GATE 2024 PH – Question 60
$L=m\dot x^2/2-\lambda x^4$ with positive lambda. At total energy E, $$T=a\int_0^{(E/\lambda)^{1/4}}\frac{dx}{\sqrt{(2/m)(E-\lambda x^4)}}.$$ The integer a is _____
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Correct answer: 4
Explanation
**Energy conservation:** $E=\tfrac12m\dot x^2+\lambda x^4$, so
$$\dot x=\pm\sqrt{\frac2m\left(E-\lambda x^4\right)} .$$
**Turning points.** The motion is between $\pm x_0$ where $E=\lambda x_0^4$, i.e. $x_0=(E/\lambda)^{1/4}$.
**Time for a part of the motion.** The time to go from $x=0$ to $x_0$ is
$$t_{0\to x_0}=\int_0^{x_0}\frac{dx}{\sqrt{(2/m)(E-\lambda x^4)}} .$$
**One full period** consists of four such quarter segments (centre → right turning point → centre → left turning point → centre), by symmetry:
$$T=4\int_0^{x_0}\frac{dx}{\sqrt{(2/m)(E-\lambda x^4)}} .$$
So the integer is $a=\mathbf{4}$.