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[QUE/CM-10002]

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  1. Show that \begin{equation} Q=-p ,\qquad P=q + Ap^2 \end{equation} (where A is any constant) is a canonical transformation,
    • (i) by evaluating \([Q,P]_{q,p}\)
    • (ii) by expressing \(pdq-PdQ\) as an exact differential \(dF(q,Q)\). Hence find the type one generating function of the transformation. To do this, you must first use the transformation to express \(p,P\) in terms of \(q,Q\).
  2. Use the relation \(F_2=F_1+PQ\) to find the type 2 generating function \(F_2(q,P)\),and check your result by showing that \(F_2\) indeed generates the transformation.

Source : Calkin

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