Why Are the EOMs for a Complex Scalar Field Not Independent?

[Milsomonk]

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 independent 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 :)

 

[TSny]

So then the equations of motion are:
$$\Box \phi^* -m^2 \phi^* +2\lambda (\phi^*)^2 \phi =0$$
$$\Box \phi -m^2 \phi +2\lambda (\phi)^2 \phi^* =0$$

Looks like a sign error in one of the terms on the left. Otherwise, I think it's OK.

You are right that the fields are not independent. They are interacting with one another.

 

[Milsomonk]

Ahh yeah I see the issue, thanks :) Yeah I just thought it was odd as one of the tutorial helpers said they were independent... but they were clearly incorrect. Thanks for your insight :)