Understand Noether's Theorem: Momentum Conservation & Exchange

[actionintegral]

I would like to understand Noether's Theorem.

Every layman's explanation of this theorem states that momentum
conservation results from
symmetry under translation. That is to say, momentum is constant as an
object moves.

But these descriptions don't discuss the exchange of momentum between
objects. That is what I would like to see follow from Noether's
Theorem. Perhaps someone can suggest a better
explanation of how Noether's Theorem demonstrates conservation of
momentum under exchange?

 

[cliowa]

I would like to understand Noether's Theorem.

Every layman's explanation of this theorem states that momentum
conservation results from
symmetry under translation. That is to say, momentum is constant as an
object moves.

I'm not sure you're understanding this correctly: symmetry under translation does NOT mean, that for the translation of the object
momentum is conserved. Symmetry under translation means, that if you move the origin of your coordinate system, as a result define a new one (obtained by translating the coordinate system you started with) and compare the formulation of physical laws, there will be no change: The description of some motion will NOT depend upon where you selected your coordinate origin (now of course I supposed you did not rotate the axis, but simply do a translation). Using this symmetry one can show that a certain quantity, which we now refer to as "momentum", is conserved.

But these descriptions don't discuss the exchange of momentum between objects.

Sure they do. You simply need to take into account ALL interactions, then there will be conservation of momentum.

That is what I would like to see follow from Noether's
Theorem. Perhaps someone can suggest a better
explanation of how Noether's Theorem demonstrates conservation of
momentum under exchange?

This is done in several textbooks. But may I ask what kind of background you have in mechanics? If you don't know the Lagrangian formalism, I would suggest you look into it.
Best regards...Cliowa

 

[actionintegral]

Cliowa:
>The description of some motion will NOT depend upon where you selected >your coordinate origin (now of course I supposed you did not rotate the >axis, but simply do a translation). Using this symmetry one can show that a >certain quantity, which we now refer to as "momentum", is conserved.

Thank you very much for responding to my question. You are saying that conservation of momentum results from freedom of choice in origin? I can't seem to make that jump. Please explain.

 

[cliowa]

actionintegral: Do you know the Lagrangian formulation of classical mechanics?
Let me know if you do, and I'll explain the conservation of momentum. You see, I don't know how to derive the conservation of momentum from symmetry under translation without using the Lagrangian formalism. I'm sorry.