GATE 2025 PH – Question 62
The Hamiltonian is $H(p,q)=p^2/(2m)+q^2A(q)$ with real $A(q)$. If $m\ddot q=-5qA(q)$ and $dA/dq=nA(q)/q$, the value of $n$ (in integer) is _____
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Correct answer: 3
Explanation
**Hamilton's equations** for $H=\dfrac{p^2}{2m}+q^2A(q)$:
$$\dot q=\frac pm,\qquad\dot p=-\frac{\partial H}{\partial q}=-\left(2qA+q^2A^\prime\right).$$
So the equation of motion is
$$m\ddot q=\dot p=-2qA-q^2\frac{dA}{dq}.$$
**Compare with the given** $m\ddot q=-5qA$:
$$2qA+q^2A^\prime=5qA\;\Rightarrow\;q^2A^\prime=3qA\;\Rightarrow\;\frac{dA}{dq}=\frac{3A}{q}.$$
Since $\dfrac{dA}{dq}=\dfrac{nA}{q}$, we have $n=\mathbf{3}$ (so $A\propto q^3$).