Subtopic: Lagrangian & Hamiltonian Dynamics & Special Relativity
Question Statement:
If (x ̇ ẏ + 𝛼xy) is a constant of motion of a two-dimensional isotropic harmonic
oscillator with Lagrangian
L = m(ẋଶ + ẏଶ)
2 - k(xଶ + yଶ)
2
then 𝛼 is
Options:
(A) + k
m
(B) - k
m
(C) - 2k
m
(D) 0
Official Answer Key:
B
Step-by-Step Solution & Explanation
### **Official Solution & Step-by-Step Derivation (GATE PH 2022 Q.42)**
**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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