Subtopic: Lagrangian & Hamiltonian Dynamics & Special Relativity
Question Statement:
The Lagrangian of a particle of mass m and charge q moving in a uniform magnetic
field of magnitude 2B that points in the z direction, is given by
L =
m
2 v2 + qB(xvy - yvx) where vx , vy , vz are the components of its velocity v .
If px , py , pz denote the conjugate momenta in the x, y, z directions and H is the
Hamiltonian, which of the following option(s) is/are correct ?
Options:
(A) dx
dt = 1
m (px - qBy)
(B) dpx
dt = qB
m (py - qBx)
(C) dpy
dt = - qB
m (px + qBy)
(D) H = 1
2m [(px + qBy)2 + (py - qBx)
2
+ pz
2]
Official Answer Key:
B
Step-by-Step Solution & Explanation
### **Official Solution & Step-by-Step Derivation (GATE PH 2025 Q.51)**
**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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