# 連心力

${\displaystyle \mathbf {F} =F{\hat {\mathbf {r} }}}$

${\displaystyle \mathbf {F} =F(r){\hat {\mathbf {r} }}}$

## 角動量恆定

${\displaystyle {\boldsymbol {\tau }}=\mathbf {r} \times \mathbf {F} =\mathbf {r} \times F{\hat {\mathbf {r} }}=0}$

${\displaystyle {\frac {d\mathbf {L} }{dt}}={\boldsymbol {\tau }}}$

## 平面運動

${\displaystyle \mathbf {r} \cdot \mathbf {L} =\mathbf {r} \cdot (\mathbf {r} \times (m\mathbf {v} ))=0}$

## 平面速度恆定

${\displaystyle L=mr^{2}{\dot {\theta }}}$

${\displaystyle {\frac {dA}{dt}}={\frac {1}{2}}r^{2}{\dot {\theta }}={\frac {L}{2m}}}$

## 連心勢

${\displaystyle \mathbf {F} =-\nabla V(\mathbf {r} )}$

${\displaystyle \nabla \times \mathbf {F} =-\nabla \times \nabla V=0}$
• 對於任何簡單的閉合迴路，連心力所做的機械功${\displaystyle W}$是0：
${\displaystyle W=\oint _{C}\mathbf {F} \cdot d\mathbf {r} =0}$

${\displaystyle \mathbf {F} =-{\frac {\partial V}{\partial r}}\ {\hat {\mathbf {r} }}}$

${\displaystyle V(r)}$連心勢。連心力也只能跟${\displaystyle r}$有關：

${\displaystyle \mathbf {F} =F(r){\hat {\mathbf {r} }}}$

## 有效勢能

${\displaystyle {\mathcal {L}}(r,\theta )={\frac {m}{2}}\left({\dot {r}}^{2}+r^{2}{\dot {\theta }}^{2}\right)-V(r)}$

${\displaystyle m{\ddot {r}}-mr{\dot {\theta }}^{2}-F(r)=0}$
${\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}(mr^{2}{\dot {\theta }})=0}$

${\displaystyle P_{\theta }\ {\stackrel {def}{=}}\ {\frac {\partial {\mathcal {L}}}{\partial \theta }}=mr^{2}{\dot {\theta }}}$

${\displaystyle {\dot {p}}_{r}=-{\frac {\mathrm {d} }{\mathrm {d} r}}\left[{\frac {P_{\theta }^{2}}{2mr^{2}}}+V(r)\right]}$

${\displaystyle V_{\rm {Eff}}(r)={\frac {P_{\theta }^{2}}{2m}}{\frac {1}{r^{2}}}+V(r)}$

${\displaystyle V_{\rm {Eff}}(r)={\frac {P_{\theta }^{2}}{2m}}{\frac {1}{r^{2}}}-{\frac {K}{r}}}$

## 有心运动的轨迹的确定

${\displaystyle u={\frac {1}{r}},h={\frac {L_{0}}{m}}}$

${\displaystyle -mh^{2}u^{2}({\frac {d^{2}u}{d\theta ^{2}}}+u)=F(u)}$

${\displaystyle r={\frac {1}{u(\theta )}}=r(\theta )}$

### 平方反比类有心力的运动轨迹方程

${\displaystyle -mh^{2}u^{2}({\frac {d^{2}u}{d\theta ^{2}}}+u)={\frac {k}{r^{2}}}}$

${\displaystyle {\frac {d^{2}u}{d\theta ^{2}}}+mf(u)=0}$

${\displaystyle r={\frac {1}{u}}={\frac {\frac {mh^{2}}{k^{2}}}{1+{\frac {mh^{2}}{k^{2}}}\cos(\theta -\theta _{0})}}}$

## 注释和参考文献

1. ^ Goldstein, Herbert. Classical Mechanics 3rd. United States of America: Addison Wesley. 1980: pp. 7. ISBN 0201657023.
2. ^ 这是个二阶常系数非齐次线性方程。
3. ^ 陈世民. 理论力学简明教程. 高等教育出版社. : 49页. ISBN 978-7-04-023918-8.
4. ^ 弹簧振子简谐振动的运动微分方程${\displaystyle {\frac {d^{2}r}{dt^{2}}}+{\frac {kx}{m}}=0}$有通解为：${\displaystyle r=A\cos(\omega t)\!}$，其中${\displaystyle A}$为积分常数，可以通过初始条件确定；${\displaystyle \omega ={\sqrt {\frac {k}{m}}}}$，为简谐振的角速度。
5. ^ 陈世民. 理论力学简明教程. 高等教育出版社. : 63页. ISBN 978-7-04-023918-8.
6. ^ 万有引力即是典型的平方反比类型的力。