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【SEOtext】0. General Formulation, [1] Classical Lagrangian Formulation, (1) Discrete System, Consider the special case: I(q)=∫dtL(q1(t),・・・,qn(t);∂0q1(t),・・・,∂0qn(t)), q∈Rn(R), L∈R(R2n) The function L is called Lagrangian of the system. (i) Euler-Lagrange Equation δI(q)/δqi(t)=0⇔L(q*(t);∂0q*(t))/∂qi(t)-d/dt∂L(q*(t);∂0q*(t))/∂[∂0qi(t)]=0 (ii) Noether's Theorem for Discrete System, Assumption, If δI(q)/δqj(t)=0 then: ∀t1,t2∈R; (∂/∂ε)I(q'*(*;q,ε);t'(t1;ε),t'(t2;ε))|ε=0=0 where: I(q;t1,t2)=∫t(2)t(1)dtL(q*(t),∂0q*(t)), q'∈Rn(R;Rn(R),R), t'∈R(R;R), q'j(t;q,0)=qj(t), t'(t;0)=t, Conclusion, If δI(q)/δqj(t)=0 then: (d/dt)Q(t;q)=0 where: Q∈R(R;Rn(R)), Q(t;q)=∂t'(t;ε)/∂ε|ε=0L(q*(t);∂0q*(t))+∂q'j(t;q,ε)/∂ε|ε=0∂L(q*(t);∂0q*(t))/∂[∂0qj(t)]