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?
Answer
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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