# 牛顿万有引力定律

（更严谨的表达请见下文中的矢量式方程。）

## 重力加速度

a1为事先已知质点的重力加速度。由牛顿第二定律知$F= m_1\ a_1$， 即$a_1=\frac{F}{m_1}$。取代前面方程中的F

$a_1 = G \frac{m_2}{r^2}$

## 向量式

$\mathbf{F}_{12} = G {m_1 m_2 \over r_{21}^2} \, \mathbf{\hat{r}}_{21}$$\mathbf{F}_{12} = - G {m_1 m_2 \over r_{21}^2} \, \mathbf{\hat{r}}_{12}$

F12: 物体2对物体1的引力
G: 万有引力常数,其值约等于6.67259×10-11 N m2/kg2
m1m2: 分别为物体1和物体2的质量
r21 = | r2r1 |: 物体2和物体1之间的距离
$\mathbf{\hat{r}}_{21} \equiv \frac{\mathbf{r}_2 - \mathbf{r}_1}{\vert\mathbf{r}_2 - \mathbf{r}_1\vert}$: 物体1物体2的单位向量

$\mathbf{a}_1 = G {m_2 \over r^2_{21}} \, \mathbf{\hat{r}}_{21}$

## 重力场

$\mathbf g(\mathbf r) = G {m_2 \over r^2} \, \mathbf{\hat{r}}$

$\mathbf{F}( \mathbf r) = m \mathbf g(\mathbf r)$

## 牛顿理论存在的问题

### 理论问题

• 没有任何征兆表明重力的传送媒介可以被识别出，牛顿自己也对这种无法说明的超距作用感到不满意（参看后文条目“牛顿定律的局限性”）。
• 牛顿的理论需要定义重力可以瞬时传播。因此给出了古典自然时空观的假设，这样亦能使约翰内斯·开普勒所观测到的角动量守恒成立。但是，这与爱因斯坦的狭义相对论理论有直接的冲突，因为狭义相对论定义了速度的极限——真空中的光速——在此速度下信号可以被传送。

### 牛顿定律的局限性

I have not yet been able to discover the cause of these properties of gravity from phenomena and I feign no hypotheses... It is enough that gravity does really exist and acts according to the laws I have explained, and that it abundantly serves to account for all the motions of celestial bodies. That one body may act upon another at a distance through a vacuum without the mediation of anything else, by and through which their action and force may be conveyed from one another, is to me so great an absurdity that, I believe, no man who has in philosophic matters a competent faculty of thinking could ever fall into it.

1. ^ 参看条目等效原理