What is the precise statement in Newtonian mechanics regarding the physical laws in all inertial frames?

Question

The first postulate of the theory of special relativity states that physical laws are the same in all inertial frames.

(1) what exactly "same" means here?

(2) Almost all the text books say that this statement holds in Newtonian mechanics also. But, by and large, they don't seem to discuss this type of statement while discussing Newtonian mechanics. But they wake up to mention this while discussing relativity.

Is there any formal statement in this regard by Newton? I want to know what exactly Newton stated in this regard.

Answer 1

Is there any formal statement in this regard by Newton? Want to know what exactly Newton stated in this regard.

It is not generally recommended to try to learn Newtonian mechanics directly from the writings of Newton. Newton had the first word on his theory, but not the last word. Many issues that were unclear or confusing in Newton's day have been clarified by subsequent researchers. So it is generally preferable to learn from modern didactical sources.

However, Newton's Principia did have several relevant quotes:

First, Newton believed in absolute space and time, so his conception of the principle of relativity was more of a practical equivalence. In the definitions he said

Absolute, true, and mathematical time, of itself, and from its own nature flows equably without regard to anything external, and by another name is called duration : relative, apparent, and common time, is some sensible and external (whether accurate or unequable) measure of duration by the means of motion, which is commonly used instead of true time ; such as an hour, a day, a month, a year.

Absolute space, in its own nature, without regard to anything external, remains always similar and immovable. Relative space is some movable dimension or measure of the absolute spaces ; which our senses determine by its position to bodies ; and which is vulgarly taken for immovable space

But then in the corollaries to his laws he states:

COROLLARY V. The motions of bodies included in a given space are the same among themselves, whether that space is at rest, or moves uniformly forwards in a right line without any circular motion

So he did recognize the principle of relativity as described by his predecessor, Galileo. Although Newton believed in the existence of absolute space and time, he recognized that it was not relevant to his laws. However, again, the understanding of this principle was refined considerably in the centuries since Newton, so a modern textbook would be a better source than the Principia.

Answer 2

Newton's 2nd Law is invariant under Galilean transformations (we forget about Lorentz transformations now as we are doing non-relativist classical mechanics). This can be easily shown by considering the Galilean transformations:

$$x'= x + vt$$

$$y'= y$$

$$z'= z$$

$$t'= t$$

This transformation helps us go frame with zero velocity to an inertial frame with a velocity of $v$. Now, since we are talking about inertial frames, $v$ should be constant. In our original frame, the equations of motion are

$$F = ma$$

In our Galilean frame, we can find the equations of motion by taking the derivative of $x'$ two times with respect to $t'$ (which means $t$). Since $v$ is constant, in the second derivative it disappears and we are left with only the second derivative of $x$. Hence,

$$a'= a$$

Hence, Newton's equations of motion are invariant under Galilean transformations and means that the same laws apply in every inertial frame.

However, all is not well if we apply these transformations to Maxwell's equations. We are then left with weird equations that are dependent on the velocity. This means that they are not invariant under Galilean transformations which causes problems as if we accept Galilean transformations as the correct transformation, there must be special frames of reference where Maxwell's equations hold as they are. This, at the time, was thought to be aether which then was disproved by Michelson-Morley experiment.

Einstein, instead of assuming aether existed, assumed that the Maxwell's equations must be same at every inertial frames and then used the Lorentz transformations to show that Maxwell's equations are indeed invariant under these transformations.

Answer 3

The 'same' means the mathematical form of an equation is unchanged when it is written in all inertial frames (IRFs). However, in Newtonian Mechanics, only the laws of mechanics, especially $F=ma$, have the same form in all IRFs while in Special Relativity, by the first postulate, all laws of physics are the same in all IRFs. The range of physical laws invariant under the transformations between the inertial frames is therefore enlarged in Special Relativity. In fact, they do discuss the invariance of laws of mechanics in all IRFs during the exposition of Newtonian mechanics, that is the Galilean principle. Einstein took a further step beyond that. That is remarkable as at that time, there is no certainty about the existence of a relativity principle for classical electrodynamics as the Maxwell equations are found to be not invariant under Galilean transformations, to which the response of the many physicists is the presupposition that there is a preferred reference frame (ether frame). In the end, it turns out that a relativity principle exists both for mechanics as well as electrodynamics, but the correct transformation between IRFs is the Lorentz transformation, instead of the Galilean transformation so the laws of mechanics has to be reformulated.

Answer 4

In his Principia Mathematica, Newton did not make a formal statement concerning the equivalence of all reference frames. He did distinguish between relative and absolute motion, which suggests that he thought the latter was a meaningful concept, but he went on to write words to the effect that detecting absolute rest might not be possible. You can read an English translation of the relevant sections here...

https://en.wikisource.org/wiki/The_Mathematical_Principles_of_Natural_Philosophy_(1846)/Definitions