GATE 2022 PH – Question 42
If $\dot x\dot y+\alpha xy$ is conserved for $L=m(\dot x^2+\dot y^2)/2-k(x^2+y^2)/2$, find alpha.
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Correct answer: (A) $k/m$
Explanation
Let $L=\tfrac m2(\dot x^2+\dot y^2)-\tfrac k2(x^2+y^2)$. The equations of motion are
$$m\ddot x=-kx,\qquad m\ddot y=-ky .$$
**Time derivative of the candidate quantity** $Q=\dot x\dot y+\alpha xy$:
$$\frac{dQ}{dt}=\ddot x\dot y+\dot x\ddot y+\alpha(\dot xy+x\dot y).$$
Substitute $\ddot x=-\dfrac kmx$ and $\ddot y=-\dfrac kmy$:
$$\frac{dQ}{dt}=-\frac km\,x\dot y-\frac km\,\dot xy+\alpha(\dot xy+x\dot y)=\left(\alpha-\frac km\right)(x\dot y+y\dot x).$$
For $Q$ to be conserved ($dQ/dt=0$ for all motions), we need
$$\alpha=\frac km\quad(\text{option A}).$$