Do we not need a Zeroth Law of motion similar to the Zeroth Law of Thermodynamics? [closed]

Is the Zeroth Law of Thermodynamics necessary?

If so, then do we need a similar law for motion?

• how would you formulate a "zeroth law of motion" ? after all it is just a matter of keeping a count, keeping historical laws in same place in the count. In addition remember, laws are physical axioms that connect measurements to the mathematical models, i.e. they are necessary to make predictive calculations, not introduced randomly. Commented Jul 8, 2016 at 6:07
• Why do we need a zeroth law of motion? Is there a theoretical connection between Newton's Laws of Motion and the Laws of Thermodynamics? (Yup, that down-vote is mine!) Commented Jul 8, 2016 at 6:19
• @sammygerbil Thank you very much! I always appreciate those who explain their reason for downvoting. Commented Jul 8, 2016 at 6:22
• What law of motion was left out of Newton's enumeration? Assuming you find one, what makes it so important and fundamental that it needs to come before the other in the numbering scheme? Commented Jul 8, 2016 at 7:03
• @lucas The question asks two very different things: 1. Why there is no zeroth law of motion (and of some reason this should have anything to do with the numbering of laws of thermodynamics?) and 2. if the zeroth law of thermodynamics is necessary. If you choose one clear question to ask, I am sure it will be better received Commented Jul 8, 2016 at 9:40

3 Answers

At the risk of causing more debate let me summarize the already posted comments like this.

We don't have a zeroeth law of mechanics because we don't need to. Newtons three laws is enough. The zeroeth law in thermodynamics was established/defined after the first and second to allow for an unambiguous definition if when three bodies are in thermal equilibrium.

For mechanics no such extra clarification is needed. An analogous law could be that of Galilean invariance, which would be formulated in "themodynic style" as

If two particles are stationary with respect to a third, then they are also stationary with respect to each other.

But the needfulness is definitely debatable, as the comments point out.

• Thank you for your attention! May you please elaborate about needfulness of the law that you have stated in your answer in comparing with zeroth law of thermodynamics? I mean why is the zeroth law of thermodynamics necessary and that law (stated by you) isn't necessary? Commented Jul 8, 2016 at 8:11
• I like your suggestion for the Zeroth Law. It is in the "thermodynamic style" - but perhaps does not quite fit with Newton 1, 2 & 3 which are about forces as well as motion. So I wonder if you have to bring forces into the Zeroth Law somehow? Commented Jul 8, 2016 at 14:37
• @sammygerbil I agree that Mikael's proposal is interesting but indeed it does not "embodies" the dynamics. In my opinion one can have something similar to a zeroth law and it is stated in terms of accelerations. Commented Jul 8, 2016 at 22:23
• @Diracology In the spirit of introducing acceleration one can instead say "If two particles have constant velocity with respect to a third, then they also have constant velocity with respect to each other." Commented Jul 9, 2016 at 9:51

If you first discover two laws and call them "the first" and "the second", and you then later discover a new law, which you would actually consider more fundamental than the others, what then? Let's stuff it in front and call it "the zeroth".

If the same happened during the formulation of the laws of motion, then there might have been a zeroth law here as well. But it didn't.

So, the answer is: Of historical reasons.

• Thank you for reply! I didn't mean historical order. Commented Jul 8, 2016 at 8:12
• I think this is an unnecessarily short sighted view of the question. The point of the zeroth law of thermodynamics is that it was an initially unappreciated assumption necessary for the three laws to work. So it is zeroth in the sense of underlying the other three. The historical order is not the only interpretation of the term. There probably exists some equivalent notion for Newton's three laws e.g. Galilean invariance or even Euclidean geometry. Commented Jul 8, 2016 at 8:18
• @JohnRennie If the laws where defined and formulated at the same time, they would naturally have been named first, second, third and fourth. The mere reason that we have zeroth, first, second and third instead is that the laws were not formulated (agreed upon to be significant enough to be formulated as laws) in the right order. If it wasn't for the historical context, there would be nothing called a zeroth law - it would have been a first law. Commented Jul 8, 2016 at 9:25

We already have something in Mechanics playing a role similar to what the Zeroth Law does for Thermodynamics. It is embodied into the three Newton's law though. Let me first clarify what the Zeroth Law of Thermodynamics is and then I will show that there is something analogous in mechanics.

Two bodies in thermal contact can, in general, change thermodynamic variables of one another. We say they are in thermal equilibrium when their thermodynamic variables no longer change with time. The zeroth law consists on the empirical fact that if $A$ is in thermal equilibrium with $B$ and $B$ is in thermal equilibrium with $C$, then $A$ and $C$ are in thermal equilibrium. This is an equivalence relation which classify a set of bodies into subsets called equivalence classes. Each class is labeled by a number $T>0$ which we call temperature. So the zeroth law allows us establish thermal equilibrium just in terms of temperature.

Now consider a set of particles in the context of Newtonian Mechanics, i.e. the obey the three Newton's laws of motion. Apply the same force $F$ to each isolated particle and measure its acceleration. This would classify particles into equivalence classes. If $A$ has the same acceleration as $B$ and $B$ has the same acceleration as $C$ then $A$ and $C$ has the same acceleration. All particles belonging to the same equivalence class has the same measured acceleration. So we can assign a label $m>0$ to this subset. We call this label inertial mass.This allows us to establish the particles' responses to forces in terms of their mass.

• Sorry, in my comment to MF I overlooked your proposal for equivalence of masses. Is it really necessary to invoke a 3rd particle? Isn't it sufficient to say "If a force $F$ generates the same acceleration $a$ in two particles of mass $m_1$ and $m_2$ then $m_1 = m_2$"? Alternatively, since forces are the subject of Newton's Laws, why not "If two forces $F_1$ and $F_2$ generate the same acceleration $a$ when acting on the same mass $m$ then $F_1 = F_2$"? However, I don't see why either version is "necessary". It is not needed for the 3 Laws to work. Commented Jul 9, 2016 at 2:47
• Thank you very much because of your time and attention! Thank you very very much because of your understanding and goodwill! So, the last paragraph of your answer is the similar law for mechanics and I think so. But you didn't answer that is it necessary as the zeroth law of thermodynamics is necessary? If it is necessary, why it is not emphasized as a separate law? Commented Jul 9, 2016 at 4:13
• @sammy gerbil the spirit of my proposal is to give a mathematical meaning to mass, just as the zeroth law does to temperature. Notice that that in my last paragraph we did not know that $F=ma$ and then we find out that there exists a label which I called mass. Of course this is not strictly necessary to Newtonian mechanics, since when assuming the second law we know what mass is. It is just as exercise of rearranging newton's law in a logical way. Commented Jul 9, 2016 at 12:18
• @lucas As I mentioned this concept is already contained in the second law. Please have a look at my comment to sammy above. Commented Jul 9, 2016 at 12:20