Lagrangian Field Theory - Maxwell's Equations

In summary: I'll keep that in mind for the future. Thanks again for your help.In summary, to calculate the partial derivative of the Lagrangian with respect to the derivative of A_v, we can rewrite the equation as a sum of terms with covariant and contravariant indices using the given equations. Then, using the relation for the partial derivative of a product, we can simplify the expression to get the correct answer of $ - \partial^\mu A^v + (\partial_\rho A^\rho) \eta^{\mu v}$.
  • #1
Woolyabyss
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1

Homework Statement


$$ L = -\frac{1}{2} (\partial_{\mu} A_v) (\partial^{\mu} A^v) + \frac{1}{2} (\partial_{\mu} A^v)^2$$

calculate $$\frac{\partial L}{\partial(\partial_{\mu} A_v)}$$

Homework Equations


$$ A^{\mu} = \eta^{\mu v} A_v, \ and \ \partial^{\mu} = \eta^{\mu v} \partial_{v}$$

The Attempt at a Solution


rewrite equation as $$ L = - \frac{1}{2} \eta^{\mu a} \eta^{v b} (\partial_{\mu} A_v) (\partial_{a} A_b) + \frac{1}{2} \eta^{v b} \eta^{v b} (\partial_{\mu} A_b) (\partial_{\mu} A_b )$$

now taking $$\frac{\partial L}{\partial(\partial_{\mu} A_v)}$$

we get $$ - \frac{1}{2} ( \eta^{\mu a} \eta^{v b} \partial_{a} A_b + \eta^{\mu a} \eta^{v b} (\partial_{\mu} A_v) \delta^{\mu v}_{a b} ) + \frac{1}{2} \eta^{v b} \eta^{v b}( \delta_{b}^{v} \partial_{\mu} A_b + \partial_{\mu} A_b \delta_{b}^{v}) $$

This is what I've got so far. I'm pretty certain I've made a number of mistakes here. On the left hand side I've got the mu's and v's repeating 3 times which probably isn't a good sign and on the right hand side the indices of the delta don't seem to be consistent.

The answer is

$$ \frac{\partial L}{\partial(\partial_{\mu} A_v)} = - \partial^{\mu} A^v + (\partial_{\rho} A^{\rho} ) \eta^{\mu v} $$

Any help would be appreciated.
 
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  • #2
You seem to have some trouble with index manipulations. There are some very simple rules that you should always follow:
  1. You can never have more than two of the same index. If you at some point write down an expression with more than two of the same index you have done something wrong and you need to work around it by using other index manipulations.
  2. If you have two of the same index, one needs to be covariant and the other contravariant. This represents a sum according to the summation convention. This has the following implication:
    • You can rename any index that appears twice as long as you rename it to something that does not appear in your expression.
  3. If you have one of a particular index, it is a free index and needs to appear on both sides of an equality. You are not allowed to rename this index.
As an example, a very common mistake is of the following nature: You have an expression that contains a scalar that can be expressed as a product ##V^2 = V_\mu V^\mu## and you want to compute ##\partial_\mu V^2##. You insert the expression for ##V^2## and get ##\partial_\mu V^2 = \partial_\mu V_\mu V^\mu##. This expression is nonsense as you have three ##\mu## indices. What went wrong is that you inserted ##V^2 = V_\mu V^\mu## in an expression that already had a ##\mu## index. However, what you can do is to use (2) to replace the ##\mu## index in the ##V^2## expression to obtain ##V^2 = V_\nu V^\nu##. Since there are no ##\nu## indices in ##\partial_\mu V^2## you can now insert this into the expression to find that ##\partial_\mu V^2 = \partial_\mu V_\nu V^\nu##, which is a valid expression.

I suggest you go through your expressions with the above in mind. Note that the last term of the Lagrangian you are given a priori violates some of the rules above. However, it is quite common to by the square denote multiplication with the same expression, but with the indices swapped from covariant to contravariant and vice versa. In other words ##(\partial_\mu A^\nu)^2## should be read as ##(\partial_\mu A^\nu)(\partial^\mu A_\nu)##, not as ##(\partial_\mu A^\nu)(\partial_\mu A^\nu)##, which is meaningless.
 
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  • #3
If you're having trouble, it might be helpful if you do not use the indices [itex]\mu, v[/itex] in your Lagrangian so that you don't get confused with the indices in the derivative that you're taking. So for your case:

[tex]
L = -\frac{1}{2} \eta^{a c} \eta^{b d}(\partial_{a} A_b) (\partial_{c} A_d) + \frac{1}{2} \eta^{a b} \eta^{c d}(\partial_{a} A_b) (\partial_{c} A_d)
[/tex]
or
[tex]
L = \frac{1}{2}\left(\eta^{a b} \eta^{c d} - \eta^{a c} \eta^{b d}\right) (\partial_{a} A_b) (\partial_{c} A_d)
[/tex]

Then use the relation
[tex]
\frac{\partial }{\partial (\partial_\mu A_v)} (\partial_{a} A_b) = \delta^{\mu}_a \delta^{v}_b
[/tex]
repeatedly. Now that you're not reusing indices in different parts of your equations it may eliminate some confusion.
 
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  • #4
Thanks a lot to the both of you. I made an error when I wrote down the Lagrangian, the term on the right was actually $$ \frac{1}{2} ( \partial_{\mu} A^{\mu} )^2 $$

Regardless, I managed to get the right answer in the end. It never occurred to me that if my derivative had indices matching indices of the Lagrangian I could run into problems.
 

1. What is Lagrangian Field Theory?

Lagrangian Field Theory is a mathematical framework used to describe the behavior of a physical system as it evolves over time. It is based on the principle of least action, which states that the path taken by a system between two points in time is the one that minimizes the action, a quantity that represents the total energy of the system.

2. What are Maxwell's Equations?

Maxwell's Equations are a set of four partial differential equations that describe the behavior of electric and magnetic fields. They were developed by James Clerk Maxwell in the 19th century and are considered one of the most fundamental laws of physics, governing the behavior of electromagnetic waves and all electromagnetic phenomena.

3. How are Lagrangian Field Theory and Maxwell's Equations related?

Lagrangian Field Theory provides a mathematical framework for understanding the behavior of physical systems, including electromagnetic fields described by Maxwell's Equations. In this framework, the equations are derived from the Lagrangian density, which describes the energy of the system at each point in space and time.

4. What is the significance of Lagrangian Field Theory in physics?

Lagrangian Field Theory is a powerful tool for understanding the behavior of physical systems, especially in the field of theoretical physics. It allows for the derivation of equations of motion for complex systems, as well as the prediction of new phenomena and the development of new theories.

5. Are there any limitations to Lagrangian Field Theory and Maxwell's Equations?

Like any scientific theory, Lagrangian Field Theory and Maxwell's Equations have their limitations. They are based on certain assumptions and may not accurately describe the behavior of systems in extreme conditions, such as at the quantum level or in the presence of strong gravitational fields. However, they have been extensively tested and are considered to be highly accurate descriptions of the physical world in most cases.

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