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Our internal energy article sez:

It excludes the kinetic energy of motion of the system as a whole and the potential energy of position of the system as a whole, with respect to its surroundings and external force fields. It includes the thermal energy, i.e., the constituent particles' kinetic energies of motion relative to the motion of the system as a whole.

But what about bulk rotational kinetic energy? Taken literally that seems to be included in the second sentence. But it doesn't seem thermodynamic at all. If you accelerate an object (without heating it) you don't increase its internal energy, and I would think the same should apply to rotational acceleration (say, of a flywheel). --Trovatore (talk) 00:05, 26 February 2025 (UTC)[reply]

If you rotate a flywheel (or any other object) fast enough, it will fly apart. That seems to me to suggest that some internal energies might have been generated by the externally applied force. {The poster formerly known as 87.81.230.195} 94.8.123.129 (talk) 12:56, 26 February 2025 (UTC)[reply]

Whether an object gets heated or not with respect to time is not relevant. Classically, at any moment in time there is a rotating reference frame for which a system's bulk rotational kinetic energy is simply zero. Then note that its total internal energy, which includes thermal energy and stresses, are equivalent to its total invariant mass. Unlike fictitious forces, its mass is not fictitious, hence its bulk rotational kinetic energy is not part of its internal energy [(with respect to its restmass)], but thermal energy is, of course. Modocc (talk) 14:51, 26 February 2025 (UTC)[reply]

But a rotating frame is not an inertial frame. I wouldn't think that would count? In any inertial frame, the bulk rotational kinetic energy is the sum of the particles' kinetic energy (at least, the part based on the rotation) with respect to the center of mass of the system. --Trovatore (talk) 18:20, 26 February 2025 (UTC)[reply]

Often, the literature does not respect the universe's absolute rotation and neither does GR's localized frames. When people head west their internal energies increase (eg. onboard clocks), and only if rotation is absent for which local curvature is zero everywhere does one really have something worth talking about as I see things. That said, in the classical setting (v<<c), the KE of non-inertial frames are still worth consideration. Modocc (talk) 19:02, 26 February 2025 (UTC)[reply]

OK, I think this discussion is not going to get to the point I'm interested in, so let me show the rest of my hand. The question arises from a very old claim, in a fortunately obscure page, that systems at absolute zero have to be "still". I think that's nonsense; an example would be a fast flywheel, which, notwithstanding its high rotational kinetic energy, can still be cooled arbitrarily close to absolute zero. The reason, I think, is that the kinetic energy isn't random and therefore not thermal (this raises interesting questions about the foundations of statistical mechanics which I have still not fully understood, and have not really seen much discussed).

A discussant at that talk page did raise an interesting point that if the object is actually at absolute zero, then you would be in an eigenstate of the Hamiltonian, therefore time invariant, which I suppose in the case of the flywheel means you would need to lose all information about the angular position of the flywheel.

There are lots of places we could go from here. I think coherent vibration is also not thermal, though it would thermalize eventually (except maybe in a superfluid or something?) whereas the flywheel's rotation would not.

Anyway, can anyone clarify these issues? --Trovatore (talk) 19:14, 26 February 2025 (UTC)[reply]

"Maxwellian energy distribution" is a term for kinetic randomness. Following up on what I said before, a gyroscopic instrument with two counter-rotating parts comes to mind with respect to an object that has additional invariant internal energy (and whether it is considered thermal energy depends on ones definition(s)) with respect to any reference frame. Our article on absolute zero states "In the quantum-mechanical description, matter at absolute zero is in its ground state, the point of lowest internal energy." In other words, its energy is not zero. Maybe that helps, maybe not. Modocc (talk) 21:35, 26 February 2025 (UTC)[reply]

I can also not emphasize enough that internal energy is simply mass [when KE=0] per Invariant mass: "The invariant mass, rest mass, intrinsic mass, proper mass, or in the case of bound systems simply mass, is the portion of the total mass of an object or system of objects that is independent of the overall motion of the system". Thus I am confident you are correct that the systems' velocities are not a factor. But what is unclear to me is what is considered to be "the point of lowest internal energy". Modocc (talk) 02:15, 27 February 2025 (UTC)[reply]

No, I don't think it's true that internal energy is the same as mass. Internal energy is a specifically thermodynamic concept, and I'm fairly sure only the "random" part counts, whatever that means. A big part of the point of this question is trying to figure out just what it does mean.

Our internal energy article says that it's determined only up to an additive constant; only changes in the internal energy are well-defined, not the exact value. That wouldn't be true if it were the same as invariant mass. --Trovatore (talk) 18:15, 27 February 2025 (UTC)[reply]

The well-defined changes are time-dependent, however the invariant mass is determined by the system's proper rest frame, for which all KE is zero, for any event. Our article on Thermal energy gives a brief description of internal energy as "the energy contained within a body of matter or radiation..." and note there are other concepts for thermal energy. Modocc (talk) 19:49, 27 February 2025 (UTC)[reply]

Well, we aren't talking about invariant mass. We're talking about internal energy. I am not persuaded that they are the same thing. (Note in passing that invariant mass definitely includes the bulk rotational KE, though that's not the point I'm primarily interested in here.) --Trovatore (talk) 20:49, 27 February 2025 (UTC)[reply]

The article states "...the portion of the total mass of an object or system of objects that is independent of the overall motion of the system." Bulk rotation is an overall motion and I took a look at Rotational energy, but it is silent on the matter. Anyway, the angular velocity of macro-systems is arbitrary and gets more complex at the atomic level. In practice, such as for the Earth's spin contribution to its KE (and not mass), is perhaps ignored, but can be calculated via a classical approximation which I used above in my first reply. Consider curve balls that are thrown different curvatures, k, at different moments in time and ignoring external events there exists local proper reference frames such that the ball's KE scalar, including its spin, is zero. If someone asked me what the ball's intrinsic mass is I would not include its rotational KE for any of its rotational energy when thrown in my calculation. It's not ignored, it's just computed separately. Recently I purchased Schwartz's Quantum Field Theory and the Standard Model. The Zero-point energy article touches upon QFT and Thermodynamic temperature has a section on internal energy and absolute zero that touches upon your query. P.S. AI's take.... Modocc (talk) 22:53, 27 February 2025 (UTC)[reply]

Our article on Thermodynamic temperature has more details and it refers to Zero-point energy and I've not read it yet. Modocc (talk) 03:26, 27 February 2025 (UTC)[reply]

Disclaimer: not a physicist so take with grain of salt: (also this could stand to be moved to the Science desk if one wants more attention; I will refrain from doing so myself) (edit: ignore I am a dum-dum and can't read)

As I understand it, thermodynamics-wise, temperature is fundamentally defined in terms of entropy. The lower a system's temperature, the less total entropy, and vice versa. Entropy is "randomness": it can be thought of as the total number of possible ways a system's internal state can be arranged, which all produce the same observables. A periodic system, like a flywheel or pendulum, has highly regular, organized, predictable motion—so, that periodic motion contributes little net entropy to the system. Real systems can't truly reach absolute zero, but they can get arbitrarily close.

Going from there into some connected topics: quantum field theory models space as containing various fields, which at every point in space are modeled by quantum harmonic oscillators. QFT models all particles as excited states of these oscillators. If we consider the textbook toy model of the particle in a box, the particle behaves as a periodic oscillator, with various vibrational modes it can have, each representing a different energy level the particle can have. Even in the ground state, the lowest-energy state the system can have, it still has internal energy distributed across its degrees of freedom, which simply can't be "gotten rid of" somehow ever (besides altering the system so it changes into a different system), any more than you can make 2 apples fill you up more than 5 identical apples if you can just somehow "try hard enough".

It helps to understand that the foundational Big Idea of quantum mechanics, is that various natural properties can only inherently have distinct discrete (countable) values or quantities, quanta: they are quantized. This is in direct contradiction to "old-school" classical mechanics, which assumes natural properties are continuous: capable of taking on an infinitely-divisible, smooth and continuous range of possible values, like the real numbers. That applies for instance to our particle in a box: its fundamental degrees of freedom can only have various distinct, countable values. If they all were at their lowest-energy values and the system were in its ground state, they simply have "nowhere else to go but up"; the system can't somehow do a limit break and awaken its latent hidden powers and smash through to achieve the "even lower than ground state" somehow. --Slowking Man (talk) 08:49, 28 February 2025 (UTC)[reply]

Suppose the system under consideration contains a steam engine using coal as fuel, with a fully loaded bunker. The system is capable of doing mechanical work, which, I think, in this context can be considered thermodynamical work, and the chemical energy in the fuel is internal energy. Suppose the system is updated; now it contains an electrical engine powered by electric batteries. Again, I think the electrical energy of the batteries is internal energy of the system. Next, the system is upgraded again; these old-fashioned forms of energy storage are replaced by a flywheel. Then, again, its rotational energy is internal energy. A helpful way of thinking about this is:

The total energy of a thermodynamic system can be divided into external and internal energy.^[1] In a formula, ${\displaystyle E_{\text{tot}}=E_{\text{ext}}+E_{\text{int}}.}$

Since the rotational energy of the flywheel is obviously not external, it is internal. I think it ultimately comes down to an accounting choice – do we choose to ascribe the energy to the system, viewed as a thermodynamic system, or is this unhelpful. If it is not reasonable, given some system, to account for any bulk rotational kinetic energy as external, it is internal energy. But for a bulky electrical battery located at the South Pole, its bulk rotational kinetic energy at a whopping spin of 1 turn per day should be considered purely external.  ​‑‑Lambiam 13:11, 5 March 2025 (UTC)[reply]

The article on the third law of thermodynamics makes the case that matter cannot be cooled to absolute zero. Vacuum fluctuations prevent it too. Modocc (talk) 19:57, 5 March 2025 (UTC)[reply]

Okay. The article says KE is not included as internal energy (which I had assumed was simply following in the tradition of defining mass as I understood it), but since that is not accurate, how should we revise the article (with references) so internal energy includes a bulk rotational KE ? Modocc (talk) 00:14, 6 March 2025 (UTC)[reply]

The article repeats the claim but adds a caveat: "Internal energy does not include the energy due to motion or location of a system as a whole. That is to say, it excludes any kinetic or potential energy the body may have because of its motion or location in external gravitational, electrostatic, or electromagnetic fields. It does, however, include the contribution of such a field to the energy due to the coupling of the internal degrees of freedom of the system with the field. In such a case, the field is included in the thermodynamic description of the object in the form of an additional external parameter." It seems to be hedging its description quite a bit... Modocc (talk) 01:00, 6 March 2025 (UTC)[reply]

I'm also annoyed, because (KE+PE) is conserved with orbits of any eccentricity and size, thus the so-called external energy that is being excluded has to be, in my view, in some sense internal energy too independent of the potential source(s) and I've always thought that was obvious, but apparently not... Modocc (talk) 01:12, 6 March 2025 (UTC)[reply]

Of course, if the references are lacking in generality and applicability, and better ones are not forthcoming, it stays written "as is". Modocc (talk) 02:56, 6 March 2025 (UTC)[reply]