Calculating derivatives of a Lagrangian density

[JPaquim]

Hey everyone,

I wasn't really sure where to post this, since it's kind of classical, kind of relativistic and kind of quantum field theoretical, but essentially mathematical. I'm reading and watching the lectures on Quantum Field Theory by Cambridge's David Tong (which you can find here), and I'm kind of stuck in calculating derivatives of the Langrangian density with respect to the derivatives of the field. For instance, the following Lagrangian (Maxwell in a vacuum):
$$ \mathcal{L} = -\frac{1}{2}(\partial_\mu A_\nu)(\partial^\mu A^\nu) + \frac{1}{2}(\partial_\mu A^\mu)^2 = -\frac{1}{2}(\partial_\mu A_\nu)(\eta^{\mu \alpha} \eta^{\nu \beta} \partial_\alpha A_\beta) + \frac{1}{2}(\eta^{\mu \nu}\partial_\mu A_\nu)^2 $$
I'm supposed to calculate $\partial \mathcal{L}/\partial(\partial_\mu A^\nu)$. Now, I have two ways to think about this, one of them is to think of each term as a homogeneous function of degree 2 of the variable $\partial_\mu A_\nu$, and using Euler's theorem on homogeneous functions, multiply by 2 and take the derivatives of each term with respect to $\partial_\mu A_\nu$ as if it wasn't being summed over, giving:
$$ \frac{\partial \mathcal{L}}{\partial(\partial_\mu A^\nu)} = -(\eta^{\mu \alpha} \eta^{\nu \beta} \partial_\alpha A_\beta) + (\eta^{\alpha \beta}\partial_\alpha A_\beta)(\eta^{\mu \nu}) = -\partial^\mu A^\nu + \eta^{\mu \nu} \partial_\alpha A^\alpha $$
This is a bit shady in my head, but apparently it works. The other way I think about it is significantly longer in terms of algebra, but basically involves actually differentiating with respect to a different set of indices, let's say $\partial \mathcal{L}/\partial(\partial_\alpha A_\beta)$, use the product rule and the fact that $\partial(\partial_\mu A_\nu)/\partial(\partial_\alpha A_\beta) = \delta^\alpha_\mu \delta^\beta_\nu$, which gives the same result, but seems unnecessarily long.

So, how do you think about it? What's the quickest but still somewhat rigorously pleasing way to think about it that gives the right answer?

Cheers

 

[dauto]

I use the second method and do all the calculations in my had to keep it from feeling unnecessarily long.

 

[JPaquim]

Ok, thank you for the feedback. BTW, the following Lagrangian gives rise to the same equations of the motion as the above one: $$ \mathcal{L} = -\frac{1}{2}(\partial_\mu A_\nu)(\partial^\mu A^\nu) + \frac{1}{2}(\partial_\nu A_\mu)(\partial^\mu A^\nu) $$
So they should differ only by a total derivative. I can test if something is a total derivative by plugging it into the Euler-Lagrange equations, and see if I get something that can be reduced to $0=0$. In this case, there should be some vector $B^\mu$ such that the difference between the two Lagrangians, is equal to $\partial_\mu B^\mu$, right? My question is, how can you find $B^\mu$ (up to the addition of a divergence-less term)?

 

[JPaquim]

Should I reask this question in the Relativity sub-forum? I don't seem to be having much success here...

 

[pmacias]

,

I would first like to commend you for your thorough and thoughtful approach to understanding and solving this problem. It is clear that you are dedicated to understanding this concept and are willing to put in the effort to fully grasp it.

In terms of calculating derivatives of a Lagrangian density, there are a few different approaches that can be taken. The first method you described, using Euler's theorem on homogeneous functions, is a valid approach and can often be the quickest and most straightforward way to solve these types of problems. However, as you mentioned, it can sometimes feel a bit "shady" and may not always provide a fully rigorous solution.

The second method you described, using the product rule and differentiating with respect to a different set of indices, is also a valid approach and can provide a more algebraically rigorous solution. However, as you also noted, it can be quite lengthy and may not always be the most efficient or intuitive way to solve the problem.

In my opinion, the best approach is to use a combination of both methods. By combining the two approaches, you can ensure both efficiency and rigor in your solution. For example, you can use Euler's theorem to quickly determine the derivative of the first term in the Lagrangian, and then use the product rule to differentiate the second term. This way, you can arrive at the correct solution in a more efficient manner while also maintaining rigor.

Ultimately, the best approach may vary depending on the specific problem and individual preferences. It is important to have a strong understanding of both methods and be able to apply them appropriately in order to solve problems in a way that is both efficient and rigorous. Keep up the good work in your studies of Quantum Field Theory!