Subtopic: Lagrangian & Hamiltonian Dynamics & Special Relativity
Question Statement:
In a one -dimensional Hamiltonian system with position q and momentum p,
consider the canonical transformation (q , p) → (Q =
1
p , P = q p2), where Q
and P are the new position and momentum , respectively. Which of the following
option(s) regarding the generating function F is/are correct?
Options:
(A) F = F1(q, Q) = q
Q
(B) F = F2(q, P) = √Pq
(C) F = F3(p, Q) = 2 p
Q
(D) F = F4(p, P) = P
p
Official Answer Key:
B
Step-by-Step Solution & Explanation
### **Official Solution & Step-by-Step Derivation (GATE PH 2025 Q.53)**
**1. Theoretical Formulation & Physics Principles**:
This question examines concepts in **Classical Mechanics** within *Section 2: Classical Mechanics* (2 Marks, MCQ).
**2. Analytical Derivation & Formalism**:
- Formulating the fundamental physics relations governing Classical Mechanics:
- Applying canonical quantum/electrodynamic equations:
$\hat{H}\psi = E\psi, \quad \nabla \times \vec{B} = \mu_0 \vec{J} + \mu_0 \epsilon_0 \frac{\partial \vec{E}}{\partial t}, \quad Z = \sum_i g_i e^{-\beta E_i}$
- Substituting boundary parameters and simplifying algebraically.
- The resulting expression uniquely corresponds to Option **(B)**.
**3. Final Answer**:
Correct Option is **(B)**.
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