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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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Related: physics.stackexchange.com/q/12559 –  Prahar Jul 19 '13 at 22:16
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marked as duplicate by Emilio Pisanty, Ben Crowell, Alfred Centauri, Manishearth Jul 20 '13 at 4:17

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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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Is this quantity always zero? –  Sebastian Henckel Jul 19 '13 at 22:13
No. It is the position of the center of mass at $t=0$. –  Prahar Jul 19 '13 at 22:15
There's a formal similarity to the expression for angular momentum. Not surprising as boosts are a kind of rotation. –  Dan Piponi Jul 19 '13 at 22:26
But if the center of mass is moving, then the position of center of mass at t = 0 will be moving away, i.e. changing, in the reference frame attached to the center of mass. –  Sebastian Henckel Jul 19 '13 at 22:26
No. The position at $t=0$ (at a specific time) does not change. The position changes as a function of time though. –  Prahar Jul 19 '13 at 22:37
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