Finding Equation of Motion Using Scalar Field Lagrangian

[HashKetchum]

Homework Statement

I must find the following equation of motion:

φ'' + 3Hφ' + dV/dφ = 0

Using the scalar field Lagrangian:

(replace the -1/2m^2φ^2 term with a generic V(φ) term though)

with the Euler-Lagrange Equation

I know that I must assume φ = φ(t) and the scale factor a = a(t)

Homework Equations

The Attempt at a Solution

 

[Asim Riaz]

The Euler-Lagrange equation for this scalar field Lagrangian is:d/dt(∂L/∂φ') = ∂L/∂φSubstituting in the Lagrangian (L) and solving for φ'' yields:φ'' = -∂L/∂φ - d/dt(∂L/∂φ')Substituting in the Lagrangian and solving for φ'' yields:φ'' = -dV/dφ - 3H(d/dt(dV/dφ))Rearranging and substituting in the definition of the Hubble parameter (H = d(ln a)/dt)φ'' + 3Hφ' + dV/dφ = 0