\magnification=\magstep1

$$
U(r) \quad = \quad K \left[  {\left({{\sigma}\over {r}}\right)}^{12} - 
                                  {\left({{\sigma}\over{r}}\right)}^{6} \right]
$$

$$
U(r) \quad = \quad \left[  {\left({{1}\over {r}}\right)}^{12} - 
                                  {\left({{1}\over{r}}\right)}^{6} \right]
$$


$$
f(x) \quad = \quad f(x_0) \ +\  (x - x_0) {{{df}\over{dx}}|_{x_0}}  \ + \ 
                   {{1}\over{2}} {(x - x_0)}^2 {{d^2f}\over{dx^2}} |_{x_0}  \ +\ 
                   {{1}\over{6}} {(x - x_0)}^3 {{d^3f}\over{dx^3}} |_{x_0}  \ +\ 
                         ...
$$

$$
f(x) \quad = \quad f(x_0) \ +\ 
              \sum_{n=1}^{\infty} {{1}\over{n!}} {(x - x_0)}^n 
                          {{{d^nf}\over{dx^n}}}|_{x_0}
$$

$$
f(x) \quad = \quad f(x_0) \ +\  0  \ + \ 
                   {{1}\over{2}} {(x - x_0)}^2 {{d^2f}\over{dx^2}} |_{x_0}  \ +\ 
                   {{1}\over{6}} {(x - x_0)}^3 {{d^3f}\over{dx^3}} |_{x_0}  \ +\ 
                         ...
$$

$$
f(x) \quad \simeq \quad f(x_0) \ +\   
                {{1}\over{2}} {(x - x_0)}^2 {{d^2f}\over{dx^2}} |_{x_0}  \ + \ ... 
$$


$$
f(x) \quad \simeq \quad f(x_0) \ +\   
                \left[ {{1}\over{2}} {{d^2f}\over{dx^2}} |_{x_0} \right] \ 
                  {(x - x_0)}^2 
$$

$$
U(x) \quad \simeq \quad U(x_0) \ +\   
                \left[ {{1}\over{2}} {{\partial^2 U}\over{\partial x^2}} |_{x_0} \right] \ 
                  {(x - x_0)}^2 
$$


$$
{\rm Force} \ F(x) \quad = \quad - \ {{\partial U}\over{\partial x}}
$$


$$
\eqalign{
{{\partial U}\over{\partial x}} \quad &= \quad  {{\partial}\over{\partial x}} 
        \biggl\{
              \quad U(x_0) \ +\   
                \left[ {{1}\over{2}} {{\partial^2 U}\over{\partial x^2}} |_{x_0} \right] \ 
                  {(x - x_0)}^2 \biggr\} \cr
    &=  \qquad \ \biggl\{  \quad 0  \quad \ + \ \ \ 2 
                \left[ {{1}\over{2}} {{\partial^2 U}\over{\partial x^2}} |_{x_0} \right] \ 
                  {(x - x_0)} \biggr\} \cr
    &=  \qquad \qquad \qquad   
                \left[ {{\partial^2 U}\over{\partial x^2}} |_{x_0} \right] \ 
                  {(x - x_0)} \cr
}
$$



$$
\left( U(x) - U(x_0) \right)  \quad \simeq  \quad 
                \left[ {{1}\over{2}} {{\partial^2 U}\over{\partial x^2}} |_{x_0} \right] \ 
                  {(x - x_0)}^2 
$$

$$
\left( {{U(x) - U(x_0)}\over{x - x_0}} \right)  \quad \simeq \quad 
                \left[ {{1}\over{2}} {{\partial^2 U}\over{\partial x^2}} |_{x_0} \right] \ 
                  {(x - x_0)}
$$

$$
F(x) \quad \simeq \quad - \left[ {{\partial^2 U}\over{\partial x^2}} |_{x_0} \right] \ 
                  {(x - x_0)}
$$

$$
F(x) \quad \simeq \quad - K (x - x_0)
$$

$$
F(x) \quad = \quad - K x
$$

$$
m \thinspace a(x) \quad = \quad - K x 
$$

$$
m \thinspace a \quad = \quad - K x 
$$


$$
m \thinspace {{d^2 x}\over{dt^2}} \quad = \quad - K x
$$

$$
{{d^2 x}\over{dt^2}} \quad = \quad - \thinspace \left( {{K}\over{m}} \right)  x
$$

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

$$
{{d^2 x}\over{dt^2}} \quad = \quad - \thinspace \omega^2  x
$$

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

$$
x(t) \quad = \quad A \sin(\omega t) + B \cos(\omega t)
$$

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


\bye
