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There have been several questions posted on the Stack Exchange along the lines of "What is energy?"   Many them have been marked as duplicates by moderators.   Among the surviving ones are: What is Energy made of?, What Is Energy? Where did it come from?, and Is energy really conserved?.   Most of the answers posted to the questions at the above links have been written by advanced practitioners, and invoke mathematical abstractions, quantum and relativistic concepts, and Noether's thereom.

However, it is almost certain that most people seeking answers to these questions are students in introductory courses.   I’ve been there.  We learn some fundamental principles and are pleased by how they fit and reinforce each other as a whole.   But we are often left somewhat confused about energy.   It seems that energy must be fundamental too, and should interlock with the other principles, and yet the educational messages we receive about it appear to have contradictions that leave energy oddly disjointed from mechanics

Part of the problem is the notion that energy exists as a separate entity somewhere.  That notion is stubborn baggage that has been attached to the energy concept since its inception in the time before science understood the atomic structure of matter, and the fundamental forces in nature.   The sense that energy exists as an entity is reinforced in our education by such standard statements as “There is only a certain, unchanging amount of energy in the universe,” “Energy can neither be created nor destroyed,” “Energy changes forms,” etc.   And while statements that energy is only a calculation can be found in many textbooks and other publications, those statements don’t penetrate effectively because they virtually always gloss over the next logical question—calculation of what, exactly?

The reification of energy in our educational messaging together with the lack of clear, explicit teaching of what energy actually is can cause confusion.   This is especially true at the intersection of intuitive understanding and mathematical understanding.   I remember puzzling over questions like, Is work energy?   Why do we need the idea of work?  If work can be negative, then energy can be negative?  The kinetic energy theorem says that $F\cdot d$ can change kinetic energy.   Why is that only true for kinetic energy?  How is kinetic energy different than other energies?

I believe that this is unfortunate, and that many people are left believing that energy is a mystery beyond their grasps.   Is it really?   I don't think so.   I decided to post this self-answered question.   I anticipate that after reading the answer the reader's reaction may be: Well,this makes sense, but if it is correct then why isn’t energy taught and written about this way everywhere?   All accepted ideas are hard to change in science-and therefore also in science education, and that's as it should be.   But besides the long-standing traditional coverage of energy in science education, there is innate satisfaction in the view that energy exists as an entity, but is unknowable except as a mathematical abstraction.   After all, this is true for most of the truths revealed in relativistic and quantum physics.   And the mathematical model of energy does work perfectly well.   The answer below may not be popular with those who have made their peace with energy in this way.   But for those who long to have both the math and the phenomena the math describes to make sense, here comes the resolution, and its correctness should be self-evident.

Can energy be defined and explained in Newtonian terms like force and motion?

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    $\begingroup$ Ehm. physics.stackexchange.com/a/217530/520. Not that my answer is in anyway special, this is discussed in every introductory textbook; it's just that you have to internalize a large group of related concepts before it becomes in any way "obvious". It's concept from the 18th century, but don't kid yourself: those guys were no slouches. $\endgroup$ – dmckee Aug 23 '18 at 3:32
  • $\begingroup$ Your third paragraph: whats wrong with.the idea that energy exists as a separate entity sonewhere (am looking at you integral E^2)... I mean thats one of the major points of field theory. $\endgroup$ – lalala Aug 26 '18 at 12:14
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Energy is as per consensus in the field defined as "The ability to do work".

And work is force-over-a-distance, as explained. This definition tells us that energy is an invented concept that covers factors that can end up doing work:

  • Something that moves can do work when impacting something else. So let's say that it carries motion eneegy, or kinetic energy.
  • A taught spring can be released and push in something else, thereby doing work. So let's say that elastic (potential) energy was stored in this spring.
  • Etc.

Energy is from this perspective just a term used to represent "the ability to do work". A "name" for that specific ability. There is no reason to think of it as more than that

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  • $\begingroup$ You have said almost exactly what I said in my answer-that energy is a concept which was invented to explain the effects of force exerted through distance. The main difference is that I give an exhaustive and original explanation of why this can be seen to be true by a physics beginner, and you fall back on the claim that it is already the consensus in the field. But if that were true, how could the great Feynman famously state that we have no idea what energy is and then not face a cacophony of challenges? And why do students come here and earnestly ask what the heck energy really is? $\endgroup$ – D. Ennis Aug 28 '18 at 20:56
  • $\begingroup$ The first three physics books I just picked up from my shelf, a high school text by Serway and Faughn, a college text by Serway alone, and a high school text by Zitzewitz, give no definition of energy at all--they hand wave around it. The fourth one, a high school freshman physical science text, does give the definition you cite. I'm not seeing the consensus. $\endgroup$ – D. Ennis Aug 28 '18 at 21:00
  • $\begingroup$ @D.Ennis Well, the consensus is that this definition is good enough to continue working with energy generally. That doesn't mean we understand much of it - but this definition seems good for our purpose. That you haven't found it clearly stated or defined like that in textbooks may not mean much - when they for the first time derive energy formula such as $K=\frac12 mv^2$ and $U=mgh$, they will most likely use this energy-is-the-ability-to-do-work definition as the premise. If they have a different definition, then I would be very interested in hearing so. $\endgroup$ – Steeven Aug 29 '18 at 5:41
  • $\begingroup$ ". . . this definition is good enough to continue working with energy generally. That doesn't mean we understand much of it ..." Right. Hence the need for the answer I posted. But there seems to be something akin to a religious belief in energy that's causing people to reject my answer out of hand without attempting to comprehend it. That's disconcerting, but interesting. $\endgroup$ – D. Ennis Aug 29 '18 at 13:54
  • $\begingroup$ @D.Ennis Your answer says the same thing and explains just as little. You pick a lot of scenarios of force-over-distance and choose to call them all energy. Why do you call them that? Because they show force-over-distance. So it explains nothing - you just use energy as an umbrella term for all such scenarios. I agree with that. Everybody does. That is the consensus. Consensus around that term for the force-over-distance ability. My answer is my own way of explaining that same thing in fewer words. $\endgroup$ – Steeven Aug 29 '18 at 14:29
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Yes, energy can be defined and explained in classical terms.   That which we call energy is force exerted through distance, and it has no existence apart from force exerted through distance.

Here's an example supporting the first part of the above claim (the second part will be dealt with below). If we compress a spring, that action will be accomplished by exerting force on it through a distance against a restoring force.   If we want to think and visualize more deeply, the restoring force is a sum of repulsive electrostatic forces between the particles of the spring as they resist being pushed closer together.   What we have in existence before us is still nothing more than a material spring and forces--nothing has been added.   Force was exerted through distance to change the configuration of the material spring, and, if we allow it, a restoring force will be exerted through distance as the spring returns to the configuration where its internal forces are in equilibrium.   It is conventional to say that there is energy stored in the compressed spring.   If we choose to say that, fine.   We can quantify the force and distance by taking their product, and call it energy.  There are many benefits to using that model.   However, what we are calling energy is quite clearly force exerted through distance.

In fact, if we examine the details in any reference to energy, including thermal, kinetic, radiant, chemical, wave, and nuclear energies, we will always see that we are referring to effects associated with force exerted through distance.   How do we give an object kinetic energy?   We exert a force on it through a distance.   What does it mean for an object to possess kinetic energy?   It has no meaning whatsoever except that the object can exert a force through a distance.   What is the thermal energy of an object or substance?   It is the kinetic and potential energies of the object's particles-their capacity to exert forces through distances on each other.   How is thermal energy transferred?   By force exerted through distance, which is the one and only way that energy is ever transferred.   How do we get energy into an electromagnetic wave?   By forcing electrons back and forth through a distance.   The evidence is overwhelming.  It constitutes what Galileo and others have called a familiar empirical regularity.   It is inescapable: That which we call energy IS force exerted through distance.

The second part of the initial claim above was that energy has no existence apart from force exerted through distance.  My defense of that proposition will probably elicit complaints about philosophizing in a physics post. But I ask the reader to note the commitment to empiricism and forgive this.  If we take the standard of what exists to be ourselves, then other objects and materials have existence as well.   Since we were toddlers we have accumulated countless experiences and observations which lead to the emperically familiar regularity that other objects and people exist in the same sense that we exist.  Another empirically familiar regularity that we come to recognize through countless experiences and observations is the existence of space, or distance between objects.   Thirdly, we come to recognize the existence of force in countless experiences and observations as we interact with objects in space.  Finally, through countless experiences and observations we come to recognize the sequencing and the cadence of events, or time.  As far as we know, as far as we have any way of knowing, objects, space, force and time are the only things that exist in the same sense that we ourselves exist.  Everything else that we can imagine as existing must be composed of objects, space, force, and/or time.  Velocity has no existence apart from space and time.  Mass has no existence apart from objects, force, space and time.  Energy has no existence apart from force and distance.

I would answer objections that might arise from modern physics considerations by noting that Einstein's theories and Schrodinger's equation necessarily began with the adoption by those men of the concept of energy as it existed in their times.   Neither of them proclaimed a new theory of energy.   Therefore, since what they referred to as energy was force exerted through distance, then this classical nature of energy is "baked in" to all references to and equations using the term energy in Relativity, Quantum Mechanics and QFT.

It is not intended here to demean the energy model.   The conservation of energy principle and the powerful calculations associated with it are indispensable in science, because the phenomena under study often involve too many forces acting on too many objects to measure and tally directly.

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    $\begingroup$ "energy [] is an invented concept" All concepts are invented, by definition. "has no actual existence" Gravity and inertia are direct consequences if energy that are readily observable. Etc. $\endgroup$ – my2cts Aug 23 '18 at 6:27
  • $\begingroup$ We are exposed to a lot of confusion and contradiction about energy in our k-16 (+) educational years. Many of us make our peace with it somehow. For instance, I used to think that that energy was the prime mover of the universe. When the truth comes along, it can offend our sensibilities. Don't just react defensively-think hard about it. $\endgroup$ – D. Ennis Aug 23 '18 at 10:47
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    $\begingroup$ I think this answer would be more acceptable without the concerns expressed above. I feel like your response is bordering on the line of philosophy. You do a great job in discussing energy as an ability to apply a force over a distance, but I don't think we can really talk about existence. For example, one can argue that space does not actually exist. It is just a concept we use to organize what we see around. Or you can argue energy does exist (@safesphere has one such example). $\endgroup$ – Aaron Stevens Aug 23 '18 at 11:05
  • $\begingroup$ People can agree to work with a definition of existence within themselves, but at the end of day, at least with how you are qualifying existence, it remains subjective. You are basing it on "existent entities which we can experience and with which we can interact with directly" (This definition can used on either side to say whether or not God exists for example), and I can see valid arguments where we experience and interact with energy directly. I think this answer would not get as much backlash if you drop the philosophy and stick with the physics. $\endgroup$ – Aaron Stevens Aug 23 '18 at 11:08
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    $\begingroup$ You may find it interesting what Richard Feynman says about energy in the Feynman Lectures in physics. "It is important to realize that in physics today, we have no knowledge of what energy is. We do not have a picture that energy comes in little blobs of a definite amount. It is not that way. However, there are formulas for calculating some numerical quantity, and when we add it all together it gives "28"'—always the same number. It is an abstract thing in that it does not tell us the mechanism or the reasons for the various formulas". $\endgroup$ – Bob D Aug 26 '18 at 0:10

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