\magnification=\magstep1

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

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

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

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

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


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

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


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


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

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

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

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

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

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

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

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

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

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

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

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

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


$$
m \thinspace {{d^2 x}\over{dt^2}} \ + \ K x \quad = \quad 0
$$


$$
\int{ \left( m \thinspace {{d^2 x}\over{dt^2}} \right) } dx \ 
             + \ \int{ \left( K x \right) } dx \quad = 
             \quad \int{ \left( 0 \right) } dx
$$


$$
\int{ \left( m \thinspace {{d^2 x}\over{dt^2}} \right) } dx \ 
             + \ {{1}\over{2}}  K x^2 \quad = 
             \quad C 
$$

$$
{ m v^2} 
$$

$$
{ {d}\over{dx}} {( m v^2)} \quad = \quad {2 m v} \ {{dv}\over{dx}}
$$

$$
 \quad {2 m v} \ {{dv}\over{dx}} \quad = 
            \quad { 2 m } \ {{dx}\over{dt}} \ {{dv}\over{dx}}
$$

$$
 \quad {2 m v} \ {{dv}\over{dx}} \quad = 
            \quad { 2 m } \ {{dx}\over{dt}} \ {{dv}\over{dx}} \quad =
            \quad { 2 m } \ {{dv}\over{dt}} \quad =
            \quad { 2 m } \ { {d^2 x}\over{dt^2} }
$$


$$
{{1}\over{2}}  m v^2 \ 
             + \ {{1}\over{2}}  K x^2 \quad = 
             \quad C 
$$

$$
\int{ \left( m \thinspace {{d^2 x}\over{dt^2}} \right) } dt \ 
             + \ \int{ \left( K x \right) } dt \quad = 
             \quad \int{ \left( 0 \right) } dt
$$

$$
\int{ \left( m \thinspace {{d^2 x}\over{dt^2}} \right) } dt \ 
             + \ \int{ \left( {\rm Force} \right) } dt \quad = 
             \quad \int{ \left( 0 \right) } dt
$$

$$
 m \thinspace v \ 
             + \ \int{ \left( {\rm Force} \right) } dt \quad = 
             \quad C
$$


$$
m ( v_f - v_i ) \ + \ \int{ \left( {\rm Force} \right) } dt \quad = \quad C
$$





\bye
