Subtopic: Lagrangian & Hamiltonian Dynamics & Special Relativity
Question Statement:
A bead is constrained to move along a long, massless, frictio nless horizontal rod
parallel to the x axis. The rod itself is moving vertically upward along the
z direction against gravity with a constant speed, starting from z = 0 at t = 0, and
remains horizontal. The conjugate momenta are denoted by px , py , pz and the
Hamiltonian by H. Which of the following option(s) is/are correct?
Options:
(A) H is the total energy of the system and is conserved
(B) H is the total energy of the system and is not conserved
(C) H is not the total energy of the system, but it is conserved
(D) H is not the total energy of the system and is not conserved
Official Answer Key:
B
Step-by-Step Solution & Explanation
### **Official Solution & Step-by-Step Derivation (GATE PH 2025 Q.52)**
**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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