Deriving equations of motion of abelian gauge field coupled to scalar

  • #1
Geigercounter
8
2
Homework Statement
Consider the following theory in three dimensions (1 time and 2 space)
i.e. an abelian gauge field coupled to a complex scalar. Here and are real numbers and Now we parametrize the spatial plane with polar coordinates and take the solutions to be of the form (this is an assumption) Here , and both go to as and to zero at the origin.
Relevant Equations
See above
I want to compute the equations of motion for this theory in terms of the functions and . My plan was to apply the Euler-Lagrange equations, but it got confusing very quickly.

Am I right that we'll have 3 sets of equations? One for each of the fields ?

So for example the equation of motion when differentiating to becomes Is this correct?

Then I'm also confused on the since we are now in spherical coordinates. Are these still derivatives with respect to and or with respect to and ? I'm having some brain lag on this part.

EDIT: I've worked some more on the problem and obtained these equations of motion:






Now plugging in our ansatz for the first one gives a large second order differential equation: This looks very messy to me...
 
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  • #2
Should your polar coordinates read ?

Have you tried rewriting your scalar fields as , and rewritten the lagrangian in terms of the real scalar fields and instead?
 
  • #3
Yes indeed. I'd edit the post but seems like I can't do that anymore. I don't think I'd need to rewrite in a real and imaginary part though, since I have an explicit solution (ansatz)...
 
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  • #4
You can always do the same trick with the ansatz.
Have you noticed what happens if you take the complex conjugate of the eq of motion for ?
What about the time parameter of the field?
 
  • #5
Well the fields are time-independent. The hermitian conjugate of the EOM for is precisely the EOM for .

I don't see ow doing your trick in the ansatz will help further. I wrote it out but that makes everything more complicated in my opinion. What's wrong with the method I'm applying now?
 
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  • #6
How can you have eq of motion with time independent fields?
Where is this problem from?
 
  • #7
This ansatz is in the form of a soliton. In easiest case it is time independent.
 
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  • #8
Geigercounter said:
This ansatz is in the form of a soliton. In easiest case it is time independent.
And this is a textbook problem? Can you provide the original problem statement?
 
  • #9
The problem statement is as I've written it here. I got it from lecture notes I studied a while back when looking at solitons.
 
  • #10
Where is the actual statement?
1684868332122.png

Is the problem to "find a closed solution to and "?
 
  • #11
Not exactly. From the equations of motion I want to find two second order differential equations in and . That statement is just below the relevant equations section.

My apologies for the bad formatting.
 
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