# 引力势能

${\displaystyle E_{p}=-{\frac {GMm}{r}}}$

## 证明

${\displaystyle m}$从无穷远处拉至${\displaystyle A}$的过程中，${\displaystyle M}$${\displaystyle m}$的间距不断减小，万有引力和外力不断增大，是变力。处理变力做功问题，需要借助积分。

${\displaystyle A}$点与${\displaystyle M}$质心相距${\displaystyle r_{A}}$先关注微元，再整体思考，可得

${\displaystyle W={\vec {F}}\cdot {\vec {x}}=\left|{\vec {F}}\right|\left|{\vec {x}}\right|\cos <{\vec {F}},{\vec {x}}>}$

${\displaystyle \Longrightarrow {\rm {d}}W={\frac {GMm}{r^{2}}}(-{\rm {d}}r)\cos \pi }$ [2]

${\displaystyle W=\int _{\infty }^{r_{A}}{\frac {GMm}{r^{2}}}(-{\rm {d}}r)(-1)}$

${\displaystyle W=-{\frac {GMm}{r_{A}}}}$

${\displaystyle E_{p}=-{\frac {GMm}{r_{A}}}}$

## 參考資料

1. ^ Alan Guth The Inflationary Universe: The Quest for a New Theory of Cosmic Origins (1997), Random House , ISBN 0-224-04448-6 Appendix A: Gravitational Energy demonstrates the negativity of gravitational energy.
2. ^ 引力势能公式是如何推出来的？ - 知乎. www.zhihu.com. [2024-02-07].
3. ^ Lev Davidovich Landau & Evgeny Mikhailovich Lifshitz, The Classical Theory of Fields, (1951), Pergamon Press, ISBN 7-5062-4256-7