- #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...
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
Then I'm also confused on the
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:
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