Understand Noether's Theorem w/ Lagrangian Example

In summary, the superscripts are confusing to me. They are related to the axes in that the x,y, and z axes are linearly independent, but I am not sure how.
  • #1
vertices
62
0
Hi

I was wondering if someone would be kind enough to help me understand an example in my class notes:

If we have a Lagrangian:

[tex]L=m(\dot{z}\dot{z^{*}})-V(\dot{z}\dot{z^{*}})[/tex]

where z=x+iy.

Why does it follow that

[tex]Q=X^{i}\frac{{\partial}L}{{\partial}\dot{q}^{i}}[/tex]

is equal to:

[tex]X\frac{{\partial}L}{{\partial}\dot{z}}+X^{*}\frac{{\partial}L}{{\partial}\dot{z^{*}}}[/tex]?

I mean, mathematically that seems wrong, why are we adding the second term (the one with the complex conjugate of [tex]\dot{z}[/tex])

Also, can I check if my understanding of the superscript 'i' is correct - does it correspond to the axes, for example i=1 corresponds to the x axis, i=2 corresponds to the y axis, etc. If z = x + iy, we are no longer talking about a 3 dimensional real space, so how are the superscripts relavent?

And is there a reason why the superscript 'i' has gone in the second line?

thanks
 
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  • #2
vertices said:
Also, can I check if my understanding of the superscript 'i' is correct - does it correspond to the axes, for example i=1 corresponds to the x axis, i=2 corresponds to the y axis, etc. If z = x + iy, we are no longer talking about a 3 dimensional real space, so how are the superscripts relavent?
The superscripts are confusing to me. e^(0 i pi/2) is 1 which is the x-axis in the complex plane, e^(1 i pi/2) is i which is the y-axis in the complex plane, e^(2 i pi/2) is -1 which is the -x axis in the complex plane, and e^(3 i pi/2) is -i which is the -y axis in the complex plane. Are the superscripts related to that somehow?
 
  • #3
DaleSpam said:
The superscripts are confusing to me. e^(0 i pi/2) is 1 which is the x-axis in the complex plane, e^(1 i pi/2) is i which is the y-axis in the complex plane, e^(2 i pi/2) is -1 which is the -x axis in the complex plane, and e^(3 i pi/2) is -i which is the -y axis in the complex plane. Are the superscripts related to that somehow?

I am not sure about this, as the x,y, and z axes should be orthogonal to each other, in other words they are linearly independent - ie. you can't define z in the way that it has been defined (z=x+iy), as a linear combination of x and y...
 
  • #4
Call me crazy, but don't you need to first have a transformation of the Lagrangian before you can define a Noether current and a conserved charge?

BTW, in my QFT class I never had to deal with a complex-valued coordinate. But I'm guessing that x and y in this context are both four vectors (and therefore z as well). I could be wrong...
 

FAQ: Understand Noether's Theorem w/ Lagrangian Example

1. What is Noether's Theorem?

Noether's Theorem is a fundamental principle in physics that states that for every differentiable symmetry of the physical system, there exists a corresponding conserved quantity. This theorem was first discovered by mathematician Emmy Noether in the early 20th century.

2. What is the significance of Noether's Theorem?

Noether's Theorem is significant because it provides a powerful tool for understanding the fundamental laws of nature. It allows us to connect symmetries of a physical system to conserved quantities, such as energy, momentum, and angular momentum, which are essential for understanding the behavior of the system.

3. What is the relationship between Noether's Theorem and Lagrangian mechanics?

Noether's Theorem and Lagrangian mechanics are closely related. Noether's Theorem can be used to derive the equations of motion from the Lagrangian of a system, providing a powerful and elegant way to understand the dynamics of a physical system.

4. Can you provide an example of Noether's Theorem in action?

One example of Noether's Theorem in action is in the conservation of energy in a system with time symmetry. If the laws of physics are the same at all times, then the energy of the system will be conserved. This is just one of the many symmetries that can be connected to conserved quantities through Noether's Theorem.

5. How is Noether's Theorem used in modern physics?

Noether's Theorem has many applications in modern physics, including in the fields of quantum mechanics, general relativity, and particle physics. It has also been used to develop new theories, such as gauge theories, which have become essential in our understanding of the universe.

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