\magnification=\magstep1

$$
U(x) \quad = \quad m g h(x) \quad = \quad 
          m g H \left[ 1 \ - \ \cos({{2 \pi x}\over{L}}) \right]
$$


$$
\eqalign{
\cos(\theta) \quad &= \quad \cos(0) \ + \ {{{\theta}^1}\over{1!}} (-\sin(0)) 
                                   \ + \ {{{\theta}^2}\over{2!}} (-\cos(0))
                                   \ + \ {{{\theta}^3}\over{3!}} (\sin(0))
                                   \ + \ {{{\theta}^4}\over{4!}} (\cos(0))
                                   \ + \ ... \cr
                   &= \quad 1      \quad + \quad 0  \quad 
                                   \ - \quad {{{\theta}^2}\over{2!}}  \quad 
                                   \ - \quad 0 \quad 
                                   \ + \quad {{{\theta}^4}\over{4!}}  \quad 
                                   \ + \quad ... \cr 
}
$$

$$
\eqalign{
\sin(\theta) \quad &= \quad \sin(0) \ + \ {{{\theta}^1}\over{1!}} (\cos(0)) 
                                   \ + \ {{{\theta}^2}\over{2!}} (-\sin(0))
                                   \ + \ {{{\theta}^3}\over{3!}} (-\cos(0))
                                   \ + \ {{{\theta}^4}\over{4!}} (\sin(0))
                                   \ + ... \cr
                   &= \quad 0      \quad + \quad {{{\theta}^1}\over{1!}}  \quad 
                                   \ - \quad 0 \quad 
                                   \ - \quad {{{\theta}^3}\over{3!}}  \quad 
                                   \ + \quad 0 \quad 
                                   \ + \quad {{{\theta}^5}\over{5!}} \quad 
                                   \ + \quad ... \cr 
}
$$


$$
\eqalign{
e^{i\theta} \quad &= \quad e^{0}  \ + \ {{{i\theta}^1}\over{1!}} (e^{0}) 
                                   \ + \ {{{i^2\theta}^2}\over{2!}} (e^{0})
                                   \ + \ {{{i^3\theta}^3}\over{3!}} (e^{0})
                                   \ + \ {{{i^4\theta}^4}\over{4!}} (e^{0})
                                   \ + ... \cr
                   &= \quad 1      \quad + \quad {{{i\theta}^1}\over{1!}}  \quad 
                                   \ - \quad {{{\theta}^2}\over{2!}} \quad 
                                   \ - \quad {{{i\theta}^3}\over{3!}}  \quad 
                                   \ + \quad {{{\theta}^4}\over{4!}} \quad
                                   \ + \quad {{{i\theta}^5}\over{5!}} \quad 
                                   \ + \quad ... \cr 
}
$$

$$
{ {d^2 y}\over{dt^2} } \quad = \quad 
            - \thinspace \left( {{k}\over{m}} \right) \thinspace y
                  \quad = \quad - \ \omega^2 y
$$

$$
\omega \quad \equiv  \quad \sqrt{ {{k}\over{m}} }
$$

$$
y(t) \quad = \quad A \cos(\omega t + \phi) 
$$

$$
y(t) \quad = \quad B \cos(\omega t) \ + \ C \sin(\omega t) 
$$

$$
y(0) \quad = \quad 0 {\rm \ m}
$$

$$
{{dy}\over{dt}}|_{t=0} \ = \ v_y(0) \quad = \quad 0.12 {\rm \ m/s} 
$$

$$
y(t) \quad = \quad (-0.04 {\rm \ m}) \cos(\omega t + {{\pi}\over{2}} ) 
$$

$$
y(t) \quad = \quad (0) \ \cos(\omega t) \ + \ (0.04 {\rm \ m})\  \sin(\omega t) 
$$

$$
y(t) \quad = \quad A e^{i(\omega t + \phi)}
$$

$$
e^{i\theta} \quad = \quad \cos(\theta) \ + \ i\thinspace \sin(\theta)
$$

$$
y(t) \quad = \quad A e^{i(\omega t + \phi)} \quad = 
                    \quad A \cos(\omega t + \phi) \ + \  
                    i\thinspace A \sin(\omega t + \phi)
$$

$$
{{dy}\over{dt}} \quad = \quad v_y(t) \quad = \quad i \omega A e^{i(\omega t + \phi)}
                        \quad = \quad i \omega y(t)
$$

$$
{{d^2y}\over{dt^2}} \quad = \quad a_y(t) \quad = \quad
                        - \omega^2 A e^{i(\omega t + \phi)}  \quad = \quad
                        - \omega^2 y(t)
$$


$$
y(t) \quad = \quad A e^{-i(\omega t + \phi)} \quad = 
                    \quad A \cos(\omega t + \phi) \ - \  
                    i\thinspace A \sin(\omega t + \phi)
$$

$$
y(t) \quad = \quad A e^{i(\omega t + \phi)}
$$

$$
y(t) \quad = \quad B e^{i(\omega t)} \ + \ C e^{-i(\omega t)} 
$$


















\bye 
