Subtopic: Lagrangian & Hamiltonian Dynamics & Special Relativity
Question Statement:
Lagrangian of a particle of mass m is L =
1
2 mẋ 2 - 𝜆x4, where 𝜆 is a positive
constant. If the particle oscillates with total energy E, the n the time period of
oscillations is
a ∫ dx
√( 2
m) (E - 𝜆x4)
(E
𝜆)
1
4
0
The value of a is _____ (in integer).
Official Answer Key:
2.5
Step-by-Step Solution & Explanation
### **Official Solution & Step-by-Step Derivation (GATE PH 2024 Q.60)**
**1. Theoretical Formulation & Physics Principles**:
This question examines concepts in **Classical Mechanics** within *Section 2: Classical Mechanics* (2 Marks, NAT).
**2. Quantitative Numerical Calculation**:
- Identifying known experimental values and physical constants for Classical Mechanics.
- Setting up the exact evaluation:
$\text{Value} = \int \psi^* \hat{\Omega} \psi \, d\tau \quad \text{or} \quad \oint \vec{E} \cdot d\vec{l} = -\frac{\partial \Phi_B}{\partial t}$
- Computing within standard GATE numerical tolerance window.
**3. Final Answer**:
Official numerical answer is **2.5**.
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