Subtopic: Hamilton's Equations & Phase Space Dynamics
Question Statement:
The Hamiltonian for a one dimensional system with mass m, position q and
momentum p is H(p, q) =
p2
2m + q2A(q), where A(q) is a real function of q. If
m
d2q
dt2 = -5qA(q), then
dA(q)
dq = n
A(q)
q . The value of n (in integer) is ______
Official Answer Key:
3 to 3
Step-by-Step Solution & Explanation
### **Official Solution & Step-by-Step Derivation (GATE PH 2025 Q.62)**
**1. Hamilton's Equations of Motion**:
- System Hamiltonian:
$H(p, q) = \frac{p^2}{2m} + q^2 A(q)$
**2. Step-by-Step Analytical Derivation**:
- Step 1: Hamilton's canonical equations:
$\dot{q} = \frac{\partial H}{\partial p} = \frac{p}{m} \implies p = m\dot{q}$
$\dot{p} = -\frac{\partial H}{\partial q} = -\frac{\partial}{\partial q}[q^2 A(q)] = -\left(2q A(q) + q^2 \frac{dA}{dq}\right)$
- Step 2: Differentiating momentum:
$\dot{p} = m \ddot{q} = m \frac{d^2q}{dt^2}$
$m \frac{d^2q}{dt^2} = -2q A(q) - q^2 \frac{dA}{dq}$
- Step 3: Equating with the given equation of motion $m \frac{d^2q}{dt^2} = -5q A(q)$:
$-2q A(q) - q^2 \frac{dA}{dq} = -5q A(q)$
$-q^2 \frac{dA}{dq} = -3q A(q)$
- Dividing both sides by $-q^2$ (for non-zero coordinate $q$):
$\frac{dA(q)}{dq} = 3\frac{A(q)}{q}$
- Comparing with $\frac{dA(q)}{dq} = n\frac{A(q)}{q}$ gives:
$n = 3$
**3. Official Verification**:
The verified official IIT answer is **3** (Numerical range: `3 to 3`).
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