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In classical mechanics, we know that laws of physics are invariant in Galilean transformations of the form:

$$ x' = x -vt$$

My question is does shifting to rotating frame also count as a Galilean transformation? If it is not so, then are laws of physics invariant when shifting between rotating frames?

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    $\begingroup$ No, since one of these frames is necessarily non-inertial: en.wikipedia.org/wiki/Galilean_transformation $\endgroup$
    – Roger V.
    Commented Sep 22, 2021 at 11:12
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    $\begingroup$ Well, to be picky, the laws of physics are invariant. But the expression of them when moving between rotating frames is not. The thing about Galilean transformations is that they leave the expressions of the laws invariant. $\endgroup$ Commented Sep 22, 2021 at 21:19

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The answers to your questions are no and no.

A full Galilean transformation will consist of a translation and a boost. In 2D (the minimum number of dimensions needed for a rotation, you can write a Galilean transformation as $$\left(\begin{matrix}t'\\x'\\y'\end{matrix}\right) = \left(\begin{matrix}t+\Delta t\\ x+\Delta x-v_xt\\y+\Delta y-v_yt\end{matrix}\right)$$ where the coefficients $\Delta t$, $\Delta x$, $\Delta y$, ​$v_x$ and $v_y$ are constant: they do not depend on $x$, $y$ or $t$. While a rotating frame will have something like $$\left(\begin{matrix}t'\\x'\\y'\end{matrix}\right) = \left(\begin{matrix}t\\ \cos(\omega t)x-\sin(\omega t)y\\\cos(\omega t)y+\sin(\omega t) x\end{matrix}\right).$$ There is no way to put this transformation in the form of a Galilean transformation.

If you go from an inertial frame to a rotating frame (or to any frame not related by a Galilean transformation) then Newton's laws will no longer apply.

There is however a larger framework in which the laws of physics are preserved for arbitrary frame transformations, namely, the theory of general relativity, but this is probably besides the level of the answer you expected.

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    $\begingroup$ The second question, as stated, has an affirmative answer, and one needs to consider it in the context of general relativity. The laws of physics can be formulated in any reference frame (classical dynamics, Maxwell's equations, etc ). They are "invariant" when formulated appropriately. For example the homogeneous Maxwell's equations read $dF=0$, and this holds whatever the reference frame you use. $\endgroup$ Commented Sep 22, 2021 at 12:28
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    $\begingroup$ @AlanGarbarz, that is correct. I edited the question to mention this. $\endgroup$
    – Andrea
    Commented Sep 22, 2021 at 12:32
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    $\begingroup$ There are expressions for fictitious forces (linear acceleration, centrifugal and coriolis) that you can insert into your coordinate transformation to make Newton's laws hold. Or, you could adopt the view that Newton's first isn't a special case of Newton's second, but rather a prescription telling you what "straight line at constant speed" looks like in your particular coordinate system, and then Newton's second law correctly tells you how to use forces to calculate deviations from these motions. And then Newton's laws still hold. $\endgroup$
    – Arthur
    Commented Sep 22, 2021 at 22:04
  • $\begingroup$ Newtons law applies perfectly well in a rotating frame without needing to learn any general relativity. Just use classic rotational kinematics to express the acceleration vector in the rotating frame. If $\vec{r}= x^i \vec{e}_i$ then just use $\dot{\vec{e}}_i= \Omega \cdot \vec{e}_i$ when differentiating. Where $\Omega(t)$ is the usual angular velocity tensor for the rotating basis (with respect to some fixed basis). $\endgroup$
    – J Peterson
    Commented Jun 3, 2023 at 6:05

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