chapter  3
8 Pages

Euler–Lagrange Equations

The Klein-Gordon equation in Lorentz covariant form [x ≡ (ct, x, y, z) ≡ (x0; x)]

gµν∂µ∂νϕ(x) + m2ϕ(x) = 0, gµν =

 

  , (3.1)

can be derived by variational calculus from an action principle

S = ∫

d4xL(ϕ, ∂µϕ, x), L(ϕ, ∂µϕ, x) = 12 (∂µϕ)2 − V(ϕ),

(∂µϕ)2 ≡ ∂µϕ∂µϕ = gµν∂µϕ∂νϕ, V(ϕ) = 12 m2ϕ2. (3.2) We assume the field to be given at the boundary of the domain Mof integration (typically assuming the field vanishes at infinity) and demand the action to be stationary with respect to any variation ϕ(x) → ϕ(x) + δϕ(x) of the field,

δS(ϕ) ≡ S(ϕ + δϕ) − S(ϕ) = ∫

M d4x

( ∂µϕ∂µδϕ − ∂V(ϕ)

∂ϕ δϕ

)

= ∫

M d4x

( −δϕ

( ∂µ∂

µϕ + ∂V(ϕ) ∂ϕ

)) +

dµσ (δϕ∂µϕ) = 0, (3.3)

where dµσ is the integration measure on the boundary ∂ M. The variation δϕ is arbitratry, except at ∂ M, where we assume δϕ vanishes, and this implies the Euler-Lagrange equation

∂µ∂ µϕ + ∂V(ϕ)

∂ϕ = 0, (3.4)

which coincides with the Klein-Gordon equation. We can also write the EulerLagrange equations for arbitrary action S(ϕ) in terms of functional derivatives

δ−S δ−ϕ(x)

= δS δϕ(x)

− ∂µ δS δ∂µϕ(x)

= 0, (3.5)

A in

where δ− stands for the total functional derivative, which is then split according to the explicit dependence of the action on the field and its derivatives (usually an action will not contain higher than first-order space-time derivatives). Please note that a functional derivative has the property δϕ(x)/δϕ(y) = δ4(x − y), which is why in the above equation we take functional derivatives of the action S and not, as one sees often, of the Lagrangian density L.