Your derivation of the equation of motion from the given Lagrangian appears to be mostly correct, but there's a small mistake in your final result. Let's go through it step by step:
Given Lagrangian:
4L=−21ϕ□ϕ+21m2ϕ2−4!λϕ4
The Euler-Lagrange equation is:
∂ϕ∂L+□(∂(□ϕ)∂L)=0
Taking the derivatives with respect to ϕ and 3∂ϕ∂L=−□ϕ+m2ϕ−3!λϕ3∂(□ϕ)∂L=−21
Now, plugging these derivatives into the Euler-Lagrange equation: −□ϕ+m2ϕ−3!λϕ3+□(−21)=0
Simplifying: −□ϕ+m2ϕ−3!λϕ3+21□=0
Rearranging terms: □ϕ−m2ϕ+3!λϕ3=0
So, the corrected equation of motion should be: □ϕ−m2ϕ+3!λϕ3=0
This is the correct equation of motion derived from the given Lagrangian.
If you want to know more about lagrang theorem,I will suggest you to visit link removed once.
