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The simplistic undergrad explanation aside. I've never really understood what energy really is. I've been told that it's something when converted from one kind of something to another kind does some "work", as defined by us, but what is that something?

Moreover, if the total amount of energy in the universe is finite and we cannot create energy. Then, where did it come from? I've learnt from thermodynamics where it goes, but where does it come from?

I know this sounds as something trivially simple, but there is so much going on over here and I just can't grasp what it is. Of course, I lack the mathematical understanding to grasp the subtle things the universe is doing, but what is it doing and how do I get to point of understanding what it's doing?

(Note: What prompted me to ask this was this answer. I'm afraid that it just puzzled me further and I sat there staring at the screen for a good 10 minutes.)

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7 Answers 7

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Energy is any quantity - a number with the appropriate units (in the SI system, Joules) - that is conserved as the result of the fact that the laws of physics don't depend on the time when phenomena occur, i.e. as a consequence of the time-translational symmetry. This definition, linked to Emmy Noether's fundamental theorem, is the most universal among the accurate definitions of the concept of energy.

What is the "something"? One can say that is a number with units, a dimensionful quantity. I can't tell you that energy is a potato or another material object because it is not (although, when stored in the gasoline or any "fixed" material, the amount of energy is proportional to the amount of the material). However, when I define something as a number, it is actually a much more accurate and rigorous definition than any definition that would include potatoes. Numbers are much more well-defined and rigorous than potatoes which is why all of physics is based on mathematics and not on cooking of potatoes.

Centuries ago, before people appreciated the fundamental role of maths in physics, they believed e.g. that the heat - a form of energy - was a material called the phlogiston. But it is a long, long time ago when experiments were done to prove that such a picture was invalid. Einstein's $E=mc^2$ partly revived the idea - energy is equivalent to mass - but even the mass in this formula has to be viewed as a number rather than something that is made out of pieces that can be "touched".

Energy has many forms - terms contributing to the total energy - that are more "concrete" than the concept of energy itself. But the very strength of the concept of energy is that it is universal and not concrete: one may convert energy from one form to another. This multiplicity of forms doesn't make the concept of energy ill-defined in any sense.

Because of energy's relationship with time above, the abstract definition of energy - the Hamiltonian - is a concept that knows all about the evolution of the physical system in time (any physical system). This fact is particularly obvious in the case of quantum mechanics where the Hamiltonian enters the Schrödinger or Heisenberg equations of motion, being put equal to a time-derivative of the state (or operators).

The total energy is conserved but it is useful because despite the conservation of the total number, the energy can have many forms, depending on the context. Energy is useful and allows us to say something about the final state from the initial state even without solving the exact problem how the system looks like at any moment in between.

Work is just a process in which energy is transformed from one form (e.g. energy stored in sugars and fats in muscles) to another form (furniture's potential energy when it's being brought to the 8th floor on the staircase). That's when "work" is meant as a qualitative concept. When it's a quantitative concept, it's the amount of energy that was transformed from one form to another; in practical applications, we usually mean that it was transformed from muscles or electrical grid or battery or another "storage" to a form of energy that is "useful" - but of course, these labels of being "useful" are not a part of physics, they are a part of the engineering or applications (our subjective appraisals).

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(Bear with me, I'm not a physicist...) Your first paragraph seems like it is circular reasoning. My understanding of Noether's theorem is once we have energy and time in a system, it tells us that - and why - energy is conserved over time. If we define energy to be "any quantity... that is conserved as the result of the fact that the laws of physics don't depend on the time", we don't need Noether's theorem, because the conservation of energy is now a tautology. But it's not a tautology - we need a definition of energy independent of its conservation to meaningfully state it is conserved. –  user1362 Jan 16 '11 at 12:31
    
(And that definition of energy is, as you say, the Hamiltonian. Everything but the first paragraph seems fine to me.) –  user1362 Jan 16 '11 at 12:33
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Thank you. For the explanation. It made my day. –  Anna Jan 16 '11 at 13:30
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Thanks for your interest and good question, Anna. And Joe: no, you don't have to "possess" energy to start with. Noether's theorem genuinely derives its form from the symmetry. You may only have equations of motion to start with. However, the theorem allows you to derive how the conserved quantity - energy - depends on anything. The theorem implies both that a conserved quantity suddenly exists - we didn't know about it "before" the symmetry was appreciated - and also its exact form. One couldn't "guess" the exact form of energy for any system before the dynamics is analyzed. –  Luboš Motl Jan 16 '11 at 15:58
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For any system, one may write down the right "guess" for the energy. For example, a harmonic oscillator has $E=p^2/2m+kx^2$. However, for every physical system, one needs to produce a different formula, a different function of different variables. There's no universal way how to "guess" the right formula for the energy. The Noether recipe is a universal method to derive what it is and it is true in any system. The formula does depend on the system - and this "flexibility" is a huge strength of the concept. Energy knows about "everything" in the system. –  Luboš Motl Jan 16 '11 at 16:00

I don't think the answer is trivially simple. I will try to give an explanation. In many problems of physics, what you are given is the initial and final states of the system. You don't know (or maybe no one does) what happens between these two states. Now there are quantities that you can measure before and after the system has undergone this change of state. The question is can you predict some of these quantities by knowing the others. Remember that we don't know the mechanism by which the system moves from these two states. But if you have something known as a conservation law, the problem becomes simple. (By saying that a quantity is conserved we mean that it doesn't change throughout some process). Suppose you have some magic function involving the quantities, which gives the same value no matter what the state of the system is, then you are done. The value of the function we call energy. And since its value doesn't change between these two states we say that its conserved.

This excerpt is from Feynman Lectures:

There is a fact, or if you wish, a law, governing all natural phenomena that are known to date. There is no known exception to this law—it is exact so far as we know. The law is called the conservation of energy. It states that there is a certain quantity, which we call energy, that does not change in the manifold changes which nature undergoes. That is a most abstract idea, because it is a mathematical principle; it says that there is a numerical quantity which does not change when something happens. It is not a description of a mechanism, or any- thing concrete; it is just a strange fact that we can calculate some number and when we finish watching nature go through her tricks and calculate the number again, it is the same. (Something like the bishop on a red square, and after a number of moves—details unknown—it is still on some red square. It is a law of this nature.)

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So, essentially, almost all of physics is based upon a quantity that no one has ever been able to truly define on it's own? –  Anna Jan 16 '11 at 10:50
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@Anna: I would guess that the vast majority of physicists are able to define energy to their satisfaction. For some people the definition will be practical, in terms of the formulas you use to calculate it, while others will have some abstract idea of what energy is. It really doesn't matter. The fact is, all physicists, even if they are bothered by an apparent lack of definition, are able to use energy to develop theories and analyze experiments together, and as far as I'm concerned, that's all a physical concept needs to be useful. If you go beyond that, you're venturing into philosophy. –  David Z Jan 16 '11 at 11:11
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Energy is never a fundamental entity of a theory as positions or wavefunctions are. Energy just shows up because our theories have some symmetries and it would always show up regardless of the kind of theory, as long as it contains the appropriate symmetries. –  Raskolnikov Jan 16 '11 at 12:32
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I would like to endorse Raskolnikov's important comment, too. Energy is not something one starts with while constructing a theory; energy is a "cherry on pie" that one can find out to be conserved. It is given by a fixed formula that can be found and when this formula is evaluated at any time, the number is always the same. It didn't have to exist at all. However, energy is really special because the information about the formula for the energy is equivalent to the information about the way how the system evolves in time - so it's a bit more fundamental than Raskolnikov suggests. –  Luboš Motl Jan 16 '11 at 18:01
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The great thing about Feynman is that when he didn't know something, he knew he didn't know something. –  Chris B. Behrens Sep 19 '11 at 20:31

To understand what energy is, it is necessary to understand the concept of work.

Work is defined as the action of a force over a path.

$$ W=\vec{F}\cdot\vec{d}$$

What does this means? It describes how "exerting" or "draining" a particular action is. For example, imagine lifting a shopping bag of mass $10kg$ vertically by $1m$. This takes work, and exactly the following amount, given by the weight of the bag times the distance.

$$W= \vec{F}\cdot\vec{d} = Fd\cos{0}=mgd=10kg\times9.8ms^{-2}\times1m=98J$$

Energy is classically defined as the capacity of a physical system to do work, or in other words: as you perform work, you exchange energy for some physical effect by doing work. Or in other terms again, by exerting a force over a distance you convert energy into work.

In our example, you need to use some form of energy to lift the shopping bag. The quantity you need is exactly the amount of work we calculated.

What happens to this work? It's converted to energy again - to gravitational potential energy:

$$U_{final} = U_{initial} + W$$

or

$$\Delta U = U_{final} = U_{initial} = W = mgd$$

which is the classical definition of gravitational potential energy.

So in practice - we never see or measure energy directly. When energy changes form, it is called work, which we can measure. So work, in a way, is a "transport" concept for energy. Energy, on the other hand is like a "reservoir" of work in potential.

Why is energy a useful quantity? After all, work seems to be a more "fundamental" quantity from an experimental point of view.

The answer to this lies in the conservation law of energy. Work in itself describes a change in energy, so it's not a conserved quantity in itself unless you embed it in the more general concept of energy which is.

In fact, we can derive large swaths of classical mechanics using conservation of energy as a prime principle, together with the principle of least action.

Caveats

In more advanced theories, conservation of energy is a much more complicated matter and does not apply as simply as in the classical sense. For example in SR, energy can be converted to apparent mass and vice versa.

There are also very interesting mathematical properties of potential energy and its relation to forces and especially fields of forces. These explanations, though are way more abstract and mathematical - I assume you want an intuitive, instinctual explanation of what energy is.

If you are looking for the former please see this question.

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Yep, I've studied that, but the thing you're overlooking is that what is this quantity? Think about it. Where does this capacity comes from? What allows us to exert a force in the first place? What is that unknown x? –  Anna Jan 16 '11 at 10:48
    
@Anna: I don't understand what is the x you are looking for. Energy is what allows us to exert a force. With "zero" energy you can't exert a force. –  Sklivvz Jan 16 '11 at 10:56
    
As LM put it I was searching for potatoes only there are none to find. –  Anna Jan 16 '11 at 13:44
    
This is the Physics 101 version, and is a good answer to have on tap for beginners. –  dmckee Jul 15 '11 at 14:55

Energy is a convenient way to account for a system's ability to do useful work. Now there are certain modern qualifications we attach to energy, mostly that total energy of a closed system is always conserved (barring cosmological effects), which is now explained by the use of symmetry and Noether's theorem (as explained by other commenters).

To try to get to a more satisfying everyday notion of energy, it is best to resort back to the concept of useful work and accounting of it. We understand that we exert effort by lifting an object from the ground to the top of a table. For practical accounting, we need to understand how much effort was expended. It was these sort of accounting problems that led early engineers to the concept of energy.

Energy is useful to us only if it has the ability to change its current form into another form. If there is a way for this to occur, we can think in terms of potential energy if it is stored form is static (like a gallon of gas) or we can think in terms of kinetic energy if the energy is stored in relative motion (like a moving car).

If the energy is in a form that is useless to us, then we measure it in terms of entropy. A closed system will have a maximum value of entropy associated with it. If the entropy of a system is lower than its maximum entropy, then that system is called "far from equilibrium" and has a lot of usable energy internal to it, so there is a lot of work that the system can do internal to itself.

In everyday terms, we only think about energy it terms of the useful work that can be derived from it. So when we talk of selling energy in an energy market, what is being traded is a commodity that can be used to do work. The way that they energy is stored is different, but in any case, when we by a certain amount of energy, we expect it to allow us to accomplish certain tasks in a predictable way.

This is a somewhat simplified discussion, there is a lot more that can be added, and several clarifications that would be needed, but I don't know your level of understanding, so I want to see if this makes sense first.

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Well energy is only a part of something else that is much more important, and that is called "Action". There is a bumper sticker that reads "Physics is where the Action is". One of the most important quantities in the universe is Planck's constant, and it has the units of action. (Joule sec). The universe is designed in such as way that the no matter how things move about or change their structure, the action changes represents the efficiency of that change. Or to put it better, the probabilities of things happening or existing can be found out by accounting for this quantity called "Action". From the principle of least action we can derive conservation laws for quantities recognisable as energy, momentum, and angular momentum, it is a consequence of the symmetries involved in the action principle. (Lubos said it better).

Now we know that of these various kinds of conserved quantities, the energy one relates to the application of force (dear old Sir Isaac Newton figured that one out) and so the nice thing about energy is that it can be stored within structures by arranging a force to be stored. And so we have food and fuel. And chemistry. And evolution.

Chemists don't often use energy directly in calculations though, they also use a kind of minimisation principle that involves both energy and another useful quantity called entropy which is a measure of the amount of freedom of choice that we allow energy to have - this measure is called the "Free energy" and this is what allows you to calculate exactly what chemical reactions will occur and to what extent. And so it goes. This free energy is not conserved, the universe is winding down like a big clockwork spring.

The big bang (if you believe in that) is simply an earlier state when the energy density was very high. It doesn't necessarily mean that the universe was a single black hole of finite size. Quantum mechanics also tells us that there is a ground state to just about everything including spacetime, so if there is a vacuum there is a ground state energy. It is generally not worth trying to create a perpetual motion machine from the vacuum though, in spite of legendary pages of Youtube videos.

One thing that energy is not, is a kind of cosmic fluid. It is just a perspective on how change can occur - Einstein relativity theory teaches us that any kind of cosmic fluid including spiritual enlightenment fluids are impossible.

Cause effect means that there is something asymmetric that has happened. Asymmetry is closely associated with the idea of information, the problem of transferring information requires that there is a net displacement in space and time. The strictures on information transfer are the same as those on energy transfer, and we find that the movement of energy suddenly becomes about the movement of bits! So knowledge, energy and time and space must be considered in the same picture.

Firstly the connection betwen energy and time is very profound. We do not understand time fully but we know that our sense of real time requires there to be a meaningful succession of different states, take away clocks and time literally loses its meaning in such a context. For pure energy, every day is groundhog day - there is an intrinsic period associated with energy states but no sense of succession.

How does "real" time enter the picture? We know that there is an opportunity in spacetime for "timelike" intervals between events, in this zone a succession of events can be established that can maintain a cause-effect relationship in all reference frames. But that doesn't give us the clock itself. Systems also entangle when creating a time ordering, but this is all unclear.

The upshot is that just as there is a energy cost to doing things in the world, there is also a "cost" for systems to even exist in the world we know - aspects of that world must be unknowable. The converse is also true, if we encounter a system that is unstable and can be stabilised by releasing specific energy, then the specificity of that energy means that the time at which the event will occur is unknowable. We simply cannot consider a concept such as energy in isolation - without understanding the nature of concepts such as knowledge and time. Its a package deal.

Finally open systems through which energy flows are better able to maintain clocks and establish a time order, so life is a phenomenon associated with unstable energy flows.

The simplest explanation that I know as to why time runs in one direction, is that events in the reverse direction are "unobservable". I know that sounds like a tautology but if you show why they are unobservable then you have a better explanation. Likewise the positive observable energy could have in its negative counterpart a reason why it is unobservable, but now I really am not qualified to comment, I have already badly exceeded my limits.

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Wow, this is going to be a risky answer. However, I think you may be looking for an answer that is more conceptual than mathematical or philosophical, so here goes:

Energy is change. That is, energy is present if we observe relationships between objects and fields changing in some way from moment to moment.

Heat energy is just a very fine-grained version of change, expressed in the motion of many molecules whose average motion is negligible.

Potential energy is the possibility of future change. It requires the additional idea that change can be absorbed by some sort of spring-like capability, stored for a length of time, then released again in the future as explicit change.

This spring-like storage effect always seems to boil down to some form of stretching or compressing fields in ways they don't want to go. Thus winding and old style clock or watch captures explicit change (winding) in the form of interesting stresses on the bonds that hold metal atoms together. For nuclear energy the fields are different, but the concept of stretching or compressing them in interesting ways remains pretty much the same.

Finally, within the idea of potential energy lies an important hint at the relationship between energy and mass. Mass is in a quite real sense the ultimate form of potential energy. In matter, the energy of the past is so well safeguarded from release that it takes an extraordinary key -- specifically an equal quantity and type of antimatter -- to unwind it fully and release all of its energy. For matter it is the various unbreakable rules of conservation, such as charge conservation, that keep this energy bound up and unavailable. But if some lock-canceling antimatter does happen to show up, watch out!

Photons, the constantly-moving quanta of changing electromagnetic fields, come close to being the purest form of energy possible, with some quibbles I won't bring up here. Not surprisingly then, photons are the majority of what is released when matter and antimatter cancel each other's locks.

With that, I should emphasize again that this is not intended to be a mathematical or philosophical answer. All I'm trying to convey is that energy is all about change. It can be ongoing change, as when objects move in large unidirectional ways (kinetic) or microscopic multidirectional ways (heat), or it can be potential change. The latter is change that was captured and stashed away sometime in the past by stressing fields. The most extreme form of potential energy, one in which the release of the energy is safeguarded by profound conservation laws, is what we call matter.

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What is energy?
Energy is the capacity of a system to do work.

Where does it come from?
It generally comes from another source of energy, as in energy gets converted from one form to another.

Where does it ultimately come from?
That my friend is a question for MetaPhysics.stackexchange.com, which sadly doesn't exist as of now. You might want to hop over to Area51 with a proposal.

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-1 Energy is the capacity of a system to do work. What is Work? Change in Energy. Isn't this cyclical? –  Bernhard Heijstek Jul 30 '11 at 22:18
    
@Bernhard en.wikipedia.org/wiki/Energy –  abel Aug 8 '11 at 16:28

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