\magnification=\magstep2


$$
\vec{F_{\rm air}} \quad = \quad -b \vec{v}
$$

$$
F_1 \quad = \quad  -k x_1 \ + \ k (x_2 - x_1) \ - \ bv_1  
      \quad = \quad -2 k x_1 \ + \ k x_2 \ - \ b v_1
$$
$$
F_2 \quad = \quad  -k (x_2 - x_1 ) \ - \ k x_2 \ - \ bv_2  
      \quad = \quad k x_1 \ - \ 2 k x_2 \ - \ b v_2
$$


$$
m {{d^2 x_1}\over{dt^2}}  \quad = \quad -2 k x_1 \ + \ k x_2 \ - \ 
                     b \thinspace {{d x_1}\over{dt}} 
$$

$$
m {{d^2 x_2}\over{dt^2}}  \quad = \quad k x_1 \ - \ 2 k x_2 \ - \ 
                     b \thinspace {{d x_2}\over{dt}} 
$$

$$
m {{d^2 (x_1 + x_2) }\over{dt^2}}  \quad = \quad - k (x_1 + x_2)  \ - \ 
                     b \thinspace {{d (x_1 + x_2)}\over{dt}} 
$$

$$
s_1 \quad \equiv \quad x_1  \ + \ x_2
$$

$$
m {{d^2 s_1}\over{dt^2}} \quad = \quad -k s_1 \ - \ b {{ds_1}\over{dt}}
$$

$$
s_1(t) \quad = \quad A_1 \cos(\omega_1 t + \phi_1) e^{-t/\tau}
$$

$$
s_1(t) \quad = \quad A_1 e^{i(\omega_1 t + \phi_1)-(t/\tau)}
$$

$$
{{d s_1}\over{dt}} \quad = \quad \left(i \omega_1 - {{1}\over{\tau}} \right) \  
                       A_1 e^{i(\omega_1 t + \phi_1)-(t/\tau)} 
$$

$$
{{d^2 s_1}\over{dt^2}} \quad = \quad 
          \left( - \omega_1^2 - i {{2 \omega_1}\over{\tau}} + {{1}\over{\tau^2}}  \right) 
                       A_1 e^{i(\omega_1 t + \phi_1)-(t/\tau)} 
$$


$$
m {{d^2 s_1}\over{dt^2}}  \ + \  b {{ds_1}\over{dt}} \ + \ k s_1 \quad = \quad 0
$$

$$
m\left(-\omega_1^2 - i {{2 \omega_1}\over{\tau}} +  {{1}\over{\tau^2}} \right)
    \ + \ 
           b \left(i \omega_1 - {{1}\over{\tau}} \right) 
    \ + \ 
           k \quad = \quad 0
$$


$$
- {{2 \omega_1}\over{\tau}} \ + \ b \omega_1 \quad = \quad 0
$$

$$
b \quad = \quad {{2 m}\over{\tau}}  
   \quad \quad \quad {\rm or \ } 
   \quad \quad \quad 
\tau \quad = \quad {{2 m}\over{b}}  
$$

$$
-\omega_1^2 \ + \ {{1}\over{\tau^2}} \ - \ {{b}\over{m \tau}} \ + \ 
         {{k}\over{m}}  \quad = \quad 0
$$

$$
\omega_1^2 \quad = \quad {{k}\over{m}} \ - \ {{b^2}\over{4 m^2}}
$$
               

$$
m {{d^2 (x_2 - x_1) }\over{dt^2}}  \quad = \quad - 3 k (x_2 - x_1)  \ - \ 
                     b \thinspace {{d (x_2 - x_1)}\over{dt}} 
$$


$$
m {{d^2 s_2}\over{dt^2}}  \ + \  b {{ds_2}\over{dt}} \ + \ 3 k s_2 \quad = \quad 0
$$

$$
s_2(t) \quad = \quad A_2 e^{i(\omega_2 t + \phi_2)-(t/\tau)}
$$

$$
\omega_2^2 \quad = \quad {{3k}\over{m}} \ - \ {{b^2}\over{4 m^2}}
$$

$$
\eqalign{
s_1 \quad &= \quad x_1 \ + \ x_2  \cr
s_2 \quad &= \quad x_2 \ - \ x_1  \cr
}
$$
               
$$
\eqalign{
x_1 \quad &= \quad {{1}\over{2}} ( s_1 \ - \ s_2  ) \cr
x_2 \quad &= \quad {{1}\over{2}} ( s_1 \ + \ s_2  ) \cr
}
$$
               
$$
x_1 \quad = \quad 
       {{1}\over{2}} \  \left( A_1 e^{i(\omega_1 t + \phi_1)-(t/\tau)} \ - \ 
                               A_2 e^{i(\omega_2 t + \phi_2)-(t/\tau)} \right) 
$$

$$
x_2 \quad = \quad 
       {{1}\over{2}} \  \left( A_1 e^{i(\omega_1 t + \phi_1)-(t/\tau)} \ + \ 
                               A_2 e^{i(\omega_2 t + \phi_2)-(t/\tau)} \right) 
$$
       
$$
x_1(t) \quad = \quad A_1 e^{i(\omega_1 t + \phi_1)-(t/\tau)}
$$

$$
{{d x_1}\over{dt}} \quad = \quad 
       \left( i \omega \ - \ {{1}\over{\tau}} \right) \ x_1
$$

$$
\eqalign{
{{d^2 x_1}\over{dt^2}} \quad &= \quad 
       { \left( i \omega \ - \ {{1}\over{\tau}} \right)}^2 \ x_1 \cr
                             &= \quad
       { \left( -\omega^2 \ - \ i {{2 \omega}\over{\tau}} 
                       \ + \ {{1}\over{\tau^2}} \right)} \ x_1 \cr
}
$$

$$
\pmatrix{
    - 2 {{k}\over{m}} + {{b}\over{m \tau}} - i {{b \omega}\over{m}} & 
           {{k}\over{m}} \cr
      \phantom{.} \cr
       {{k}\over{m}} &
         - 2 {{k}\over{m}} + {{b}\over{m \tau}} - i {{b \omega}\over{m}} \cr 
}
  \pmatrix{
      x_1 \cr  \phantom{.} \cr x_2 \cr
  }
     =
    \left( -\omega^2 - i {{2 \omega}\over{\tau}} + {{1}\over{\tau^2}} \right)
  \pmatrix{
      x_1 \cr \phantom{.} \cr x_2  \cr
  }
$$
  

$$
\pmatrix{
    - 2 {{k}\over{m}} + {{b}\over{m \tau}} - i {{b \omega}\over{m}} & 
           {{k}\over{m}} \cr
      \phantom{.} \cr
       {{k}\over{m}} &
         - 2 {{k}\over{m}} + {{b}\over{m \tau}} - i {{b \omega}\over{m}} \cr 
}
  \pmatrix{
      p \cr  \phantom{.} \cr q \cr
  }
     =
    \left( -\omega^2 - i {{2 \omega}\over{\tau}} + {{1}\over{\tau^2}} \right)
  \pmatrix{
      p \cr \phantom{.} \cr q  \cr
  }
$$
  
$$
s_1 \quad = \quad p x_1 \ + \ q x_2 
$$

$$
s_1 \quad = \quad 1 x_1 \ + \ 1 x_2  
   \quad \quad \Rightarrow 
   \quad \quad
   s_1 \quad = \quad \pmatrix { 1 \cr 1 } 
$$
       

$$
s_2 \quad = \quad -1 x_1 \ + \ 1 x_2  
   \quad \quad \Rightarrow 
   \quad \quad
   s_2 \quad = \quad \pmatrix { -1 \cr 1 } 
$$
       


       







\bye
