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GATE 2024 PH – Question 11

Classical Mechanics · canonical transformations: Poisson bracket. Special theory of relativity: Lorentz transformations, relativistic kinematics, mass-energy equivalence. · 1 mark · Multiple choice

If $F_1(Q,q)=Qq$ generates a canonical transformation from $(p,q)$ to $(P,Q)$, which relation is correct?

  1. $p/P=Q/q$
  2. $P/p=Q/q$
  3. $p/P=-Q/q$
  4. $P/p=-Q/q$

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Correct answer: (C) $p/P=-Q/q$

Explanation

For a type-1 generating function $F_1(q,Q)$ of a canonical transformation, the relations are
$$p=\frac{\partial F_1}{\partial q},\qquad P=-\frac{\partial F_1}{\partial Q}.$$

**With $F_1=Qq$:**
$$p=\frac{\partial(Qq)}{\partial q}=Q,\qquad P=-\frac{\partial(Qq)}{\partial Q}=-q .$$

**Ratio:**
$$\frac pP=\frac{Q}{-q}=-\frac Qq\quad(\text{option C}).$$

(This transformation simply exchanges the roles of coordinate and momentum: $Q=p$, $P=-q$.)