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Noether's Theorem states that if a Lagrangian is symmetric for a certain transformation, this leads to an invariant: Symmetry of translation gives momentum conservation, Symmetry of time gives Energy conservation etc.

The Galilean principle stating that all reference frames that move with constant speed relative to each other are equivalent is also a symmetry principle: Setting up a physical system that is identical to the original except for a constant velocity (boost) added will have the same behaviour.

Shouldn't there be an invariant associated with this symmetry? If yes, what is that invariant?

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The conserved quantity corresponding to boost symmetry is $$ \int d^3 x (P_0 x_i - P_i t) $$ which is the relativistic analogue of $x_{CM} - v_{CM} t$, the position of the center of mass at $t=0$. It is quite a useless conserved quantity, and that is why people don't talk about it.

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  • $\begingroup$ Is this quantity always zero? $\endgroup$
    – yippy_yay
    Commented Jul 19, 2013 at 22:13
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    $\begingroup$ No. It is the position of the center of mass at $t=0$. $\endgroup$
    – Prahar
    Commented Jul 19, 2013 at 22:15
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    $\begingroup$ No. The position at $t=0$ (at a specific time) does not change. The position changes as a function of time though. $\endgroup$
    – Prahar
    Commented Jul 19, 2013 at 22:37
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    $\begingroup$ Right. That's the reason this conserved quantity is useless. Its simply telling us that the center of mass at time $t=0$ is preserved. But that is obviously true! $\endgroup$
    – Prahar
    Commented Jul 19, 2013 at 23:01
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    $\begingroup$ @SergioPrats we only work in natural units here. $\endgroup$
    – Prahar
    Commented May 20, 2023 at 19:45

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