Does energy exist, or is it just a relationship between other fundamental things?

Could anyone tell me what energy really is? I searched for it, and some people said that energy doesn't exist physically and it is not fundamental, but it is a relationship between other fundamental things, and there is not energy by itself, so it should be related to something else.

So could anyone help me understand it?

I think term energy has slightly different meanings in different branches of physics. I prefer to think of it as the quantity that is conserved due to time-invariance of the equations of motion, i.e. my notion of energy is related to the Largangian mechanics (https://en.wikipedia.org/wiki/Lagrangian_mechanics), and energy is basically the Hamiltonian.

• +1 because I like the idea of symmetry. But it might be too advanced for the OP in this case. – S V Jun 7 at 1:44
• Also, as far as I know energy is not a conserved quantity in General Relativity. But we still use it there. – S V Jun 7 at 1:45
• I must say that my GR is not as good as I would like it to be, but I am ok with SR. What I like there, in SR, is that energy can be obtained from some basic arguments about the Lagrangian and Noethers Theorem. Same trick works in electromagetism. – Cryo Jun 7 at 6:52
• I'm sorry, but can someone please point out in which branch of physics that energy has a different meaning? Also, the Hamiltonian of a system will not be its energy in the case of open systems, velocity dependent potentials, or time dependent coordinate systems. – user400188 Jun 7 at 9:48
• Different meaning: I would imagine most applied disciplines may care less about time-invariance and more about the actual equation. Example, I would expect optical scientists, to talk about the optical energy density in the as ($\epsilon_0E^2/2+B^2/(2\mu_0$) even when there are non-linear effects that break time-invariance (e.g. a signal beam will not experience time-invariant environment if there is a strong pump-beam changing the refractive index of the medium). Sure this is not quite correct, but amongst themselves these scientists will know what is going on. – Cryo Jun 7 at 14:27

"What energy is" is a philosophical question. It turns out its impossible for science to talk about what "reality" is like, other than to say that science forms models which have an "energy" term in them and they seem to be pretty good predictors. If you're interested in that line of reasoning, I highly recommend looking into the philosophy of science.

However, we can find energy as a meaningful thing in our models. One of the foremost ways of modeling our reality for scientific purposes is in the idea of "action." The idea of action is formed from this question:

Given a path that a system may take from state 1 to state 2, what path does it take? In other words, if someone throws a ball (state 1) and later someone catches it (state 2), what did it do along the way?

What we have noticed through decades (and even centuries) of observation is that you can phrase this as a minimization problem (more formally, a stationary problem, which is a wider concept, but minimization is easier to think about). We noticed that you can define a function for a system, called the Lagrangian, such that if you integrate it across the entire path the system takes through time, it's at a minimum (at a stationary point, in the complete version). This integration across all time is called the "action" for the path taken. Interestingly, this function works for all configurations the problem might take on. You can find an action describing that ball flying through the air which works not only for your thrower and catcher as they are, but a thrower and catcher anywhere on the field!

This is a very abstract concept, and it's okay if it doesn't make 100% sense when you first work with it. But what makes it important was that we came to this concept of Action without invoking forces or energy, or any of those other terms. We just pointed out that the paths objects take tend to be the one which minimizes action across the entire path. Or, more generally, we determined that you could find a Lagrangian for which the "correct" path is always found by solving this optimization problem that minimizes the action. Actually figuring out a Lagrangian function which does this is another matter, what matters is that one exists!

Now should you accept this declaration that there always exists a Lagrangian function such that the correct path of objects is always found by minimizing the action? Perhaps not. Don't take my word for it. Science is an empirical art, not a purely mathematical art. It's the observation of scientists over the centuries that say "We can always describe the motion of particles this way!"

Now once you have this, we then can invoke one of the most powerful mathematical formalisms in all of physics: Nother's Theorem. This theorem shows that if you have a system which is described by this optimization problem, this minimization of action, and it has a continuous symmetry, then there is some conserved value. This is neat because it takes some very abstract mathematical concepts, like continuous symmetries and action, and ties it directly to the idea of conserved values.

One continuous symmetry that's very important to physicists is time symmetry. Basically that says that the laws of physics don't change over time. We're only looking at laws that stay constant forever, from the big bang to however we end. The laws of physics being the same at all times is formally termed as "time symmetry." If you do something at one time, or do it 5 seconds later, the laws of physics will be the same in both cases.

This continuous time symmetry must have an associated conserved value, by Nother's theorem. We call that conserved value "energy." And if you actually go through all the fancy Calculus of Variations, you find that the thing that we conserve when we conserve energy is precisely what we told you was "energy" all along.

So, down in the weeds, that's the neat nature of energy. Energy is the thing that must be conserved if the laws of physics are immutable over time.

• Cort, I really like your approach. An explanation of energy should not be such as to scare someone from learning and thinking more about it. – Bob D Jun 7 at 15:12

Well, maybe it's not important to know what energy is, as much as it is to know what energy can do. The typical definition of energy is that energy is the capacity to do work. If something has energy, it has the capacity to move things, lift things, heat things, and so forth. The other thing that is important to know is that energy can never be "created" or "destroyed". In other words, total energy is always conserved. It simply morphs into different forms as it is transferred between things in the form of either heat or work.

Hope this helps.

• But in General Relativity energy is not conserved, right? So saying it can't be created or destroyed might be slightly misleading. – S V Jun 7 at 1:47
• @SV I don't know that that is correct. It's my understanding it is an open question. Wiki on the subject states "In general relativity, energy-momentum conservation is not well-defined except in certain special cases". It concludes with the statement "The theory of general relativity leaves open the question of whether there is a conservation of energy for the entire universe". As far as I am aware, Einstein himself didn't reach that conclusion. In any event I don't believe it is "misleading" to omit something that is largely a matter of conjecture, as with so many things in cosmology. – Bob D Jun 7 at 13:37
• I believe this idea arises from the fact that the universe appears to be expanding with a virtually constant energy density (necessary for spacetime to be flat), but this does face the problem of whether the universe is infinite or not. – S V Jun 7 at 14:53

The previous answers are more than adequate expressions of the idea of 'energy' as a tool in physics--but I think you are asking more than that. You may wish to investigate the original proposition of 'vis viva' or 'living force' by Leibniz. What you will often hear in such discussions of 'energy', as was said here, is that physics or physicists cannot say ultimately what 'energy' is, and are constrained to this practical application of the very idea of energy to the notion of its conservation, which while it may in general be true (that a measurable quantity of energy functions as a parameter of continuous symmetry), can say no more about it.

However, this view is not necessarily correct, that it is somehow impossible to say what energy truly is; on the contrary, it is only the approach of physics which impedes if not prohibits the consideration and resolution of that question, and many others. And you are quite intuitively correct to suppose that energy may eventually be defined at least as the function of a more fundamental relation.

Energy is ubiquitous in reality, and as we are told in QFT, even the vacua of its quantum fields are continuously energetic: viz. there is no state which can realistically be said to be without energy. How then can we explain this; and if indeed energy is a function of motion, what ultimately is moving? One is entitled to ask then, since energy is classically and in any real context only the motion of a force over a spatial distance--i.e. the spatial integration of the force--, what force may be conceived to act ubiquitously in this way to imply ubiquitous energy? And it is reasonable to say that orthodox physics is effectively stalled at an impasse between the perspectives on this question embodied in QFT and GRT, essentially because it prohibits itself from formulating precisely and explicitly such a question; the arcane musings of M-theory notwithstanding.

In order to address such a question however, what is required--and what the sciences have come to regard as anathema-- is first simply to propose a priori the existence of a unitary universal substance or fabric, and to understand that because we too are comprised of it, not only is it impossible to discern its ultimate nature, but that due to this very constraint, its essence becomes effectively equivalent to the ubiquitous force holding the universal unity and entirety together. All of reality then is effectively only the dynamic action of this singular force in various contexts of its more-or-less local distribution; and one proceeds to consider how it is that the existence of such a force acting everywhere at once permits the world of matter and space, perpetually inflated, as we perceive it.

At the same time, there is no need to dispense with the magnificent theories of QED and the mathematical methods perfected in the structure of the SM; for it is only the more fundamental physical basis of these theories which we seek, and which, if a viable model of such a foundation is indeed revealed by the correct pursuit of the reasoning suggested, is quite amenable to the same rigorous calculus.

To reiterate then, were one to suppose that the universe is composed entirely and exclusively of a singular inviolate substance, it would be quite impossible for us, also comprised of it, to determine its ultimate nature. Since however we are embroiled in its dynamics, we are therefore capable of, if also constrained to, an understanding of its principles --operating continuously all around and within--, commencing with some meditation on the self-evident postulate, more properly an innate perception, that a unitary force must bind that inviolate universal unity or substance together.

That is, as the ancients understood it, the universal entirety, in whatever it eventually consists, must by very definition hold itself together; so that with this self-evident knowledge, a priori, one is first required to inquire how it is able to do this without collapsing upon itself; to which the answer is that, since this 'cohesive effect' is universal and must operate at or through every conceivable spatial point, that force of 'cohesion' must be acting in disparate components at any and all such points; the holding together one way tends to balance approximately the holding together in the opposite direction at any point. When imagined with respect to a common archetype--say the purely geometrically-defined interval on a universal cubic lattice (comprising infinite mutually 'cohesive loci' in perfect equilibrium)--, these disparate or asymmetrical components of a 'cohesive force' are effectively vibratory in origin and in reality always slightly out of phase with each other within a range defined by half that interval; and any 'field theory' attempting to describe these vectors of cohesive force and their relations is of necessity bound fundamentally by constraints of such a 'phase relation'.

What results from this interaction between points defined by disparate resultants in such a 'cohesive force' then is a ceaseless motion as these components (arising eventually from the interior dimension of 'spatial depth' itself) act one against the other throughout universal reality in an inherent tendency to (impossible) equilibrium or 'cohesive symmetry'. That motion we call 'energy', whether in the relative vacuum of space or in the midst of what we describe as 'matter' by virtue of our similitude with it, and consequently our capacity to perceive some condition of resonance with it (within a definitive range); and the distribution of that 'cohesive force', of which all ostensible forces are aspects, constitutes the properties of reality, mass, charge, momentum, and energy when that is understood as an oscillatory principle in that distribution. The reciprocal cubic lattice model of 'cohesive loci' mentioned is capable of representing these basic quantities for the electron and photon (for example) as functions of linear and planar area distribution of such cohesive force; and is further capable of explaining in those terms the essential physical basis of QED.

So, you see, the most fundamental and indeed the only real self-evident or axiomatic truth which can be conceived in the human mind is that of the inviolate universal unity of existence; from which it follows that it is comprised of a singular substance, motivated by a singular cause, process and will (or 'metaphysical will' as Schopenhauer puts it); and indeed that it is the singular 'cohesive force' operating exclusively and distributed throughout it which is its most fundamental conceivable property. 'Energy' is one function of its 3-D distribution.

That of course, along with the rest of the above, is only my opinion; of which, if you are at all interested, you may find more elaborate expressions if you look.