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ySEOtextz0. General Formulation,Notice that if Ý0qi(t)=Vi(q*(t),p*(t)) then:I(q)=çdt[pi(t)Ý0qi(t)-H(q*(t),p*(t))]=0.Then we know the equations before eÌf mean vanishing of variation of a functional which is connected to the action by:Ý0qi(t)=Vi(q*(t),p*(t)).(v)Poisson bracket,Poisson bracket {}PB is defined as a mapping which maps an element of (R(Rn,Rn),R(Rn,Rn)) to an element of R(Rn,Rn) as follows.{F,G}PB(x,y)=ÝF(x,y)/ÝyiÝG(x,y)/Ýxi-ÝF(x,y)/ÝxiÝG(x,y)/Ýyi,Canonical equations can be written in terms of Poisson bracket as follows.(d/dt)F(q*(t),p*(t))={H,F}PB(q*(t),p*(t)),To know this equation contains the original canonical equations, we have only to consider the special cases:F=Qi,Pi where Qi,Pi¸R(Rn,Rn) are defined by the equations:Qi(x,y)=xi,Pi(x,y)=yi(i=1,2,EEE,n).The Poisson bracket between Qj and Pk is:{Qj,Pk}PB(x,y)=-ƒÂjk.(vi)Canonical Transformations,Prove that conserved quantities in Noether's theorem play a role of generators in canonical transformations.(vii)Treatment of Complex Variables(n=2) |
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