Subtopic: Lagrangian & Hamiltonian Dynamics & Special Relativity
Question Statement:
The Lagrangian
L0 = 1
2 mq̇ 2 - 1
2 m𝜔2q2
with the generalized coordinate q is transformed to L = L0 + 𝛼
df(q)
dt . Consider the
following statements:
(i) Expression for the canonical momentum does not change.
(ii) The equation of the motion does not change.
Which of the following options is correct for the above statements?
Options:
(A) Both (i) and (ii) are correct.
(B) Both (i) and (ii) are not correct.
(C) (i) is correct and (ii) is not correct.
(D) (i) is not correct and (ii) is correct.
Official Answer Key:
B
Step-by-Step Solution & Explanation
### **Official Solution & Step-by-Step Derivation (GATE PH 2026 Q.46)**
**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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