Euler-Lagrange equations and symmetries

In summary: In the SU(N) theory, the right side of the equations of motion, which represents the source term, is "rotated" after the transformation. This means that the source term is now expressed in a different basis, but it still describes the same physical quantity. This is because the symmetry transformation does not change the underlying physics, only the way it is represented mathematically. In summary, the equations of motion must be invariant under the symmetry transformation, but the Lagrangian may contain a total derivative term that leads to non-invariance. This ensures consistency with the conservation laws and the physical significance is that the underlying physics remains unchanged.
  • #1
michael879
698
7
I'm hoping this is a really simple question, but I can't seem to find a definitive answer anywhere!

If the action is invariant under some symmetry transformation, do the equations of motion need to be invariant as well?

The trouble I'm having relates to SU(N) yang-mills theories where I'm finding that even if the action is invariant under the SU(N) local gauge transformation, the equations of motion are not. I believe this is related to the fact that the lagrangian is not invariant, but changes by a total derivative term. However I'm not sure if I'm doing something wrong or this is actually possible...

*edit* also, if someone could explain the physical significance of this that would be amazing, because I am completely confused. For example in the SU(N) theory, the right side of the EOM (which is just the source term) is "rotated" after the transformation. i.e. [itex][D_\nu,F^{\mu\nu}]=J^\mu \rightarrow [D_\nu,F^{\mu\nu}]=U^\dagger J^\mu U[/itex]
 
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  • #2
. What does this mean?The answer to your question is yes, the equations of motion need to be invariant under the symmetry transformation. This is because the equations of motion are derived from the action, so if the action is invariant then so must the equations of motion. In the case of SU(N) Yang-Mills theories, the action is indeed invariant under the SU(N) local gauge transformation, but the equations of motion are not. This is because the Lagrangian contains a total derivative term which changes under the transformation. This term is known as the "anomaly" and it is responsible for the non-invariance of the equations of motion. The physical significance of this is that it ensures that the system is consistent with the conservation of energy, momentum and charge.
 

1. What are Euler-Lagrange equations?

Euler-Lagrange equations are a set of differential equations used to find the extrema of a functional. They are derived from the principle of least action, which states that a physical system will follow the path of least action between two points.

2. How are Euler-Lagrange equations related to symmetries?

Euler-Lagrange equations and symmetries are closely related. Symmetries are transformations that leave the functional unchanged, and the solutions to the Euler-Lagrange equations are invariant under these transformations.

3. What is the significance of symmetries in physics?

Symmetries play a crucial role in physics, as they can lead to conservation laws and help simplify complex systems. They also provide a deeper understanding of the underlying principles and patterns in nature.

4. What are the applications of Euler-Lagrange equations and symmetries?

Euler-Lagrange equations and symmetries have various applications in fields such as classical mechanics, quantum mechanics, and field theory. They are used to solve problems in dynamics, electromagnetism, and other areas of physics.

5. Are there any limitations to using Euler-Lagrange equations and symmetries?

While Euler-Lagrange equations and symmetries are powerful tools, they have limitations. They can only be applied to systems with a well-defined action and may not work for highly complex or non-linear systems. Additionally, they may not provide a unique solution, and further analysis may be required.

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