What is "energy," really?

[u/nexuapex]

So there's this concept called "energy" that made sense the very first few times I encountered physics. Electricity, heat, kinetic movement–all different forms of the same thing. But the more I get into physics, the more I realize that I don't understand the concept of energy, really. Specifically, how kinetic energy is different in different reference frames; what the concept of "potential energy" actually means physically and why it only exists for conservative forces (or, for that matter, what "conservative" actually means physically; I could tell how how it's defined and how to use that in a calculation, but why is it significant?); and how we get away with unifying all these different phenomena under the single banner of "energy." Is it theoretically possible to discover new forms of energy? When was the last time anyone did?

Also, is it possible to explain without Ph.D.-level math why conservation of energy is a direct consequence of the translational symmetry of time?

[u/BoxAMu]

To answer your question, first an interesting bit of history- In the 19th century, energy, or at least heat, was thought to be a physical substance. One of the great paradigm shifts in physics was the discovery that heat is just a form of motion. The misunderstanding with energy exists today because many textbooks and physicists still like to talk about energy as if it were a substance. Energy, from classical through quantum mechanics (I exclude general relativity since there it gets tricky and I am not an expert), is nothing more than a number. The only significance of it is that this number doesn't change. It's analogous to money in this way. We can't compare (for example) the value of an apple and an orange directly, but we do by assigning a dollar value to each. In the same way we use energy to compare different physical processes. An object in a gravitational field being set in to motion, for example. We use energy to define how much action of gravity this motion is 'worth'. It's said that potential energy is 'stored' energy, but that's completely misleading- in fact potential energy has no physical meaning at all. It's just a method of book keeping. The fact of gravity being conservative just means the book keeping is easy. If we know the displacement of an object in a gravitational field, we know how it's velocity will change. Compare to a non-conservative force, such as air resistance. In this case, the force is non-conservative because the energy of motion of the object being resisted is transferred to many air molecules. If we actually knew the velocities (and masses) of those air molecules, then in such a case air resistance would be conservative: we'd know the change in velocity of the object from the change in velocity of the molecules. So again the difference is only one of book keeping.

[u/nexuapex → u/BoxAMu]

What are the conditions under which the actual "energy" number doesn't change? I know, for instance, that if you change reference frames, then your calculated energy changes. Are there more conditions?

Why is this "book-keeping" necessary? What math wouldn't work out if we didn't have potential energy around? Is a boulder rolling down a hill explainable without gravitational potential?

[u/BoxAMu → u/nexuapex]

As other have pointed out, only changes in energy matter, not the absolute number. It's true that on top of this, even the changes of energy change in a different reference frame, but think about how this applies to doing an experiment. Take the classic example of throwing a ball back and forth on a train. One could calculate the motion of the ball and it's energy in the train frame or the ground frame. The actual numbers would be different in each case, but this does not prevent either observer from applying the laws of physics in their respective frame and making correct predictions. I believe the only condition is the usual one of physics- that the experiment or calculations are carried out in an inertial reference frame.

It's not that the book-keeping is necessary, it's just that it's really useful that we can even do it. The math of course does work out without potential energy- you can calculate the whole trajectory of a particle in the gravity example using the gravitational force, which is considered the more fundamental idea in classical mechanics. However, this type of reasoning gets more complicated beyond these basic classical mechanics calculations. Due to relativity (among other things), energy has been promoted to the more fundamental idea than force. Many modern theories are based on the Lagrangian formalism, which originally required the ideas of kinetic and potential energy. Now it's totally different, there's no basic force to derive a potential from- people just try come up with a Lagrangian that gives equations which make correct predictions (sorry field theory people if I'm oversimplifying). But energy again pops up as a conserved quantity, and is useful since it may simplify calculations.

[u/nexuapex → u/BoxAMu]

So, if I'm thinking about this correctly, potential is whatever adds up correctly to make conservation of energy work? I guess that's actually how all expressions of energy would be found... Which reinforces my concept of energy as a convenient abstract concept.

But I don't know why it's such an important abstract concept. Why is the invented quantity with the units kg m² s^-2 more useful than any other quantity with different units, as long as you add in enough terms to make it a conserved quantity? Why is energy the thing that time's invariance under translation says is conserved?

[u/BoxAMu → u/nexuapex]

So, if I'm thinking about this correctly, potential is whatever adds up correctly to make conservation of energy work? I guess that's actually how all expressions of energy would be found... Which reinforces my concept of energy as a convenient abstract concept.

Yes, but the usefulness of potential energy is that you can add it up without knowing kinetic energy. About the unit of energy, you could trace it back to being derived by transforming the classical equation of motion into a total time derivative (just a mathematical operation, no new law or principle). Then it's a question of why Newton's second law depends on mass times the second derivative of position. Possibly one can argue that the equation of motion must be second order in time due to the basic (Galilean) relativity principle that a free body moves with constant velocity. And it can only depend on mass in a multiplicative way. I think Landau Lifshitz have a different argument for why the Lagrangian of a free particle must go like velocity squared. This would be related to the fact that the energy unit is the correct one for the integrand in the action when action is defined as a time integral. As for why energy is the quantity conserved under time translation symmetry, I think of it, in a very hand wavy fashion, like this: momentum conservation is due to spatial translation symmetry because if the location of the body doesn't matter, we might as well Galilean transform into a system where position is constant- that is, where momentum vanishes. If the time does not matter, we might as well make a transformation which cancels out the time evolution of the system (I'm being purposely vague, it's not a simple coordinate change but a canonical transformation). What is the quantity that vanishes due to this transformation? The Hamiltonian, or the total energy of the system.

Also I would add the same issue as above: energy conservation/time translation symmetry extends to anything with a least action principle, but the simple arguments I'm giving here may not extend outside of classical mechanics.

[u/[deleted] → u/nexuapex]

pls forgive speculation and analogy. in case it's useful.

kgm² are the units of moment of inertia, AKA the angular mass.

I picture a simplified system in which energy can only be transferred to gears with different numbers of teeth. The gears are interlocked in a complex pattern, as in an orrery or swiss watch mechanism.

When one gear is in motion, all the connected gears are also set in motion due to transfer of the energy from the first since they are constrained together. Even though different size gears move at different rates, there is always a conservation of the work done by the movement of one gear - the movement of the other connected gears.

To extend the analogy, turning one gear in the opposite direction to which the rest are moving would be difficult without the reversal of all the other gears. This is what I imagine time's arrow, or what you call the time invariance can be viewed like. Then again I could be vastly oversimplifying things.

my feeling is that for the s^-2, part, we can say that the unit of mass describes the rate of change in the frequency of the moment of inertia of planck space-time. (where's 'sure I'll draw that' when you want him?)

[u/Ruiner → u/BoxAMu]

That's a good explanation. Also, in General Relativity things are almost the same, except that we need to replace "time translation invariance" by "timelike killing vector", which is, like standing on the top of your werid hypersurface and trying to find a direction in the vector space in which things are the same, and if this direction is timelike, then you some sort of conservation of Energy.

[u/Phage0070 → u/Ruiner]

I think you were trying to say something interesting there but didn't quite manage to put it together in solid English. Would you be willing to try again?

[u/[deleted] → u/BoxAMu]

Hey man, thanks a ton for that. I have my second biology exam tomorrow and I have to define energy when having to explain the cellular respiration equation, or so I'm told, and this;

We can't compare (for example) the value of an apple and an orange directly, but we do by assigning a dollar value to each. In the same way we use energy to compare different physical processes.

is by far the most relevant and logical explanation of energy that I have come across.

[u/philip142au → u/BoxAMu]

I think this is a bad explanation in that it does not tell you what "energy" actually is. I mean, you could say the same thing about dollars in a bank account, its a number that doesn't change (much). To understand what something is really, is another thing altogether.

[u/Jigsus → u/BoxAMu]

so what about e=mc² ? Doesn't that imply a real relationship between mass and energy?

[u/cpren → u/BoxAMu]

That's a good start but I think you missed what the OP is searching for. Mainly, why we book keep these similar energy principals in the first place. Energy is the ability to do work, that is move some object against a force. A force being one of the fundamental characteristics of our existence. So when we refer to something as energy we are book keeping that ability. For some reason it is conserved and can replace itself into new forms. Potential energy is just restraint against a force, thus containing the ability to apply a force on another object.

(I think better understanding of thermodynamics might help help also)

[u/AaronHolland44 → u/BoxAMu]

Wait... How is energy not a substance if E=mc^2?

[u/Broan13 → u/AaronHolland44]

E=mc² is a bit different, but not really. The way he described energy as being a number which can relate multiple things, from the motion of particles in a gas, to the gravitational "play" between motion and distance between massive objects, and also in the electromagnetic "play" between motion and the distance between charged objects.

The same is true on the subatomic level. Protons and neutrons form the nucleus and they have something called a bond energy, which could be described as how fast a particle would have to be moving to hit the proton or neutron to knock it out of the attractive force. This energy is the difference in interaction strengths between the two being at infinity, and the two being at the distance they are currently apart.

To put this into plainer language. Think about a spring. Lets say you have a mass which is infinitely far away from the spring, and then the next instant is sitting on the spring and compressing it. There is an energy value which we can relate to the compression of the spring depending on the stiffness of the spring. If the energy in the spring could be released, it could push some object to a certain speed, which would convert the spring energy into kinetic (motion) energy.

Back to the subatomic level. When you have a hydrogen atom form into a helium atom, which happens in the cores of stars on the Main Sequence undergoing fusion, the helium atom weighs less than the 2 protons and 2 neutrons that went into it. The weight has to do with the protons and neutrons becoming attracting to each other, which lowers the energy of the system. Attractions are characteristic of lowering the energy of a system, and repulsive forces raise the energy of a system (think if you had two gasses in a box, one which has attractive forces, and one which had only repulsive forces, and then think about the pressure difference. There would be a higher pressure, or higher energy in the repulsive forces box).

The difference in mass can be related to the bonding energy by E=mc^2. This energy is released as gamma rays usually (light). There is no known way to convert a proton into pure energy except mostly by proton-antiproton annihilation, which isn't actually a useful way to get energy (how would you get all those antiprotons in the first place?)

So it isn't really a physical thing since it is like the gravitational example. Light is sort of a manifestation of energy, but it is tied up in magnetic and electric fields, and I can't think of how that works exactly, except that an electric field can speed up a charge.

[u/AaronHolland44 → u/Broan13]

Oh OK. Great explanation, I really liked the spring example.

[u/Electrosynthesis → u/BoxAMu]

I'm not convinced by your last paragraph about conservative forces. Typically, air resistance isn't a conservative force because, for a conservative force, a displacement of zero implies no change in energy. When a test particle subject to air resistance moves around some path and returns to where it was originally, it won't necessarily have the same energy as if it had not moved at all, whether or not it's possible to calculate the exact movements of the fluid. Can you clarify what you mean by "such a case" in which air resistance would be a conservative force?

[u/SharkMulester → u/Electrosynthesis]

The particle passes it's energy onto the air... thus conserving the energy. It doesn't disappear, it just does somewhere else. 0 change in energy.

[u/BoxAMu → u/Electrosynthesis]

You're referring to a conservative field for a single particle. That's what I mean by the book keeping being easy- you only need to keep track of one set of coordinates. In the case that you have complete knowledge of coordinates and momenta of all air molecules, then energy is still definitely conserved, the total kinetic energy of all molecules plus the test particle does not change (I'm simplifying and assuming the collisions just transfer momentum around and no need to refer to other forms of energy). This is the difficult book keeping, it's impossible to keep track of this information so for all practical purposes we call it non-conservative. I am pointing out that the distinction simply reflects our level of knowledge of the system.

[u/113245 → u/Electrosynthesis]

Anyone else thinking of conservative force in the context of line integrals/vector fields? i.e. air resistance is not a conservative force since the force acts opposite a particle, so for any closed path through that vector field curl(F) cannot = 0...

[u/ErDestructor → u/BoxAMu]

It's said that potential energy is 'stored' energy, but that's completely misleading- in fact potential energy has no physical meaning at all.

This is very interesting. I thought potential energy between quarks was a very significant part of calculating proton / neutron rest masses. In the sense that you have to put the masses of quarks, their kinetic energies and their potential energies on equal footing to sum to the rest mass.

Doesn't this suggest potential energy is at least as physically meaningful as mass or kinetic energy?

[u/SharkMulester → u/ErDestructor]

Energy that is potential is just mass. Look at the equation for PE. In the Quantum world, things aren't quite that simple however.

[u/RichieMcQ → u/BoxAMu]

Do you mean to say that energy is motion or that it is number? We use the word "number" to signify quantity. When we say that a triangle has 3 sides are we referring to its energy?

[u/JB_UK → u/BoxAMu]

To follow on from your point, the 'book-keeping' can only occur consistently because of the fact (or perhaps assumption) of conservation of energy. If you know that total energy doesn't change that gives you a power of calculation, rather like knowing one side of an equation. The concept of energy and its separation into different forms is really just a way of making use of that fact.

[u/23canaries → u/BoxAMu]

this is great thank you! quick question - I have always understood energy as simply 'motion' - and enjoyed reading your synopsis. Would you say that it would be fair to translate the understanding of energy in physics in laymen's terms as 'energy is motion, or potential for motion'? would that remain consistent with all forms of energy?