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ySEOtextz0. General Formulation, (iii) Treatment of Complex Variables, Consider the special case: n=2, L(x1,x2;y1,y2)=Lc(z(x1,x2),z(x1,x2);z(y1,y2),z(y1,y2)) where: LcC(C4), zC(R2), Lc(,́;,Ł)R, ,ŁC, z(x1,x2)=(x1+ix2)/2, a. Euler-Lagrange Equation, I(q)/qj(t)=0{Lc(u(t),u(t);0u(t),0u(t))/u(t)-d/dtLc(u(t),u(t);0u(t),0u(t))/[0u(t)]=0,Lc(u(t),u(t);0u(t),0u(t))/u(t)-d/dtLc(u(t),u(t);0u(t),0u(t))/[0u(t)]=0 where: u(t)=z(q1(t),q2(t)), b. Conserved Charge, Q(t;q)=Qc(t;u)=t'(t;)/݃|=0Lc(u(t),u(t);0u(t),0u(t))+u'(t;u,)/݃|=0Lc(u(t),u(t);0u(t),0u(t))/[0u(t)]+u'(t;u,)/݃|=0Lc(u(t),u(t);0u(t),0u(t))/[0u(t)] where: u(t)=z(q1(t),q2(t)), u'(t;u,)=z(q'1(t;q,),q'2(t;q,)) |
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