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【SEOtext】0. General Formulation, (2) Continuous System, Consider the special case: I(φ)=∫dtL(φ*(t,*),∂0φ*(t,*)),φ∈Rn(R4),L∈R(Rn(R3),Rn(R3)) The functional L is called Lagrangian of the system. (i) Euler-Lagrange Equation δI(φ)/δφα(x)=0⇔δL(φ*(t,*),∂0φ*(t,*))/δ[φ*(t,*)]α(x)-d/dtδL(φ*(t,*),∂0φ*(t,*))/δ[∂0φ*(t,*)]α(x)=0 (ii) Local Field Theory, Consider the special case: L(φ,ψ)=∫d3xL(φ*(x);ψ*(x),∂φ*(x)), φ,ψ∈Rn(R3); L∈R(Rn;Rn,R3×n) The function L is called Lagrangian density of the system. a. I(φ)=∫d4xL(φ*(x);∂*φ*(x)) b. δI(φ)/δφα(x)=0⇔∂L(φ*(x);∂*φ*(x))/∂φα(x)-∂/∂xμ∂L(φ*(x);∂*φ*(x))/∂[∂μφα(x)]=0 c. Noether's Theorem for Local Field Theory, Assumption, If δI(φ)/δφα(x)=0 then: ∀Ω⊂R4; (∂/∂ε)I(φ'*(*;φ,ε);Ω'(Ω,ε))|ε=0=0 where: I(φ;Ω)=∫x∈Ωd4xL(φ*(x),∂*φ*(x))