I am trying to do a problem from Arnold; Mathematical methods of Classical mechanics. But I didn't get the desired result mentioned by the author.
Let $E_0$ be the value of the potential function at a minimum point $\xi$. Find the period $T_0 = \lim_{E\to E_0} T(E)$ of small oscillations in a neighbourhood of the point $\xi$.
Answer: $\frac{2\pi}{\sqrt{U''(\xi)}}$, Where $U(\xi)$ is the potential energy.
It's an important result with a large scope.
Let $U$ be the potential with a minimum for $x=\xi$. Taylor expansion near $\xi$ is:
$$U(x)\simeq U(\xi)+(x-\xi)U'(\xi)+\frac{1}{2}(x-\xi)^2U''(\xi)$$
Since $\xi$ is the location of a minimum, you have both:
$$U'(\xi)=0\\ U''(\xi)>0$$
Conservation of mechanical energy yields, with $k=U''(\xi)$:
$$\frac{d}{dt}\left(\frac{1}{2}\,m\dot{x}^2+U(x)\right)=0 \quad\Rightarrow\quad m\ddot{x}+k(x-\xi)=0 $$
You can read directly in this harmonic oscillator equation the angular frequency:
$$\omega^2=\frac{k}{m}$$
and the associated period:
$$T=\frac{2\pi}{\omega}=2\pi\sqrt{\frac{m}{U''(\xi)}}$$
$m$ is missing in the result you mention. Perhaps $U$ is redefined to contain $m$, or $U$ is a gravitational potential instead of a potential energy. But you get the idea.
