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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 = qb

What are the values of a and b?

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CSIR UGC (NET) Mathematical Science: Held On (7 June 2023)
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  1. a = −1, b = 0
  2. a = −1, b = 1
  3. a = 1, b = 0
  4. a = 1, b = −1

Answer (Detailed Solution Below)

Option 4 : a = 1, b = −1
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Detailed Solution

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Concept:

The generalized coordinate transformation from (p, q) to (P, Q) is canonical if  = 1

Explanation:

Given Q = pq(a+1), P = qb is canonical if  = 1

Now,  = q(a+1),  = (a+1)pqa,  = 0,  = bqb-1 

 = 1

⇒  = 1

⇒  = 1

⇒ - bqa+b = 1

Only option (4) is satisfying the above relation.

Hence option (4) is corret

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