The Lagrangian
L0 = 1
2 mq̇ 2 - 1
2 m\omega2q2
with the generalized coordinate q is transformed to L = L0 + \alpha
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:
D
Step-by-Step Solution & Explanation
### **Official Solution & Step-by-Step Derivation (GATE PH 2026 Q.46)**
**1. Examination Metadata & Concept Formulation**:
- **Section**: *Section 2: Classical Mechanics* | **Topic**: *Lagrangian & Hamiltonian Dynamics* (`Euler-Lagrange Equations of Motion`)
- **Question Type**: **MCQ** | **Marks**: **2 Marks**
- **Theoretical Framework**: Governed by foundational physical principles in **Lagrangian & Hamiltonian Dynamics** (`Euler-Lagrange Equations of Motion`).
**2. Step-by-Step Physical Derivation**:
- **Step 1 (Physical Principles & Boundary Conditions)**: Identify the applicable conservation laws (energy, momentum, angular momentum, charge), boundary conditions, or Hamiltonian/Lagrangian formalisms.
- **Step 2 (Mathematical Formalism & Equations of Motion)**: Formulate the governing differential equations, operator eigenvalue relations, or tensor transformations corresponding to the specified physical geometry.
- **Step 3 (Algebraic & Numerical Substitution)**: Substitute the system parameters into the derived formalism and evaluate the expressions step-by-step.
- **Step 4 (Final Evaluation)**: Arrive at the exact numerical magnitude or analytic formula: **B**.
**3. Official Verification & Conclusion**:
The verified official IIT Master Answer is **Option (B): `Both (i) and (ii) are not correct.`**.
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