Subtopic: Lagrangian & Hamiltonian Dynamics & Special Relativity
Question Statement:
The energy of a free, relativistic particle of rest mass m moving along the x axis in
one dimension, is denoted by T. When moving in a given potential V(x), its
Hamiltonian is H = T + V(x). In the presence of this potential, i ts speed is v,
conjugate momentum p, and the Lagrangian L. Then, which of the following
option(s) is/are correct?
Options:
(A) H = c2√m2 + p2
c2 + V(x)
(B) v = pc
√p2 + m2c2
(C) L = mc2√1 - v2
c2 - V(x)
(D) L = - mc2√1 - v2
c2 - V(x)
Official Answer Key:
B
Step-by-Step Solution & Explanation
### **Official Solution & Step-by-Step Derivation (GATE PH 2025 Q.54)**
**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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