Equation of motion for interacting fields

[>>S/Z<<]

Homework Statement

2N classical, real, scalar fields \phi_i (x^{\nu}) \psi_i (x^{\nu}) ,where i=1,...,N

I have to find the equations of motions for \phi_i (x^{\nu}) \psi_i (x^{\nu})

Lagrange density is given by

L= \sum_{i=1}^N (\frac{1}{2}(\partial_{\mu} \phi_i \partial^{\mu} \phi_i+\partial_{\mu} \psi_i \partial^{\mu} \psi_i-m^2(\phi_i \phi_i + \psi_i \psi_i) - \lambda \phi_i \psi_i))

Homework Equations

Euler-Lagrange equation

\partial_{\mu} \frac{\partial L}{\partial (\partial_{\mu} \phi_i)} - \frac{\partial L_0}{\partial \phi_i} = 0

The Attempt at a Solution

The problem for me is the interaction term? Can I just put in the Euler-Lagrange equation as well, or what do I do with it? I found some equations by disregarding the interaction term

\partial_{\mu} \partial^{\mu} \phi_i + m^2 \phi_i = 0
\partial_{\mu} \partial^{\mu} \psi_i + m^2 \psi_i = 0

Is this right?

 

[diazona]

It's easier to read the math if you actually wrap it in [tex ] and [ /tex] tags I've done that in the quote below for everyone's reference:

Homework Statement

2N classical, real, scalar fields $\phi_i (x^{\nu})$, $\psi_i (x^{\nu})$, where i=1,...,N

I have to find the equations of motions for $\phi_i (x^{\nu})$, $\psi_i (x^{\nu})$

Lagrange density is given by

$$L= \sum_{i=1}^N (\frac{1}{2}(\partial_{mu} \phi_i \partial^{\mu} \phi_i+\partial_{mu} \psi_i \partial^{\mu} \psi_i-m^2(\phi_i \phi_i + \psi_i \psi_i) - \lambda \phi_i \psi_i))$$

Homework Equations

Euler-Lagrange equation

$$\partial_{\mu} \frac{\partial L}{\partial (\partial_{\mu} \phi_i)} - \frac{\partial L_0}{\partial \phi_i} = 0$$

The Attempt at a Solution

The problem for me is the interaction term? Can I just put in the Euler-Lagrange equation as well, or what do I do with it? I found some equations by disregarding the interaction term

$$\partial_{mu} \partial^{\mu} \phi_i + m^2 \phi_i = 0$$
$$\partial_{mu} \partial^{\mu} \psi_i + m^2 \psi_i = 0$$

Is this right?

You may recognize the equations you found as the Klein-Gordon equation which governs the evolution of a free scalar field. So it is exactly what you'd expect to find if you discard the interaction term, i.e. pretend that the fields do not interact. But it's not the answer you need. You can't just disregard that term and pretend that it doesn't change anything.

Just include the interaction term in the Lagrangian, as you do with the other terms.

 

[>>S/Z<<]

It's easier to read the math if you actually wrap it in [tex ] and [ /tex] tags I've done that in the quote below for everyone's reference:

Sorry about that! My first post so not familiar with it all yet

You may recognize the equations you found as the Klein-Gordon equation which governs the evolution of a free scalar field. So it is exactly what you'd expect to find if you discard the interaction term, i.e. pretend that the fields do not interact. But it's not the answer you need. You can't just disregard that term and pretend that it doesn't change anything.

Just include the interaction term in the Lagrangian, as you do with the other terms.

Okay so just use the Euler-Lagrange equation on the full Lagrange.

Thanks for the reply:)!