Subtopic: Lagrangian & Hamiltonian Dynamics & Special Relativity
Question Statement:
Consider the Lagrangian L = mẋẏ - m𝜔0
2xy. If px and py denote the generalized
momenta conjugate to x and y, respectively, then the canonical equations of motion
are
Options:
(A) ẋ = px
m , ṗx = -m𝜔0
2x , ẏ = py
m , ṗy = -m𝜔0
2y
(B) ẋ = px
m , ṗx = m𝜔0
2x , ẏ = py
m , ṗy = m𝜔0
2y
(C) ẋ = py
m , ṗx = -m𝜔0
2y , ẏ = px
m , ṗy = -m𝜔0
2x
(D) ẋ = py
m , ṗx = m𝜔0
2y , ẏ = px
m , ṗy = m𝜔0
2x
Official Answer Key:
B
Step-by-Step Solution & Explanation
### **Official Solution & Step-by-Step Derivation (GATE PH 2024 Q.38)**
**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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