Understand Noether's Theorem: Momentum Conservation & Exchange

In summary, Noether's Theorem explains conservation of momentum by showing that the symmetry under translation in the Lagrangian formulation of classical mechanics results in the conservation of momentum. This means that the description of a motion is independent of the chosen coordinate origin, and by taking into account all interactions, momentum is conserved. The exchange of momentum between objects can also be explained using this theorem. It is recommended to have knowledge of the Lagrangian formalism in order to fully understand the conservation of momentum.
  • #1
actionintegral
305
5
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?
 
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  • #2
actionintegral said:
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.


actionintegral said:
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.

actionintegral said:
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
 
  • #3
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.
 
  • #4
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.
 

FAQ: Understand Noether's Theorem: Momentum Conservation & Exchange

What is Noether's Theorem and why is it important?

Noether's Theorem is a fundamental principle in physics that states that for every continuous symmetry in a physical system, there exists a corresponding conserved quantity. This means that if a physical system remains unchanged under certain transformations, such as rotations or translations, then there must be a corresponding physical quantity that remains constant. This theorem is important because it provides a deep understanding of the underlying symmetries in nature and helps explain the conservation laws that govern many physical phenomena.

How does Noether's Theorem relate to momentum conservation?

Noether's Theorem states that for every continuous translational symmetry in a physical system, there exists a conserved quantity known as momentum. This means that if a physical system remains unchanged under translations in space, then the total momentum of the system must be conserved. This provides a mathematical explanation for why momentum is conserved in many physical processes, such as collisions and interactions between particles.

Can Noether's Theorem be applied to other conservation laws?

Yes, Noether's Theorem can be applied to other conservation laws in addition to momentum conservation. For example, it can also be used to explain the conservation of energy, angular momentum, and electric charge. This is because each of these conservation laws is related to a specific symmetry in a physical system, and Noether's Theorem shows that these symmetries are connected to conserved quantities.

Is Noether's Theorem applicable to all physical systems?

No, Noether's Theorem is only applicable to physical systems that exhibit continuous symmetries. This means that it cannot be used to explain the conservation of quantities in systems with discrete symmetries, such as crystal structures or quantum states. However, it is a powerful tool in understanding the symmetries and conservation laws in many physical systems.

How has Noether's Theorem impacted our understanding of physics?

Noether's Theorem has had a significant impact on our understanding of physics by providing a deep connection between symmetries and conservation laws. It has helped to explain various physical phenomena and has been used to derive important equations and principles, such as the conservation of energy and the laws of motion. Noether's Theorem also plays a crucial role in many areas of modern physics, including quantum mechanics and relativity.

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