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I'm just starting to learn about symmetry. I understand that physical laws are symmetric under translation and consequently (is it not a consequence?) under uniform velocity in a straight line.

I see that the laws are symmetric under rotation. but are the laws also symmetric under uniform rotation about a fixed axis? You surely can tell if your lab is under a constant rotation about an axis, can't you?

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The invariance of the laws of physics under transformations between frames that move with constant speed with respect to each other is not a consecuence of the invariance under translations. This kind of transformations is called a boost in general and in the case of non-relativistic mechanics the specific name is Galilei boost.

The spacetime symmetries in non-relativistic mechanics are generated by three types of transformations: translations, rotations and Galilei boosts. Changes between systems of reference that rotate uniformly with respect to one another are neither of any of these types nor a combination of some of them and therefore they're not symmetries of physical laws (if we accept the premises).

More physically, a frame that rotates uniformly at angular velocity $\omega$ with respect to an inertial one sees a radial force $F=mr\omega^2$ on any body with mass $m$ and distance $r$ to the rotation center. Another inertial force appears: the Coriolis force. They're both present in the rotating frame even when there are no forces appearing in the inertial ones. Thus, the laws of physics are not invariant under the transformations we're considering.

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Yes you can. Uniform rotation about a fixed axis creates centrifugal acceleration so your coordinate system won't be inertial anymore. Acceleration means change in speed OR in direction. Uniform rotation equals a constant change in direction. You will be able to feel this rotation as you feel a force when sitting in a car which is moving on a circle.

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