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.
Thank you in advance.
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".
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.
To Mods: Notice that I've already seen the following links but did not get my answer:
What Is Energy? Where did it come from?
What's the real fundamental definition of energy?
So please don't mark my post as a duplicate.
Answer
Let $\vec{F}$ be a conservative force field, that is $$ \nabla \times \vec{F} = 0 $$ or alternatively $$ \phi := -\int_\gamma \vec{F} \cdot d\vec{a} $$ does not depend on the path $\gamma$ and its parametrization $\vec{a}(s)$, but only on the "endpoints".
Take Newton's equation of motion: $$ m \ddot{\vec{r}} = \vec{F} $$ multiply by $\dot{\vec{r}}$ and integrate over time $$ m \int_{t_0}^t \frac{d \dot{\vec{r}}}{dt} \cdot \dot{\vec{r}} ~dt = \int_{t_0}^t \vec{F} \cdot \dot{\vec{r}} ~dt$$ $$ m \int_{\vec{v}_0}^{\vec{v}} \dot{\vec{r}} \cdot d \dot{\vec{r}} = \int_{\vec{r}_0}^{\vec{r}} \vec{F} \cdot d \vec{r} $$ $$ m \int_{v_0}^{v} v ~dv + m \int_{\vec{n}_0}^{\vec{n}} v^2 \vec{n} \cdot d \vec{n} = -\phi(\vec{r}) + \phi(\vec{r}_0) $$ $$ \frac{1}{2} m v^2 - \frac{1}{2} m v_0^2 + m \int_{\vec{n}_0}^{\vec{n}} v^2 \vec{n} \cdot d \vec{n} = -\phi(\vec{r}) + \phi(\vec{r}_0) $$ with $\vec{n}$ the unit vector pointing in the direction of the velocitiy $\vec{v} = \dot{\vec{r}} = v\cdot\vec{n}$. The remaining integral is a path integral over a path on the unit circle / sphere. That is: $\vec{n} \perp \frac{d\vec{n}}{ds}$ for any parametrization $\vec{n}(s)$. Thus, we have: $$ \int_{\vec{n}_0}^{\vec{n}} v^2 \vec{n} \cdot d \vec{n} = 0 $$ and we are left with:
$$ \frac{1}{2} m v^2 + \phi(\vec{r}) = \frac{1}{2} m v_0^2 + \phi(\vec{r_0}) = \text{const.} $$
Thus we have
$$ E := \frac{1}{2} m v^2 + \phi(\vec{r}) $$
where $\phi(\vec{r})$ is called potential energy and $T=\frac{1}{2} m v^2$ is called kinect Energy. The total energy $E$ is therefore a constant (over time). Work $W$ is the difference
$$ W = \phi(\vec{r}) - \phi(\vec{r}_0) = - \int_{\vec{r}_0}^{\vec{r}} \vec{F} \cdot d\vec{r} $$ and can even be defined (in terms of the integral) if the force $\vec{F}$ is not conservative.
In conlusion: Total energy $E$ is a conserved quantity. Work is the energy needed to go from a point $\vec{r}_0$ to $\vec{r}$ (and end up with the same velocity), e.g. the difference in potential energy. Kinetic energy is associated with movement and potential energy is associated with the presence of external forces.
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