Discussion
The condition for generating function F, defining canonical transformations, is that the differential dF is exact. Which of the following relations does not satisfy this condition ?
Answer: B
The relation that does not satisfy the condition is \(dF = \sum_{i}q_{i}\,\delta p_{i} + \sum_{i}Q_{i}\,\delta P_{i}\).
A transformation is canonical if \(\sum p\,dq - \sum P\,dQ\) is an exact differential \(dF_{1}\). Legendre transformations give the other standard forms:
- \(\sum p\,dq + \sum Q\,dP = d(F_{1} + \sum PQ)\): exact (type 2).
- \(-\sum q\,dp - \sum P\,dQ = d(F_{1} - \sum pq)\): exact (type 3), so its negative \(\sum q\,dp + \sum P\,dQ\) is exact too.
- \(-\sum q\,dp + \sum Q\,dP\) is the type 4 form.
\(\sum q\,dp + \sum Q\,dP\) has the sign of the \(Q\,dP\) term reversed relative to type 4; it differs from an exact form by \(2\sum Q\,dP\), which is not exact in general, so it does not define a canonical transformation.
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