# 重力位

## 位能

${\displaystyle V={\frac {U}{m}},}$

${\displaystyle \Delta U\approx mg\Delta h.}$

## 數學形式

${\displaystyle V(\mathbf {r} )=-{\frac {GM}{r}}}$

### 推導

${\displaystyle \mathbf {F} =-{\frac {GMm}{r^{2}}}\mathbf {\hat {r}} }$

• ${\displaystyle \mathbf {F} }$ ：質量${\displaystyle m}$ 的質點受到的萬有引力
• ${\displaystyle G}$ 萬有引力常數
• ${\displaystyle m}$ ：質點1的質量
• ${\displaystyle M}$ ：質點2的質量
• ${\displaystyle r}$ ：兩個物體之間的距離
• ${\displaystyle \mathbf {\hat {r}} }$ ：由${\displaystyle M}$ 指向${\displaystyle m}$ 的單位向量

${\displaystyle \mathbf {g} ={\frac {\mathbf {F} }{m}}=-G{\frac {M}{r^{2}}}\mathbf {\hat {r}} }$

${\displaystyle W=\int _{\mathbf {a} }^{\mathbf {b} }-\mathbf {g} \cdot \mathbf {dl} }$

${\displaystyle \mathbf {g} \cdot \mathbf {dl} =-{\frac {GM}{r^{2}}}\mathbf {d} r}$

{\displaystyle {\begin{aligned}W&=\int _{\mathbf {a} }^{\mathbf {b} }-\mathbf {g} \cdot \mathbf {dl} \\&=\int _{\mathbf {a} }^{\mathbf {b} }{\frac {GM}{r^{2}}}\mathbf {d} r\\&=GM({\frac {1}{r_{a}}}-{\frac {1}{r_{b}}})\end{aligned}}}

${\displaystyle V(\mathbf {r} )=-\int _{\mathcal {O}}^{\mathbf {r} }\mathbf {g} \cdot \mathbf {dl} }$

${\displaystyle V(\mathbf {r} )}$ 就稱為重力位。只要預先設定一個標準參考點${\displaystyle {\mathcal {O}}}$ ${\displaystyle V}$ 的值就可以由${\displaystyle \mathbf {r} }$ 來決定。

${\displaystyle V(\mathbf {r} )=GM({\frac {1}{\infty }}-{\frac {1}{r}})=-{\frac {GM}{r}}}$

{\displaystyle {\begin{aligned}W&=\int _{\mathbf {a} }^{\mathbf {b} }-\mathbf {g} \cdot \mathbf {dl} \\&=-\int _{\mathbf {\mathcal {O}} }^{\mathbf {b} }\mathbf {g} \cdot \mathbf {dl} -\int _{\mathbf {a} }^{\mathbf {\mathcal {O}} }\mathbf {g} \cdot \mathbf {dl} \\&=-\int _{\mathbf {\mathcal {O}} }^{\mathbf {b} }\mathbf {g} \cdot \mathbf {dl} +\int _{\mathbf {\mathcal {O}} }^{\mathbf {a} }\mathbf {g} \cdot \mathbf {dl} \\&=V(\mathbf {b} )-V(\mathbf {a} )\end{aligned}}}

### 重力位的梯度

${\displaystyle V(\mathbf {b} )-V(\mathbf {a} )=-\int _{\mathbf {a} }^{\mathbf {b} }\mathbf {g} \cdot \mathbf {dl} }$

${\displaystyle V(\mathbf {b} )-V(\mathbf {a} )=\int _{\mathbf {a} }^{\mathbf {b} }(\mathbf {\nabla } V)\cdot \mathbf {dl} }$

${\displaystyle \int _{\mathbf {a} }^{\mathbf {b} }(\mathbf {\nabla } V)\cdot \mathbf {dl} =-\int _{\mathbf {a} }^{\mathbf {b} }\mathbf {g} \cdot \mathbf {dl} }$

${\displaystyle \mathbf {g} =-\mathbf {\nabla } V}$

### 疊加

${\displaystyle V(\mathbf {r} )=\sum _{i=1}^{n}-{\frac {Gm_{i}}{|\mathbf {r} -\mathbf {r_{i}} |}}.}$

${\displaystyle V(\mathbf {r} )=-\int _{\mathbf {R} ^{3}}{\frac {G}{|\mathbf {r} -\mathbf {r} \prime |}}\,\mathbf {d} m(\mathbf {r} \prime ),}$

${\displaystyle V(\mathbf {r} )=-\int _{\mathbf {R} ^{3}}{\frac {G}{|\mathbf {r} -\mathbf {r} \prime |}}\,\rho (\mathbf {r} \prime )\mathbf {d} \tau (\mathbf {r} \prime ).}$

## 泊松方程

${\displaystyle \mathbf {\nabla } \cdot \mathbf {g} =-4\pi G\rho .}$

${\displaystyle {\nabla }^{2}V=4\pi G\rho .}$

## 球形對稱

${\displaystyle V(r)={\frac {2}{3}}\pi G\rho (r^{2}-3R^{2}),\qquad r\leq R.}$

## 多極展開

${\displaystyle V(\mathbf {r} )=-\int _{\mathbf {R} ^{3}}{\frac {G}{|\mathbf {r} -\mathbf {r} \prime |}}\,\rho (\mathbf {r} \prime )\mathbf {d} \tau (\mathbf {r} \prime )}$

## 註腳

1. ^ Solivérez, C.E. Electrostatics and magnetostatics of polarized ellipsoidal bodies: the depolarization tensor method 1st English. Free Scientific Information. 2016. ISBN 978-987-28304-0-3.
2. ^ Marion, J.B.; Thornton, S.T. Classical Dynamics of particles and systems 4th. Harcourt Brace & Company. 1995: 192. ISBN 0-03-097302-3.
3. ^ Arfken, George B.; Weber, Hans J. Mathematical Methods For Physicists International Student Edition 6th. . 2005: 72. ISBN 978-0-08-047069-6.
4. ^ Sang, David; Jones, Graham; Chadha, Gurinder; Woodside, Richard; Stark, Will; Gill, Aidan. Cambridge International AS and A Level Physics Coursebook illustrated. Cambridge University Press. 2014: 276. ISBN 978-1-107-69769-0.
5. ^ Muncaster, Roger. A-level Physics illustrated. . 1993: 106. ISBN 978-0-7487-1584-8.