- #1
Milsomonk
- 96
- 17
Homework Statement
Find the equations of motion for the Lagrangian below:
$$ L=\partial_\mu \phi^* \partial^\mu \phi - V( \phi,\phi^* ) $$
Where :
$$ V( \phi,\phi^* )= m^2 \phi^* \phi + \lambda (\phi^* \phi)^2 $$
Homework Equations
Euler Lagrange equation:
$$ \partial_\mu \dfrac {\partial L} {\partial (\partial_\mu \phi)} -\dfrac {\partial L} {\partial \phi} =0 $$
The Attempt at a Solution
So I have calculated the equations of motion for each field but I'm surprised to find they're not independant of each other so I'm wondering if I've made a mistake somewhere? Here are my workings:
$$ \dfrac {\partial L} {\partial \phi} =m^2 \phi^* +2\lambda (\phi^*)^2 \phi $$
$$\dfrac {\partial L} {\partial (\partial_\mu \phi)} = \partial_\mu \phi^* $$
So then the equations of motion are:
$$\Box \phi^* -m^2 \phi^* +2\lambda (\phi^*)^2 \phi =0$$
$$\Box \phi -m^2 \phi +2\lambda (\phi)^2 \phi^* =0$$
Any suggestions would be appreciated :)