Lagrangian symmetry for the bullet and gun

[gionole]

TL;DR

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.

 

[PeroK]

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 suspect you have misunderstood the concept of spatial symmetry. You say you understand all that, but your example shows that you haven't.

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.

If you understand the concept, why do you not even know where 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.

You should study some more appropriate examples of Lagrangian mechanics. And post your working if you get stuck. There must be some in your textbook.

 

[gionole]

I suspect you have misunderstood the concept of spatial symmetry. You say you understand all that, but your example shows that you haven't

Can you give me a pointer what I dont understand, so that I can try to work on it first ?

 

[PeroK]

Can you give me a pointer what I dont understand, so that I can try to work on it first ?

You haven't posted anything to show what you do and don't understand. You have simply picked a bizarre example of an explosion fuelled by the release of chemical energy. Is that a typical example of Lagrangian mechanics? Where does the release of chemical energy fit into what you have learned about Lagrangian mechanics?

 

[gionole]

Well, As I stated, in the symmetry, we say: "would we get dL = 0 if we shift the system by some epsilon. Well, I'm trying to see in real world use case where without our interaction(without our shift), Lagrangian doesn't change. Symmetry is something you can make on the system so that action doesn't change which in turn means L doesn't change. I'm not sure what it is you think that I don't understand till now.

But most of the time, things happen on its own. Car hits another car and ofc, I wasn't sitting in it and calculating Lagrangian when it would give no change in 2 different coordinates of space. It just happened on its own. I'm trying to see in such cases that conservation law of momentum trully comes from the symmetry(dL=0). To me, this seems a legit question and I just picked an example of gun/bullet and want to see why momentum is conserved in terms of Lagrangian. @PeroK

 

[PeroK]

Well, As I stated, in the symmetry, we say: "would we get dL = 0 if we shift the system by some epsilon. Well, I'm trying to see in real world use case where without our interaction(without our shift), Lagrangian doesn't change. Symmetry is something you can make on the system so that action doesn't change which in turn means L doesn't change. I'm not sure what it is you think that I don't understand till now.

But most of the time, things happen on its own. Car hits another car and ofc, I wasn't sitting in it and calculating Lagrangian when it would give no change in 2 different coordinates of space. It just happened on its own. I'm trying to see in such cases that conservation law of momentum trully comes from the symmetry(dL=0). To me, this seems a legit question and I just picked an example of gun/bullet and want to see why momentum is conserved in terms of Lagrangian. @PeroK

I suggest you find an example problem from your textbook and post it in the homework forum. Note that "I don't even know where to start" is not acceptable in terms of the Homework guidelines:

https://www.physicsforums.com/threads/homework-help-guidelines-for-students-and-helpers.635513/

Here are some common mistakes to avoid:

• Don't simply say "I have no clue," "I have no idea where to start," or "I'm completely lost."
  These don't qualify as attempts. Instead, it suggests you haven't put much effort into reading and understanding your textbook and lecture notes, going over similar examples, etc.

 

[gionole]

I suggest you find an example problem from your textbook and post it in the homework forum.

I am curious about gun/bullet. Are you saying this cant be explained with lagrangian ? If so, why does the momentum get conserved there ? Well, you might explain this in terms of newtonian but trying to see it with Lagrangian.

I will wait maybe someone else can understand this.

 

[vanhees71]

The point of Noether's theorem is that momentum conservation must hold for all of physics as long as it's describable by Newtonian (or special relativistic) physics since the space time model of both Newtonian ans special-relativstic physics implies that space (in SR for an inertial observer) is homogeneous, i.e., spatial translations must be a symmetry of all equations describing closed systems. So not only mechanics but also the chemistry behind the explosion of your gun powder (which is an application of non-relativsitic quantum mechanics when viewed from a fundamental point of view) implies that momentum conservation holds true. Of course this cannot be described by Lagrangian mechanics.

The point of Emmy (sic!) Noehter's theorem is that based on the given symmetry group of Newtonian spacetime you can pretty much derive the form of the Lagrangian for a closed system of particles. Assuming in addition that all fundamental forces are pair forces only you get
$$L=\sum_{k=1}^{N} \frac{m_k}{2} \dot{\vec{x}}^2 - \frac{1}{2} \sum_{j=1}^N \sum_{k \neq j} V(|\vec{x}_j-\vec{x}_k|).$$
It's a bit more simple to use this Lagrangian to derive the conservation laws from the space-time symmetry, i.e., the full group of Galilei transformations. The spatial translations are given by
$$\vec{x}_k'=\vec{x}_k+\delta \vec{a}, \quad t'=t, \quad \delta \vec{a}=\text{const}.$$
Since the $\delta \vec{a}$ are arbitrary, and since the Lagrangian is obviously invariant under these transformation, the analysis a la Noether leads to the conservation of the total momentum
$$\vec{P}=\sum_{k=1}^N m_k \dot{\vec{x}}_k.$$