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

#### HamedBegloo

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 $$\displaystyle p := mv$$, 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:

$$\displaystyle E := something$$

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:

$$\displaystyle E := -\frac{dS}{dt}$$

Where $$\displaystyle S$$ is the action and $$\displaystyle t$$ 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 $$\displaystyle [ML^{2}T^{-2}]$$) 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".

#### Woody

Energy is an imbalance or tension within a system,
it indicates that the system is not in equilibrium.
What is observed as energy is the system re-organising itself toward equilibrium.

#### HamedBegloo

Energy is an imbalance or tension within a system,
it indicates that the system is not in equilibrium.
What is observed as energy is the system re-organising itself toward equilibrium.
So what about a system which is reached its maximum entropy. It's in equilibrium but still contains energy.

#### Woody

Are we perhaps rather loosely using two different definitions of energy?
The energy indicated in the "work" definition is "Relative" Energy
(the potential energy at the top of the hill is higher relative to that in the valley).
And that relative definition seems to cover the non-relativistic Newtonian mechanics pretty well.

It seems only when we start to talk about E=mc^2 that we start having to consider some "absolute" measure of Energy.
Now we are talking about energy in terms of the curvature of spacetime.
Thus a transfer of energy indicates a change in the curvature of spacetime.

Note, I am making this up as I go, but hopefully not talking total #####
(insert an appropriate word or phrase according to your personal taste).

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#### HamedBegloo

Are we perhaps rather loosely using two different definitions of energy?
The energy indicated in the "work" definition is "Relative" Energy
(the potential energy at the to of the hill is higher relative to that in the valley).
And that relative definition seems to cover the non-relativistic Newtonian mechanics pretty well.

It seems only when we start to talk about E=mc^2 that we start having to consider some "absolute" measure of Energy.
Now we are talking about energy in terms of the curvature of spacetime.
Thus a transfer of energy indicates a change in the curvature of spacetime.

Note, I am making this up as I go, but hopefully not talking total #####
(insert an appropriate word or phrase according to your personal taste).
Correct. But what I'm talking about is "Energy" must have a "quantitative definition" much like "Momentum" which had the definition: $p := mv$. Don't you think?

#### Woody

What units would one define the curvature of spacetime in?

#### HallsofIvy

I think of "energy" as a book keeping device! We start with the definition of energy as "work"- force times distance. Then, since that work may result in motion, to maintain "conservation of energy", we introduce "kinetic energy", and since it may result in moving an object upward, "potential energy". And if there is friction, the work may cause a heating rather than all going to kinetic or potential energy, we count heat as energy.

#### Woody

In the original post of this thread there was the statement
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.
I would perhaps argue that this system does NOT contain energy,
or at least that the energy is spread so thinly as to be negligible at all points within the system.
spacetime will then be "flat" at all points in the universe.
Perhaps this brings in again the concept of relative energy, work cannot be done if all energy levels throughout the system are identical.

Just to be contentious:
We learn at school that energy cannot be created or destroyed, just converted to another form.
This is seems to be stated as a fundamental axiom of physics, but is it based on anything other than experimentally derived observations?

#### topsquark

Forum Staff
In the original post of this thread there was the statement

I would perhaps argue that this system does NOT contain energy,
or at least that the energy is spread so thinly as to be negligible at all points within the system.
spacetime will then be "flat" at all points in the universe.
Perhaps this brings in again the concept of relative energy, work cannot be done if all energy levels throughout the system are identical.

Just to be contentious:
We learn at school that energy cannot be created or destroyed, just converted to another form.
This is seems to be stated as a fundamental axiom of physics, but is it based on anything other than experimentally derived observations?
The definition of energy will almost certainly not cover all the bases. Just like distance, energy is not really a well defined concept. (Try to define what the unit meter refers to without a circular argument.) We can define energy mathematically in the sense that we can track what happens to it in a process, but it is a fundamental quantity and basically has the form of an axiom ala Euclid. In introductory Physics we tend to call it by three names: work, kinetic energy, and various potential energies. Thermodynamics adds another way...roughly speaking we can relate it to heat and temperature. (And various associated free energies of a system.)

As far as conservation, Special Relativity has a broader rule: conservation of energy and momentum all at one go. The momentum 4-vector has the form $$\displaystyle \left ( \frac{E}{c}, p_x, p_y, p_z \right )$$. It is this 4-vector that is actually conserved.

-Dan

#### HamedBegloo

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

Assume Space($$\displaystyle \vec{r}$$), Time($$\displaystyle t$$), Mass($$\displaystyle m$$) and Charge($$\displaystyle q$$) 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 $$\displaystyle \vec{v}:=\frac{d\vec{r}}{dt}$$), Momentum(Mass multiplied by velocity $$\displaystyle \vec{p}:=m\vec{v}$$), Force(Rate of change of momentum $$\displaystyle \vec{F}:=\frac{d\vec{p}}{dt}$$), Current Intensity(Rate of change of charge $$\displaystyle I:=\frac{dq}{dt}$$), Angular momentum(Moment of momentum $$\displaystyle \vec{L}:=\vec{r}\times \vec{p}$$), 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: $$\displaystyle E := something$$

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