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I've been reading Arnold's book on Classical Mechanics. I understand that most "classical" forces such as gravity, spring are supposed to be Galilean invariant. But what if I start a rocket, and then at certain time, adjust the thrust vector of the rocket differently (i.e. control.). This rocket as a mechanical system is obviously not time invariant because the "F" depends explicitly on time. Is there something I am not understanding here?

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Yes, you're misunderstanding what "invariance" means.

Galilean transformations relate observations between different reference frames.

What it sayys is that "physics are the same" in any intertial frame.

This doesn't mean everybody measures exactly the same results. This means that physical laws are the same.

If I am observer $A$ and I see the rocket accelerates at $t_0$, suppose there's another observer $B$ who starts counting when I measure $t_0$.

If I see the rocket crashes at $t_F$, $B$ will see the crash at $t_0-t_F$. YEs, the time in both frames is different, but not random: they are related by Galilean transformations.

That's what transformations do: they relate measurements from different observers.

If they must be related it's because they are different, but not randomly different, because the physical laws hold the same way, and therefore we'll get the same conclusions.

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  • $\begingroup$ I understand that the results are not randomly different. However, in chapter one of Arnold, he concludes that the "force field" of the mechanical system cannot depend explicitly on time. For example, the gravitational force in a n-body system does not depend on time explicitly (only through the position of the particles). But here, we have an artificial force that does depend on time. Perhaps because this artificial force is not a kind of fundamental mechanical law. $\endgroup$ – Shuheng Zheng Jul 23 '18 at 5:22
  • $\begingroup$ That's correct, it refers to "natural forces". Time dependence is broken when energy is not conserved. $\endgroup$ – FGSUZ Jul 23 '18 at 9:29

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