- #1

gionole

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- TL;DR Summary
- Emma noether’s conservation law

I have been learning emma noether’s theorem where every symmetry results in conservstion law of something, sometimes momentum, sometimes angular momentum, sometimes energy or sometimes some kind of quantity.

Every explanation that I have listened to or learned from talks about homogeneity of the space and says that if we have a system of particles and we shift each one of them by some infinetisemal distance(we dont have to shift all of them by same amount, but could be we shift first particle by 2epsilon, where second particle by 3epsilon). This all depends what their lagrangian is and whether this shift gives the possibility that dL=0.

While I understood all that, I have a hard time imagining this in real world usecase. In real life, we dont care about shifting them, they just move without our interaction.

I want to solve the following: imagine a gun which pulls a bullet and bullet hits a box. As we know from school physics, impulse is conserved - total momentum before pulling was 0 and the final momentum of the gun and bullet must be 0 as well. So we write p1+p2=0. I want to see/prove that lagrangian does not change for my example because if we have momentum conservation, there must have been symmetry such as dL=0. I dont even know how to start. Before pulling, L seems 0 as no kinetic and potential energy exists in the system but do I still have to write them in the formula ? Would appreciate a nudge or even better, thorough analysis.

Every explanation that I have listened to or learned from talks about homogeneity of the space and says that if we have a system of particles and we shift each one of them by some infinetisemal distance(we dont have to shift all of them by same amount, but could be we shift first particle by 2epsilon, where second particle by 3epsilon). This all depends what their lagrangian is and whether this shift gives the possibility that dL=0.

While I understood all that, I have a hard time imagining this in real world usecase. In real life, we dont care about shifting them, they just move without our interaction.

I want to solve the following: imagine a gun which pulls a bullet and bullet hits a box. As we know from school physics, impulse is conserved - total momentum before pulling was 0 and the final momentum of the gun and bullet must be 0 as well. So we write p1+p2=0. I want to see/prove that lagrangian does not change for my example because if we have momentum conservation, there must have been symmetry such as dL=0. I dont even know how to start. Before pulling, L seems 0 as no kinetic and potential energy exists in the system but do I still have to write them in the formula ? Would appreciate a nudge or even better, thorough analysis.