Identifying conserved quantities using Noether's theorem

In summary, the conversation discusses the conserved quantities for three different potentials: U(r) = U(x^2), U(r) = U(x^2 + y^2), and U(r) = U(x^2 + y^2 + z^2). For each potential, it is determined that energy and linear momentum in the y and z direction are conserved, but there is uncertainty about the conservation of angular momentum. It is suggested that the lagrangian is invariant under rotations around certain axes, leading to the conclusion that angular momentum is conserved in those directions. However, further discussion reveals that this conclusion may not be entirely accurate and that angular momentum is actually conserved in all directions due to the independence of the potential
  • #1
Snydes
5
0
I've been asked to find the conserved quantities of the following potentials: i) U(r) = U(x^2), ii) U(r) = U(x^2 + y^2) and iii) U(r) = U(x^2 + y^2 + z^2). For the first one, there is no time dependence or dependence on the y or z coordinate therefore energy is conserved and linear momentum in the y and z direction are conserved. I'm having trouble with the angular momentum. It would seem to me that since there is only a dependence on x, that the lagrangian would be invariant under rotations around the y and z-axis and thus angular momentum in those directions is conserved. Similar approach for the other two potentials. Can anyone give me any more depth or background on this, what would be a concrete way to approach these types of problems so I can be more confident in my answer
 
Physics news on Phys.org
  • #2
Snydes said:
since there is only a dependence on x, that the lagrangian would be invariant under rotations around the y and z-axis and thus angular momentum in those directions is conserved.
You are halfway there, but draw the wrong conclusion. Which axis can you rotate around without changing x?
 
  • #3
Orodruin said:
You are halfway there, but draw the wrong conclusion. Which axis can you rotate around without changing x?

If I rotate around the x axis, the x coordinate should remain unchanged then should it not, therefore the angular momentum in the x direction is conserved?

For the second case, would this mean that we are solely dealing with the xy plane, thus a rotation around the z axis leaves the system invariant and angular momentum in the z direction is conserved?
 
  • #4
Correct.
 
  • #5
Orodruin said:
Correct.

What about the third case? my first though would be that since U(r(vector))=U(r^2(scalar)) then it is independent of any rotational angle (theta, phi) then angular momentum is conserved for all space. Is this true?
 
  • #6
Snydes said:
What about the third case? my first though would be that since U(r(vector))=U(r^2(scalar)) then it is independent of any rotational angle (theta, phi) then angular momentum is conserved for all space. Is this true?

Angular momentum is conserved in all directions (rather than "all space"). A general rotation has three angles (see Euler angles).
 
  • Like
Likes Snydes

1. What is Noether's theorem and why is it important in science?

Noether's theorem is a fundamental principle in physics that relates symmetries in a system to conserved quantities. It is important in science because it provides a powerful tool for understanding the underlying laws and principles of nature.

2. How is Noether's theorem used to identify conserved quantities?

Noether's theorem states that for every continuous symmetry in a physical system, there exists a corresponding conserved quantity. By analyzing the symmetries present in a system, physicists can use Noether's theorem to identify the conserved quantities and their associated laws.

3. Can Noether's theorem be applied to all physical systems?

Yes, Noether's theorem can be applied to all physical systems, including classical and quantum systems. It has been used to successfully explain conservation laws in various areas of physics, such as mechanics, electromagnetism, and quantum field theory.

4. What are some examples of conserved quantities that can be identified using Noether's theorem?

Some examples of conserved quantities that can be identified using Noether's theorem include energy, momentum, angular momentum, and charge. These quantities are conserved due to the symmetries present in the corresponding physical systems.

5. How does Noether's theorem impact our understanding of the laws of nature?

Noether's theorem provides a deep insight into the fundamental laws of nature by relating symmetries and conservation laws. It has contributed to the development of modern physics and continues to be a valuable tool for scientists in understanding the underlying principles of the universe.

Similar threads

I Why isn't the Lagrangian invariant under ##\theta## rotations?
    • Classical Physics
Replies
5
Views
732
I Variant and Invariant Physical Quantities....
    • Classical Physics
Replies
10
Views
3K
A How Does Noether's Theorem Extend Beyond Conservation in Hamiltonian Systems?
    • Classical Physics
Replies
7
Views
788
I Conservative forces and internal and external forces
    • Classical Physics
Replies
9
Views
1K
Find the Conserved Quantity of a Lagrangian Using Noether's Theorem
    • Calculus and Beyond Homework Help
Replies
5
Views
2K
I Conservation of Quantity: Noether's Theorem
    • Beyond the Standard Models
Replies
19
Views
3K
Using Noether's Theorem to get conserved quantities
    • Advanced Physics Homework Help
Replies
2
Views
2K
The relation between Lie algebra and conservative quantities
    • Classical Physics
Replies
14
Views
2K
I Noether’s theorem -- Question about symmetry coordinate transformation
    • Mechanics
Replies
4
Views
870
A The tautological 1-form: Lagrange vs. Hamilton formalism
    • Classical Physics
Replies
27
Views
2K

Hot Threads

Recent Insights

Back
Top