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\subsubsection*{Equilibrium Points}

A particle placed at an \emph{equilibrium point} $x_0$ will stay there for all time.
\[ m \ddot{x} = -\dfrac{V}{x} \]
implies equilibrium points obey
\[ \left. \dfrac{V}{x} \right|_{x_0} \]
i.e. critical points of $V$. We can look at motion
near equilibrium point. Taylor expanding,
\[ V(x) \approx V(x_0) + \half (x - x_0)^2 V''(x)
+ \cdots \]
\begin{itemize}
    \item $V''(x) > 0 \implies$ minimum of $V$,
        potential of harmonic oscillator
        \[ m \ddot{x} \approx V''(x_0)(x - x_0) \]
        This point is \emph{stable}. Particle
        oscillates with frequency $\omega =
        \sqrt{\frac{V''(x_0)}{m}}$.
    \item $V''(x_)) \implies$ maximum of $V
        \implies$ point is \emph{unstable}.
        \[ x - x_0 \approx A e^{\alpha t} + B
        e^{-\alpha t} \]
        with
        \[ \alpha = \sqrt{-\frac{V''(x_0)}{m}} .\]
    \item $v''(x_0) = 0 \implies$ more work
        needed.
\end{itemize}

\begin{example*}[the pendulum]
    The equation of motion is
    \[ \ddot{\theta} = -\frac{g}{x} \sin \theta \]
    The energy is
    \[ E = \half ml^2 \dot{\theta}^2 - mgl \cos
    \theta .\]
    \begin{itemize}
        \item $E > mgl \implies \dot{\theta} \neq 0$
            for all $t$.
        \item $E < mgl \implies \dot{\theta} = 0$
            at some point $\theta_0$. $\implies$
            oscillates back and forth and
            \[ E = -mgl \cos \theta_0 .\]
    \end{itemize}
    Using the general solution for 1-dimensional
    system,
    \[ T = \int_0^{\theta_0} \frac{\dd
    \theta}{\sqrt{\frac{2E}{ml^2} + \left(
    \frac{2g}{l} \right) \cos\theta}} =
    4\sqrt{\frac{l}{g}} \int_0^{\theta_0}
    \frac{\dd \theta}{\sqrt{2\cos\theta -
    2\cos\theta_0}} \]
    but, for small $\theta$, $\cos\theta \approx 1
    - \frac{\theta^2}{2}$:
    \[ T \approx 4 \sqrt{\frac{l}{g}}
    \int_0^{\theta_0} \frac{\dd
    \theta}{\sqrt{\theta_0^2 - \theta^2}} = 2\pi
    \sqrt{\frac{l}{g}} \]
    (Note: independent of $\theta_0$). \\
    This is the result for the harmonic
    oscillator, of course.
\end{example*}
