same meaning in the classical mechanics(such that a dot particle without any volume can have a non-zero spin) so we take it as an intrinsic quantity.

But I'm talking about just one single theory(namely classical mechanics) which I expect to work completely in its own scope (not outside its range of validity).

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

And you will never get one, for the simple reason that it is a universal quantity that exists in a multitude of different forms.

And thanks to frames of reference considerations, we cannot even agree on a zero value, so it is only meaningful to discuss changes in energy.

Ultimately, you end up with 'That property of a closed system that is conserved as a result of the First Law of Thermodynamics'.

Delta E equals zero.

How you conceptualise E is purely a matter of personal preference. But it must encompass the universal element - ie a recognition of the quantitative equivalence of heat, work, kinetic energy, gravitational/electrical/chemical potential, rest mass etc etc etc.

Personally, I view it as the sum total of all motion or potential to move. Works for me. Others may differ.

But doesn't theories supposed to be designed in an axiomatic way.

I know Physics is not math but I know a physical theory is a mathematical framework that fits best with our descriptions of nature. On the other hand mathematics is always constructed in an "Axiom-Theorem" way for the propositions claimed and in a "Primitive notion-Well defined notion" way for the concepts introduced. So it's a real pain when you talk about a so important concept in a theory and have no precise definition of it. Yes I can talk about Work, Heat, Kinetic Energy, Potential Energy, Thermal Energy and so on but I know "Energy" as a whole concept should be defined. For example when we say "The energy of a system is conserved" we are talking about energy itself not its various forms. Maybe physicists are smart enough to conceptualise "Energy" without a rigorous definition but for a noob like me it doesn't work.

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

Uh, oh... it seems it gone wrong First and foremost stop shouting.

I never shout anything as I know here is a community of educated people and everyone must talk with each other with respect. I just wanted to show that what my main request is and reveal it distinctively. Anyway if it bothered you I apologize deeply.

Edit: I searched the web and I understood it mean shouting. Again apologies.

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

I read your previous post but I didn't catch your definition. However maybe it's just me who couldn't get the definition and it's my fault.

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.

I know it's not common but many places I see people use the "Colon Equal($:=$)" notation for defining new quantities. See here(Link below)

http://mathworld.wolfram.com/Defined.html

It's some how like the equivalent sign($\equiv$)

P.S. I know the "Colon Equal" operator is used for variable assignment in programming.

Fourthly definitions do not require an equals or assignment operator.

Yes. But they could be defined by an equation(if they are not primitive).

Plus as you see I also stated the language-based definitions of quantities. For example:

Velocity: Rate of change of Spatial position (Lingual definition)

or

$\vec{v}:=\frac{d\vec{r}}{dt}$ (Mathematical definition)

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

I still don't know How I did an infringement. But as stated early. If I did something wrong I'm so sorry.

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

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: $E := something$
13. ### What is the rigorous quantitative definition of the concept of "Energy"?

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.

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

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.

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

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

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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