# Doubt from Arnold; Mathematical methods of classical mechanics (page 20)

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.

• Hint: The period is just $P=2 \pi/ \omega$, where $\omega$ is the pulsation. For a potential of the form $U=U'' x^2/2$... (the mass is 1). Commented Jun 14, 2022 at 6:36

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.

• Thanks a lot, thanks for the simple explanation @Miyase Commented Jun 14, 2022 at 8:13