Next Page
Previous Page
Home
【SEOtext】0.General Formulation,∂Hc(u(t),u†(t);v(t),v†(t))/u†(t)=-∂0v(t)Now∂0qi(t)=Vi(q*(t),p*(t))reduces to:∂0u(t)=Vc(u(t),v(t)).d.Poisson Bracket,Poisson Bracket{}cPB is defined as a mapping which maps an element of (C(C4),C(C4)) to an element of C(C4) as follows.{Fc,Gc}cPB(ξ,ξ';η,η')=∂Fc(ξ,ξ';η,η')/∂η'・∂Gc(ξ,ξ';η,η')/∂ξ-∂Fc(ξ,ξ';η,η')/∂ξ・∂Gc(ξ,ξ';η,η')/∂η'+∂Fc(ξ,ξ';η,η')/∂η・∂Gc(ξ,ξ';η,η')/∂ξ'-∂Fc(ξ,ξ';η,η')/∂ξ'・∂Gc(ξ,ξ';η,η')/∂η,If:Fc(z(x),z†(x);z(y),z†(y))=F(x,y),Gc(z(x),z†(x);z(y),z†(y))=G(x,y),x,y∈R2,then:{Fc,Gc}cPB(z(x),z†(x);z(y),z†(y))={F,G}PB(x,y).Canonical equations reduce to:(d/dt)Fc(u(t),u†(t);v(t),v†(t))={Hc,Fc}cPB(u(t),u†(t);v(t),v†(t))where:u(t)=z(q1(t),q2(t)),v(t)=z(p1(t),p2(t))Let's define fundamental functions as follows.Q+c(ξ,ξ';η,η')=ξ, P+c(ξ,ξ';η,η')=η,Q-c(ξ,ξ';η,η')=ξ',P-c(ξ,ξ';η,η')=η'