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ySEOtextz0.General Formulation,a.Velocity,Velocity is defined as a function VcC(C2) by the equation:Vc(z(x1,x2),z(y1,y2))=z(V1(x,y),V2(x,y)),x,yR2.Then it follows that:̓,ŁC;{=Lc(,́;Vc(,),Vc(,))/Vc(,),Ł=Lc(,́;Vc(,),Vc(,))/Vc(,)These conditions determine Vc completely.b.Hamiltonian,Hamiltonian is defined as a function HcC(C4) such that:x,yR2;Hc(z(x1,x2),z(x1,x2);z(y1,y2),z(y1,y2))=H(x,y)Then it follows that:̓,ŁC;Hc(,́;,Ł)=ŁVc(,)+Vc(,)-Lc(,́;Vc(,),Vc(,)),c.Canonical Equations,If u(t)=z(q1(t),q2(t)) and v(t)=z(p1(t),p2(t)) then the canonical equations (iii) are equivalent to the following equations.{Hc(u(t),u(t);v(t),v(t))/v(t)=0u(t),Hc(u(t),u(t);v(t),v(t))/v(t)=0u(t),Hc(u(t),u(t);v(t),v(t))/u(t)=-0v(t) |
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