# 分析力学/牛顿运动定律的抽象化

${\displaystyle {\boldsymbol {F}}={\dot {\boldsymbol {p}}}}$

${\displaystyle {\boldsymbol {F}}}$为空间上仅关于坐标 ${\displaystyle {\boldsymbol {r}}}$的场，而 ${\displaystyle {\boldsymbol {r}}}$是仅关于 ${\displaystyle t}$的函数，则上式变为

${\displaystyle {\boldsymbol {F}}({\boldsymbol {r}}(t))=m{\ddot {\boldsymbol {r}}}(t)}$

{\displaystyle {\begin{aligned}{\boldsymbol {F}}&=-\nabla U\\&=-\left({\partial U \over \partial x}{\boldsymbol {e}}_{x}+{\partial U \over \partial y}{\boldsymbol {e}}_{y}+{\partial U \over \partial z}{\boldsymbol {e}}_{z}\right)\end{aligned}}}

${\displaystyle U}$称为质点的势能.

{\displaystyle {\begin{aligned}m{\boldsymbol {\dot {r}}}&={\partial T \over \partial {\boldsymbol {\dot {r}}}}\\&={\partial T \over \partial {\dot {x}}}{\boldsymbol {e}}_{x}+{\partial T \over \partial {\dot {y}}}{\boldsymbol {e}}_{y}+{\partial T \over \partial {\dot {z}}}{\boldsymbol {e}}_{z}\end{aligned}}}

${\displaystyle F_{x}=m{\ddot {x}}={\operatorname {d} \! \over \operatorname {d} \!t}(m{\dot {x}})}$

${\displaystyle -{\partial U \over \partial x}={\operatorname {d} \! \over \operatorname {d} \!t}{\partial T \over \partial {\dot {x}}}}$

${\displaystyle y,z}$分量亦同理. 记 ${\displaystyle x,y,z}$${\displaystyle q_{1},q_{2},q_{3}}$，则相应方程组可记为

${\displaystyle -{\partial U \over \partial q_{i}}={\operatorname {d} \! \over \operatorname {d} \!t}{\partial T \over \partial {\dot {q_{i}}}}}$

${\displaystyle -{\partial E \over \partial q_{i}}={\operatorname {d} \! \over \operatorname {d} \!t}{\partial E \over \partial {\dot {q_{i}}}}}$

${\displaystyle {\partial L \over \partial q_{i}}={\operatorname {d} \! \over \operatorname {d} \!t}{\partial L \over \partial {\dot {q_{i}}}}}$

${\displaystyle L}$ 被称为拉格朗日量.