Can somebody show me a non-trivial exmple of Noether Theorem?

In summary, Noether Theorem states that a system's Lagrangian will have a conserved quantity if it remains unchanged under a continuous coordinate transformation. It does not necessarily have to have a cyclic coordinate. An example of a non-trivial application of this theorem is the derivation of the conserved Runge-Lenz vector in a 1/r potential, which may be challenging to understand.
  • #1
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Can somebody show me a "non-trivial" exmple of Noether Theorem?

Noether Theorem states that if the Lagrangian of a system is invariant under some continuous coordinate transformation, then there's a conserved quantity.But does it simply mean the Lagrangian has to have a cyclic coordinate? Or a cyclic coordinate is just a special "trivial" case of Noether theorem? If so could somebody show me a "non-trivial" example? I mean a Lagrangian with no cyclic coordinates but we can apply Noether Theorem on.
 
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  • #2


You can show the Runge-Lenz vector is a conserved quantity in a 1/r potential. However, I warn you - after wrestling with this derivation, you might wish for the trivial examples again.
 
  • #3


Yes, there are numerous non-trivial examples of Noether Theorem in action. One example is in the field of quantum mechanics, where the time-independent Schrödinger equation is invariant under time translations. This leads to the conservation of energy, which is a fundamental quantity in quantum mechanics.

Another example is in classical mechanics, where the Lagrangian for a system with rotational symmetry is invariant under rotations. This leads to the conservation of angular momentum, which is a fundamental quantity in understanding the dynamics of rotating objects.

In general, Noether Theorem can be applied to any system with a continuous symmetry, such as translational symmetry, rotational symmetry, or gauge symmetry in field theories. These symmetries can lead to the conservation of various quantities, such as energy, momentum, angular momentum, and charge.

Furthermore, Noether Theorem has also been applied in other fields of physics, such as general relativity, where the spacetime metric is invariant under diffeomorphisms. This leads to the conservation of energy and momentum in gravitational systems.

In summary, Noether Theorem is a powerful tool that allows us to connect symmetries in a system to the conservation of important physical quantities. It has numerous non-trivial applications in various areas of physics and continues to be a fundamental principle in understanding the laws of nature.
 

Related to Can somebody show me a non-trivial exmple of Noether Theorem?

Question:

What is the Noether Theorem?

Answer:

The Noether Theorem is a fundamental concept in physics that states that for every continuous symmetry in a physical system, there exists a corresponding conserved quantity.

Question:

What is a non-trivial example of Noether Theorem?

Answer:

A non-trivial example of Noether Theorem is the conservation of angular momentum in a system with rotational symmetry. This means that if the laws of physics remain unchanged under rotations, then the total angular momentum of the system will be conserved.

Question:

How does Noether Theorem apply to relativity?

Answer:

In relativity, Noether Theorem can be used to explain the conservation of energy and momentum. This is because the laws of physics remain unchanged under translations in space and time, and thus the corresponding conserved quantities are energy and momentum.

Question:

What is the significance of Noether Theorem?

Answer:

Noether Theorem is significant because it provides a deep understanding of the relationship between symmetries and conservation laws in physics. It has also been instrumental in the development of modern theories, such as relativity and quantum mechanics.

Question:

How is Noether Theorem applied in other fields besides physics?

Answer:

Noether Theorem has applications in other fields, such as mathematics and engineering. It has been used to study the symmetries of differential equations and to develop numerical methods for solving them. It has also been applied in engineering to study the symmetries of physical systems and to design control algorithms.

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