https://www.zhihu.com/question/21540160
牛顿定律
,Fi是第i个物体受的合力
![\Rightarrow \mathbf{F }_{i }\, \mathrm{d } \, t \, =m _{i }\, \mathrm{d } \, \dot{\mathbf{r }}_{i } \,](https://www.zhihu.com/equation?tex=%5CRightarrow+%5Cmathbf%7BF+%7D_%7Bi+%7D%5C%2C+%5Cmathrm%7Bd+%7D+%5C%2C+t++%5C%2C+%3Dm+_%7Bi+%7D%5C%2C+%5Cmathrm%7Bd+%7D+%5C%2C+%5Cdot%7B%5Cmathbf%7Br+%7D%7D_%7Bi+%7D+%5C%2C+)
![\Rightarrow \mathbf{F }_{i }\cdot \dot{\mathbf{r }}_{i }\, \mathrm{d } \, t \, =\mathbf{F }_{i }\cdot \, \mathrm{d } \, \mathbf{r }_{i } \, =m _{i }\dot{\mathbf{r }}_{i }\cdot \, \mathrm{d } \, \dot{\mathbf{r }}_{i } \,](https://www.zhihu.com/equation?tex=%5CRightarrow+%5Cmathbf%7BF+%7D_%7Bi+%7D%5Ccdot+%5Cdot%7B%5Cmathbf%7Br+%7D%7D_%7Bi+%7D%5C%2C+%5Cmathrm%7Bd+%7D+%5C%2C+t++%5C%2C+%3D%5Cmathbf%7BF+%7D_%7Bi+%7D%5Ccdot+%5C%2C+%5Cmathrm%7Bd+%7D+%5C%2C+%5Cmathbf%7Br+%7D_%7Bi+%7D+%5C%2C+%3Dm+_%7Bi+%7D%5Cdot%7B%5Cmathbf%7Br+%7D%7D_%7Bi+%7D%5Ccdot+%5C%2C+%5Cmathrm%7Bd+%7D+%5C%2C+%5Cdot%7B%5Cmathbf%7Br+%7D%7D_%7Bi+%7D+%5C%2C+)
![\Rightarrow \mathbf{F }_{i }\cdot \, \mathrm{d } \, \mathbf{r }_{i } \, =\, \mathrm{d } \left( \frac{1 }{2 }m _{i }{\dot{\mathbf{r }}_{i }}^{2 }\right) \,](https://www.zhihu.com/equation?tex=%5CRightarrow+%5Cmathbf%7BF+%7D_%7Bi+%7D%5Ccdot+%5C%2C+%5Cmathrm%7Bd+%7D+%5C%2C+%5Cmathbf%7Br+%7D_%7Bi+%7D+%5C%2C+%3D%5C%2C+%5Cmathrm%7Bd+%7D+%5Cleft%28+%5Cfrac%7B1+%7D%7B2+%7Dm+_%7Bi+%7D%7B%5Cdot%7B%5Cmathbf%7Br+%7D%7D_%7Bi+%7D%7D%5E%7B2+%7D%5Cright%29++%5C%2C+)
咱们用T表动能
![\Rightarrow \sum _{i }\mathbf{F }_{i }\cdot \, \mathrm{d } \, \mathbf{r }_{i } \, \; =\sum _{i }\, \mathrm{d } \left( \frac{1 }{2 }m _{i }{\dot{\mathbf{r }}_{i }}^{2 }\right) \, \; =\sum _{i }\, \mathrm{d } \, T _{i } \, \;](https://www.zhihu.com/equation?tex=%5CRightarrow+%5Csum+_%7Bi+%7D%5Cmathbf%7BF+%7D_%7Bi+%7D%5Ccdot+%5C%2C+%5Cmathrm%7Bd+%7D+%5C%2C+%5Cmathbf%7Br+%7D_%7Bi+%7D+%5C%2C+%5C%3B+%3D%5Csum+_%7Bi+%7D%5C%2C+%5Cmathrm%7Bd+%7D+%5Cleft%28+%5Cfrac%7B1+%7D%7B2+%7Dm+_%7Bi+%7D%7B%5Cdot%7B%5Cmathbf%7Br+%7D%7D_%7Bi+%7D%7D%5E%7B2+%7D%5Cright%29++%5C%2C+%5C%3B+%3D%5Csum+_%7Bi+%7D%5C%2C+%5Cmathrm%7Bd+%7D+%5C%2C+T+_%7Bi+%7D+%5C%2C+%5C%3B+)
而
i表内,e表外
![=\, \mathrm{d } \, W _{外 } \, +\, \mathrm{d } \, W _{内 } \,](https://www.zhihu.com/equation?tex=%3D%5C%2C+%5Cmathrm%7Bd+%7D+%5C%2C+W+_%7B%E5%A4%96+%7D+%5C%2C+%2B%5C%2C+%5Cmathrm%7Bd+%7D+%5C%2C+W+_%7B%E5%86%85+%7D+%5C%2C+)
![\Rightarrow \, \mathrm{d } \, W _{外 } \, +\, \mathrm{d } \, W _{内 } \, =\sum _{i }\, \mathrm{d } \, T _{i } \, \;](https://www.zhihu.com/equation?tex=%5CRightarrow+%5C%2C+%5Cmathrm%7Bd+%7D+%5C%2C+W+_%7B%E5%A4%96+%7D+%5C%2C+%2B%5C%2C+%5Cmathrm%7Bd+%7D+%5C%2C+W+_%7B%E5%86%85+%7D+%5C%2C+%3D%5Csum+_%7Bi+%7D%5C%2C+%5Cmathrm%7Bd+%7D+%5C%2C+T+_%7Bi+%7D+%5C%2C+%5C%3B+)
积个分
这是动能定理
如果非弹性碰撞的话,物体在受挤压力的时候向内凹陷,一定有内力负功,还有很少摩擦力,按物理语言,这些功转成了热
![W _{外 }+W _{内 }=-Q < 0](https://www.zhihu.com/equation?tex=W+_%7B%E5%A4%96+%7D%2BW+_%7B%E5%86%85+%7D%3D-Q+%3C+0+)
所以动能关系
![{\left( \sum _{i }T _{i }\; \right) }_{末 }< {\left( \sum _{i }T _{i }\; \right) }_{初 }](https://www.zhihu.com/equation?tex=%7B%5Cleft%28+%5Csum+_%7Bi+%7DT+_%7Bi+%7D%5C%3B+%5Cright%29+%7D_%7B%E6%9C%AB+%7D%3C+%7B%5Cleft%28+%5Csum+_%7Bi+%7DT+_%7Bi+%7D%5C%3B+%5Cright%29+%7D_%7B%E5%88%9D+%7D)
![\mathbf{F }_{i }\, \mathrm{d } \, t \, =m _{i }\, \mathrm{d } \, \dot{\mathbf{r }}_{i } \,](https://www.zhihu.com/equation?tex=%5Cmathbf%7BF+%7D_%7Bi+%7D%5C%2C+%5Cmathrm%7Bd+%7D+%5C%2C+t++%5C%2C+%3Dm+_%7Bi+%7D%5C%2C+%5Cmathrm%7Bd+%7D+%5C%2C+%5Cdot%7B%5Cmathbf%7Br+%7D%7D_%7Bi+%7D+%5C%2C+)
![\Rightarrow \sum _{i }\mathbf{F }_{i }\, \mathrm{d } \, t \, \; =\sum _{i }m _{i }\, \mathrm{d } \, \dot{\mathbf{r }}_{i } \, \;](https://www.zhihu.com/equation?tex=%5CRightarrow+%5Csum+_%7Bi+%7D%5Cmathbf%7BF+%7D_%7Bi+%7D%5C%2C+%5Cmathrm%7Bd+%7D+%5C%2C+t++%5C%2C+%5C%3B+%3D%5Csum+_%7Bi+%7Dm+_%7Bi+%7D%5C%2C+%5Cmathrm%7Bd+%7D+%5C%2C+%5Cdot%7B%5Cmathbf%7Br+%7D%7D_%7Bi+%7D+%5C%2C+%5C%3B+)
![\sum _{i }\mathbf{F }_{i }\, \mathrm{d } \, t \, \; =\sum _{i }{\mathbf{F }_{i }}^{{\left( i \right) }}\, \mathrm{d } \, t \, \; +\sum _{i }{\mathbf{F }_{i }}^{{\left( e \right) }} \, \mathrm{d } \, t \, \;](https://www.zhihu.com/equation?tex=%5Csum+_%7Bi+%7D%5Cmathbf%7BF+%7D_%7Bi+%7D%5C%2C+%5Cmathrm%7Bd+%7D+%5C%2C+t++%5C%2C+%5C%3B+%3D%5Csum+_%7Bi+%7D%7B%5Cmathbf%7BF+%7D_%7Bi+%7D%7D%5E%7B%7B%5Cleft%28+i+%5Cright%29+%7D%7D%5C%2C+%5Cmathrm%7Bd+%7D+%5C%2C+t++%5C%2C+%5C%3B+%2B%5Csum+_%7Bi+%7D%7B%5Cmathbf%7BF+%7D_%7Bi+%7D%7D%5E%7B%7B%5Cleft%28+e+%5Cright%29+%7D%7D+%5C%2C+%5Cmathrm%7Bd+%7D+%5C%2C+t++%5C%2C+%5C%3B+)
而物体间内力是成对出现的
![\sum _{i }{\mathbf{F }_{i }}^{{\left( i \right) }}\; =\sum _{i \neq j }\mathbf{F }_{i j }\;](https://www.zhihu.com/equation?tex=%5Csum+_%7Bi+%7D%7B%5Cmathbf%7BF+%7D_%7Bi+%7D%7D%5E%7B%7B%5Cleft%28+i+%5Cright%29+%7D%7D%5C%3B+%3D%5Csum+_%7Bi+%5Cneq+j+%7D%5Cmathbf%7BF+%7D_%7Bi+j+%7D%5C%3B+)
我们还有牛顿第三定律:
![\sum _{i \neq j }\mathbf{F }_{i j }\; =0](https://www.zhihu.com/equation?tex=%5Csum+_%7Bi+%5Cneq+j+%7D%5Cmathbf%7BF+%7D_%7Bi+j+%7D%5C%3B+%3D0+)
所以合力的冲量元即为合外力的冲量元
![\sum _{i }\mathbf{F }_{i }\, \mathrm{d } \, t \, \; =\sum _{i }{\mathbf{F }_{i }}^{{\left( e \right) }} \, \mathrm{d } \, t \, \;](https://www.zhihu.com/equation?tex=%5Csum+_%7Bi+%7D%5Cmathbf%7BF+%7D_%7Bi+%7D%5C%2C+%5Cmathrm%7Bd+%7D+%5C%2C+t++%5C%2C+%5C%3B+%3D%5Csum+_%7Bi+%7D%7B%5Cmathbf%7BF+%7D_%7Bi+%7D%7D%5E%7B%7B%5Cleft%28+e+%5Cright%29+%7D%7D+%5C%2C+%5Cmathrm%7Bd+%7D+%5C%2C+t++%5C%2C+%5C%3B+)
用Ii表示第i个物体受合外力的冲量
![\, \mathrm{d } \, \mathbf{I }_{i } \, ={\mathbf{F }_{i }}^{{\left( e \right) }}\, \mathrm{d } \, t \,](https://www.zhihu.com/equation?tex=%5C%2C+%5Cmathrm%7Bd+%7D+%5C%2C+%5Cmathbf%7BI+%7D_%7Bi+%7D+%5C%2C+%3D%7B%5Cmathbf%7BF+%7D_%7Bi+%7D%7D%5E%7B%7B%5Cleft%28+e+%5Cright%29+%7D%7D%5C%2C+%5Cmathrm%7Bd+%7D+%5C%2C+t++%5C%2C+)
![\sum _{i }\mathbf{F }_{i }\, \mathrm{d } \, t \, \; =\sum _{i }\, \mathrm{d } \, \mathbf{I }_{i } \, \; =\, \mathrm{d } \, \mathbf{I } \,](https://www.zhihu.com/equation?tex=%5Csum+_%7Bi+%7D%5Cmathbf%7BF+%7D_%7Bi+%7D%5C%2C+%5Cmathrm%7Bd+%7D+%5C%2C+t++%5C%2C+%5C%3B+%3D%5Csum+_%7Bi+%7D%5C%2C+%5Cmathrm%7Bd+%7D+%5C%2C+%5Cmathbf%7BI+%7D_%7Bi+%7D+%5C%2C+%5C%3B+%3D%5C%2C+%5Cmathrm%7Bd+%7D+%5C%2C+%5Cmathbf%7BI+%7D+%5C%2C+)
所以动量定理的微分式我们也有了,I是体系合外力的冲量
![\, \mathrm{d } \, \mathbf{I } \, =\sum _{i }m _{i }\, \mathrm{d } \, \dot{\mathbf{r }}_{i } \, \;](https://www.zhihu.com/equation?tex=%5C%2C+%5Cmathrm%7Bd+%7D+%5C%2C+%5Cmathbf%7BI+%7D+%5C%2C+%3D%5Csum+_%7Bi+%7Dm+_%7Bi+%7D%5C%2C+%5Cmathrm%7Bd+%7D+%5C%2C+%5Cdot%7B%5Cmathbf%7Br+%7D%7D_%7Bi+%7D+%5C%2C+%5C%3B+)
碰撞的话,一瞬间质点系的合外力为零
![\mathbf{I }=\mathbf{0 }](https://www.zhihu.com/equation?tex=%5Cmathbf%7BI+%7D%3D%5Cmathbf%7B0+%7D)
自然有
![\sum _{i }m _{i }\, \mathrm{d } \, \dot{\mathbf{r }}_{i } \, \; =\mathbf{0 }](https://www.zhihu.com/equation?tex=%5Csum+_%7Bi+%7Dm+_%7Bi+%7D%5C%2C+%5Cmathrm%7Bd+%7D+%5C%2C+%5Cdot%7B%5Cmathbf%7Br+%7D%7D_%7Bi+%7D+%5C%2C+%5C%3B+%3D%5Cmathbf%7B0+%7D)
所以积分后就有我们的动量守恒律:
![\sum _{i }m _{i }\dot{\mathbf{r }}_{i }\; =常](https://www.zhihu.com/equation?tex=%5Csum+_%7Bi+%7Dm+_%7Bi+%7D%5Cdot%7B%5Cmathbf%7Br+%7D%7D_%7Bi+%7D%5C%3B+%3D%E5%B8%B8+)
![\sum _{i }m _{i }\mathbf{v }_{i }\; =守恒量](https://www.zhihu.com/equation?tex=%5Csum+_%7Bi+%7Dm+_%7Bi+%7D%5Cmathbf%7Bv+%7D_%7Bi+%7D%5C%3B+%3D%E5%AE%88%E6%81%92%E9%87%8F+)
作者:沈飞
链接:https://www.zhihu.com/question/21540160/answer/469870033
来源:知乎
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