Consider the constants a and b such that the following generalized coordinate transformation from (p, q) to (P, Q) is canonical

Q = pq^(a+1), P = q^b

What are the values of a and b?

Concept:

The generalized coordinate transformation from (p, q) to (P, Q) is canonical if \(\frac{\partial (P, Q)}{\partial (p,q)}\) = 1

Explanation:

Given Q = pq(a+1), P = qb is canonical if \(\frac{\partial (P, Q)}{\partial (p,q)}\) = 1

Now, \(\frac{\partial Q}{\partial p}\) = q(a+1), \(\frac{\partial Q}{\partial q}\) = (a+1)pqa, \(\frac{\partial P}{\partial p}\) = 0, \(\frac{\partial P}{\partial q}\) = bqb-1 

\(\frac{\partial (P, Q)}{\partial (p,q)}\) = 1

⇒ \(\begin{vmatrix}\frac{\partial P}{\partial p}&\frac{\partial P}{\partial q}\\\frac{\partial Q}{\partial p}&\frac{\partial Q}{\partial q}\end{vmatrix}\) = 1

⇒ \(\begin{vmatrix}0&bq^{b-1}\\q^{a+1}&(a+1)pq^a\end{vmatrix}\) = 1

⇒ - bq^a+b = 1

Only option (4) is satisfying the above relation.

Hence option (4) is corret