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Galilean invariance and conserved quantities 
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#1
Nov409, 03:13 AM

P: 47

Hi I have a simple question
what is the conserved quantity corresponding to the symmetry of galilean invariance? and Lorentz invariance? cheers M 


#3
Nov409, 06:08 AM

P: 46

As I understand it (but I may be wrong), not every symmetry in the laws of physics leads to a conserved quantity, only those that leave the Lagrangian unchanged. Hence there is no conserved quantity associated with Galilean or Lorentz invariance. 


#4
Nov409, 06:34 AM

Sci Advisor
P: 1,261

Galilean invariance and conserved quantities
Momentum for Galileo; momentum and energy for Lorentz. Angular momentum follows from rotational invariance.



#5
Nov409, 06:37 AM

Mentor
P: 16,386

Gallilean and Lorentz invariance describe how translations through space, time and angle affect (or, in this case, don't) the Lagrangian.



#6
Nov409, 06:56 AM

HW Helper
PF Gold
P: 1,961

*Formally, this can be obtained from Noether's theorem. The killing vector of lorentz transformations in the xt plane is μ = x∂_{t} + t∂_{x}, and the consrved quantity is dS⋅μ = xE + tP_{x}. 


#7
Nov409, 02:52 PM

Mentor
P: 16,386

This may be a little advanced, but fundamentally what is happening is that Lorentz boosts, translations and rotations form what is called the "Poincaré Group". These transformations are, of course, intertwined  a rotation and a translation is equivalent to a (different) translation and (different) rotation, and if you boost something, after a period of time it's now translated.
You can do the same thing with Newtonian mechanics, and I am pretty sure what you get is the Euclidian Group. (I am less sure because this doesn't describe the world we live in, so I haven't thought about it very hard). In both cases, the conserved quantities associated with these symmetries are energy, momentum and angular momentum. However, the algebraic expressions for these quantities (i.e. the way they are interrelated) are different, because the symmetries are different. Quantities that are invariant (which is different than being conserved) are also different. One can pick off these conserved quantities one by one, but it's often more efficient to exploit the whole symmetry in one go. If this is more confusing than enlightening, just ignore it. 


#8
Nov409, 06:26 PM

Sci Advisor
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#9
Nov409, 11:02 PM

P: 47




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