# 作用量本文重定向自 作用量

## 历史

1744年，皮埃尔·莫佩尔蒂在一篇论文《The agreement between the different laws of Nature that had, until now,seemed incompatiable》中，发表了最小作用量原理：光选择的传播路径，作用量最小。他定义作用量为移动速度与移动距离的乘积。用这原理，他证明了费马原理：光传播的正确路径，所需的时间是极值；他也计算出光在反射与同介质传播时的正确路径。1747年，莫佩尔蒂在另一篇论文《On the laws of motion and of rest》中，应用这原理于碰撞，正确地分析了弹性碰撞与非弹性碰撞；这两种碰撞不再需要用不同的理论来解释。

## 作用量形式

### 作用量（泛函）

${\displaystyle {\mathcal {S}}[\mathbf {q} (t)]=\int _{t_{1}}^{t_{2}}L[\mathbf {q} ,\ {\dot {\mathbf {q} }},\ t]\,\mathrm {d} t\,\!}$

### 简略作用量（泛函）

${\displaystyle {\mathcal {S}}_{0}=\int \mathbf {p} \,\mathrm {d} \mathbf {q} \,\!}$

### 哈密顿主函数

${\displaystyle S(\mathbf {q} ,\ \mathbf {P} ,\ t)=\int L[\mathbf {q} ,\ {\dot {\mathbf {q} }},\ t]\,\mathrm {d} t\,\!}$

${\displaystyle S(\mathbf {q} ,\ \mathbf {P} ,\ t)=S(\mathbf {q} ,\ \mathbf {a} ,\ t)\,\!}$

### 哈密顿特征函数

${\displaystyle H=\alpha \,\!}$

${\displaystyle W(\mathbf {q} ,\ \mathbf {a} )=S(\mathbf {q} ,\ \mathbf {a} ,\ t)-\alpha t\,\!}$

${\displaystyle {\frac {dW}{dt}}={\frac {\partial W}{\partial \mathbf {q} }}{\dot {\mathbf {q} }}=\mathbf {p} {\dot {\mathbf {q} }}\,\!}$

${\displaystyle W(\mathbf {q} ,\ \mathbf {a} )=\int \mathbf {p} {\dot {\mathbf {q} }}dt=\int \mathbf {p} \,d\mathbf {q} \,\!}$

### 作用量-角度坐标

${\displaystyle J_{k}=\oint p_{k}\mathrm {d} q_{k}\,\!}$

## 数学导引

${\displaystyle {\mathcal {S}}\ {\stackrel {\mathrm {def} }{=}}\ \int _{t_{1}}^{t_{2}}L(\mathbf {q} ,\ {\dot {\mathbf {q} }},\ t)\,dt\,\!}$

${\displaystyle {\boldsymbol {\varepsilon }}(t_{1})={\boldsymbol {\varepsilon }}(t_{2})\ {\stackrel {\mathrm {def} }{=}}\ 0\,\!}$

${\displaystyle \delta {\mathcal {S}}=\int _{t_{1}}^{t_{2}}\;\left[L(\mathbf {q} +{\boldsymbol {\varepsilon }},\ {\dot {\mathbf {q} }}+{\dot {\boldsymbol {\varepsilon }}})-L(\mathbf {q} ,\ {\dot {\mathbf {q} }})\right]dt=\int _{t_{1}}^{t_{2}}\;\left({\boldsymbol {\varepsilon }}\cdot {\frac {\partial L}{\partial \mathbf {q} }}+{\dot {\boldsymbol {\varepsilon }}}\cdot {\frac {\partial L}{\partial {\dot {\mathbf {q} }}}}\right)\,dt\,\!}$

${\displaystyle \delta {\mathcal {S}}=\left[{\boldsymbol {\varepsilon }}\cdot {\frac {\partial L}{\partial {\dot {\mathbf {q} }}}}\right]_{t_{1}}^{t_{2}}+\int _{t_{1}}^{t_{2}}\;\left({\boldsymbol {\varepsilon }}\cdot {\frac {\partial L}{\partial \mathbf {q} }}-{\boldsymbol {\varepsilon }}\cdot {\frac {d}{dt}}{\frac {\partial L}{\partial {\dot {\mathbf {q} }}}}\right)\,dt\,\!}$

${\displaystyle \delta {\mathcal {S}}=\int _{t_{1}}^{t_{2}}\;{\boldsymbol {\varepsilon }}\cdot \left({\frac {\partial L}{\partial \mathbf {q} }}-{\frac {d}{dt}}{\frac {\partial L}{\partial {\dot {\mathbf {q} }}}}\right)\,dt\,\!}$

${\displaystyle \delta {\mathcal {S}}=\int _{t_{1}}^{t_{2}}\;{\boldsymbol {\varepsilon }}\cdot \left({\frac {\partial L}{\partial \mathbf {q} }}-{\frac {d}{dt}}{\frac {\partial L}{\partial {\dot {\mathbf {q} }}}}\right)\,dt=0\,\!}$

${\displaystyle {\frac {\partial L}{\partial \mathbf {q} }}-{\frac {d}{dt}}{\frac {\partial L}{\partial {\dot {\mathbf {q} }}}}=\mathbf {0} \,\!}$

${\displaystyle p_{k}\ {\stackrel {\mathrm {def} }{=}}\ {\frac {\partial L}{\partial {\dot {q}}_{k}}}\,\!}$

${\displaystyle {\frac {\partial L}{\partial q_{k}}}=0\,\!}$

## 参考文献

• Cornelius Lanczos, "The Variational Principles of Mechanics",（Dover Publications, New York, 1986）, ISBN 0-486-65067-7.这领域最常引用的参考书。
• 列夫·朗道and E. M. Lifshitz, "Mechanics, Course of Theoretical Physics", 3rd ed., Vol. 1,（Butterworth-Heinenann, 1976）, ISBN 0-7506-2896-0.这本书一开始就讲解最小作用量原理。
• Herbert Goldstein "Classical Mechanics", 2nd ed.,（Addison Wesley, 1980）, pp. 35-69。
• Thomas A. Moore "Least-Action Principle" in Macmillan Encyclopedia of Physics, Volume 2,（Simon & Schuster Macmillan, 1996）, ISBN 0-02-897359-3, OCLC 35269891, pages 840–842。
• Robert Weinstock, "Calculus of Variations, with Applications to Physics and Engineering",（Dover Publications, 1974）, ISBN 0-486-63069-2。非常好的古早书。
• Dugas, René, "A History of Mechanics",（Dover, 1988）, ISBN 0-486-65632-2, pp. 254-275。