# 懸鏈線

${\displaystyle y=a\cosh {\frac {x}{a}}}$或者簡單地表示為${\displaystyle y={\frac {a\left(e^{\frac {x}{a}}+e^{-{\frac {x}{a}}}\right)}{2}}}$

${\displaystyle {\frac {L}{a}}=\sinh {\frac {d}{a}}}$

## 方程的推導

${\displaystyle T\sin \theta =mg}$

${\displaystyle T\cos \theta =H}$

${\displaystyle \tan \theta ={\frac {\mathrm {d} y}{\mathrm {d} x}}={\frac {mg}{H}}}$

${\displaystyle mg=\rho s}$，　其中${\displaystyle s}$是右段${\displaystyle AB}$繩子的長度，${\displaystyle \rho }$是繩子線重量密度，${\displaystyle \tan \theta }$為切線方向，記${\displaystyle a={\frac {\rho }{H}}}$, 代入得微分方程${\displaystyle {\frac {\mathrm {d} y}{\mathrm {d} x}}=as}$;

${\displaystyle p'={\frac {\rho }{H}}{\sqrt {1+p^{2}}}\ \cdots \cdots \ (2)}$

${\displaystyle \int {\frac {dp}{\sqrt {1+p^{2}}}}=\int adx}$

${\displaystyle ln(p+{\sqrt {1+p^{2}}})=ax+C}$，即${\displaystyle \mathrm {arsinh} p=ax+C}$

${\displaystyle x=0}$時，${\displaystyle {\frac {dy}{dx}}=p=0}$

## 工程中的應用

${\displaystyle y=a\ \left(\cosh {\frac {x}{a}}-1\right)}$

${\displaystyle L=a\ \sinh {\frac {x}{a}}}$
${\displaystyle \tan \alpha =\sinh {\frac {x}{a}}}$
${\displaystyle F_{0}=a\ \gamma }$