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.
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".

EDIT(Question clarification):
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:  $E := something$

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.
 A: 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.
A: I'm going to take somewhat of a philosophical angle in this answer. The problem you are having is that you are asking for a quantitative definition of a physical concept. Any quantitative expression will necessarily contain symbols for other physical concepts that themselves will be quantitatively defined in terms of other physical concepts. Since there are a finite number of such concepts, eventually you will circle back to one already mentioned. This is similar to how, in a dictionary, words are defined in terms of other words and you eventually reach a loop.


*

*mountain: a large hill

*hill: a small mountain


Like words in a dictionary, the equations of science give the relations between concepts. The concepts themselves are defined in terms of lived, physical experience. People have noted that, when wood is burned, they feel warm when standing near it. This experience creates the notion of something inside the wood being released and transferred to the skin of someone nearby. This notion is given the name heat. Later, this heat was found to be useful in applying to machines to complete tasks. Thus, some relation between heat and work was suspected and formalized into the concept of energy that moves from place to place and form to form. The equations of energy, work, heat, kinetic energy, potential energy, etc. are descriptions of relations between concepts, not definitions.
Even Marcel's answer starts with a conservative force field. A conservative force field is defined as one that has a potential energy associated with it (this is equivalent to saying it has zero curl or path-independent line integrals), so it does no better in avoiding the circular-definition problem. The way to think about energy and work is that they are not defined in terms of each other, but correspond to physical experience separately and then later are related by discovered equations.
A: You're up against an insurmountable problem, the fact that no one knows what energy is.  We can only describe how it behaves in relation to other things, and quantify it.
The best description, I think, is that the transfer of energy is always associated with a change in system properties, and vice-versa.
Energy transfer can be quantified as $$E = Fd$$
I know, that probably strikes you as trivial, and/or evokes the objection, "no that's only for work done, or for changes in kinetic energy."  Not so.  That's part of the damage done by the work concept (but don't get me started).  Thermal energy is transferred by force-through-distance collisions between particles in a statistical war between the warmer system and the cooler system.  The warmer system wins because there are many more angles available in the set of random collision angles for a faster particle to collide with a slower particle in such a way as to speed up the latter, than vice versa.  Electromagnetic energy is transferred into and out of an electromagnetic wave by force exerted through distance.  Gravitational and other "potential" energy types are transferred into and out of fields by force exerted through distance.  I've yet to encounter a valid counterexample in classical physics.  And energy, force and space being what they are, I doubt that there are translatable counterexamples in the quantum realm.
A: Energy is a physical quantity that is constant for a closed system.
$$E = const.$$
As such it is not unique. There are other physical quantities that are constant for closed systems but named differently, e.g. charge.
