I Laws of Motion for New Lagrangian: Partial Differential Equations

Maniac_XOX
Messages
86
Reaction score
5
TL;DR Summary
Is it possible to simplify this complicated equation??
$$\partial^\beta F_{\beta\alpha} +\partial^\beta A_\mu A^\mu \delta^\alpha_\sigma \delta^\rho_\beta+\mu^2 A_\alpha = 2A_\mu (\partial_\rho A^\rho) +\frac {4\pi}{c}J_\alpha$$
 
Physics news on Phys.org
You don't have the same free indices on each of your terms. There are three free indices on the second one on the left and you have a free ##\mu## instead of ##\alpha## in the first term on the right. Correct your typos and we may be able to help.
 
  • Like
Likes Maniac_XOX
Ibix said:
You don't have the same free indices on each of your terms. There are three free indices on the second one on the left and you have a free ##\mu## instead of ##\alpha## in the first term on the right. Correct your typos and we may be able to help.
Hmm you're right, the complicated part where i most def went wrong was finding the euler lagrange equation $$\partial^\beta \frac{\partial L}{\partial(\partial^\beta A^\alpha)}=\frac {\partial L}{\partial A^\alpha}$$
to the following section of the lagrangina i used: ##-\beta A_\mu A^\mu (\partial_\rho A^\rho)##

Need help finding the answer to this.

I found that $$\partial^\beta \frac{\partial L}{\partial(\partial^\beta A^\alpha)}=\partial^\beta(-\beta A_\mu A^\mu \delta^\alpha_\sigma \delta^\rho_\beta)$$

and also that $$\frac {\partial L}{\partial A^\alpha}= -2\beta A_\alpha (\partial_\rho A^\rho) $$

Could u help me out?
 
Maniac_XOX said:
I found that ∂β∂L∂(∂βAα)=∂β(−βAμAμδσαδβρ)
This is quite obviously wrong as you do not have the same free indices on both sides. Can you find your error and/or tell us how you arrive at this?
 
Orodruin said:
This is quite obviously wrong as you do not have the same free indices on both sides. Can you find your error and/or tell us how you arrive at this?
hello yeah i realized actually, I've come to the right equation being $$\partial^\beta(A_\mu A^\mu g_{\rho\sigma} \delta^\sigma_\beta \delta^\rho_\alpha)=(\partial_\rho A^\rho) g_{\mu\gamma} (\delta_\alpha^\gamma A^\mu + A^\gamma \delta^\mu_\alpha)$$ from euler lagrange $$\partial^\beta \frac{\partial L}{\partial(\partial^\beta A^\alpha)}=\frac {\partial L}{\partial A^\alpha}$$
When put inside the whole complete equation it becomes:
$$\partial^\beta F_{\beta\alpha} + \partial^\beta(A_\mu A^\mu g_{\rho\sigma} \delta^\sigma_\beta \delta^\rho_\alpha) + \mu^2 A_\alpha =(\partial_\rho A^\rho) g_{\mu\gamma} (\delta_\alpha^\gamma A^\mu + A^\gamma \delta^\mu_\alpha) + \frac {4\pi}{c} J_\alpha$$
Can you help me simplify this?
 
Last edited:
I would start by using the summation properties of the Kronecker delta.
 
I'd move the ##g_{\mu\lambda}## inside the bracket before following Orodruin's advice.
 
Ibix said:
I'd move the ##g_{\mu\lambda}## inside the bracket before following Orodruin's advice.
Orodruin said:
I would start by using the summation properties of the Kronecker delta.
Okay so I've tried lowering the indices on the right hand side and using the property $$g_{\mu\gamma}g^{\mu\nu}=\delta^\nu_\gamma$$
therefore on the right hand side I get
$$(\partial_\rho A^\rho) (\delta_\gamma^\nu \delta_\alpha^\gamma A_\nu + \delta_\gamma^\lambda \delta^\mu_\alpha A_\lambda) + \frac {4\pi}{c} J_\alpha$$
which should be able to simplify as:
$$(\partial_\rho A^\rho) (\delta_\alpha^\nu A_\nu + \delta_\gamma^\lambda \delta^\mu_\alpha A_\lambda) + \frac {4\pi}{c} J_\alpha$$

Is this correct? If so how do i go from there?
 
The point is that ##g_{\mu\gamma}\delta^\gamma_\alpha=g_{\mu\alpha}##.
 
  • #10
Ibix said:
The point is that ##g_{\mu\gamma}\delta^\gamma_\alpha=g_{\mu\alpha}##.
OH i get it, so i should be getting this$$\partial^\beta F_{\beta\alpha} + \partial^\beta(A_\mu A^\mu g_{\rho\beta} \delta^\rho_\alpha) + \mu^2 A_\alpha =(\partial_\rho A^\rho) (g_{\mu\alpha} A^\mu + g_{\gamma\alpha} A^\gamma) + \frac {4\pi}{c} J_\alpha$$which finally becomes$$\partial^\beta F_{\beta\alpha} + \partial^\beta(A_\mu A^\mu g_{\alpha\beta}) + \mu^2 A_\alpha = (A_\alpha + A_\alpha) + \frac {4\pi}{c} J_\alpha$$
so then the right hand side becomes ##2A_\alpha + \frac {4\pi}{c} J_\alpha## ?
 
  • #11
You seem to have lost a ##\partial_\rho A^\rho## on the right hand side. Otherwise, yes.
 
  • #12
Ibix said:
You seem to have lost a ##\partial_\rho A^\rho## on the right hand side. Otherwise, yes.
Right i forgot that and also ##-\beta## factors on both sides while focusing on the other stuff. A question i have involving the LHS is:
Does ##\partial^\beta(-\beta \times g_{\alpha\beta}A_\mu A^\mu)##
mean the same as $$\frac {\partial (-\beta \times g_{\alpha\beta}A_\mu A^\mu)}{\partial A^\beta}$$ ? Cuz then wouldn't that bit simplify to ##-2\beta A_\alpha## as well?
 
Back
Top