# 守恆量

## 動量

$\mathbf{F}=0$

$\mathbf{F}=\frac{\mathrm{d}\mathbf{p}}{\mathrm{d}t}$

## 角動量

$\boldsymbol{\tau}=0$

$\boldsymbol{\tau}=\frac{\mathrm{d}\boldsymbol{\ell}}{\mathrm{d}t}$

## 能量

$E=T+V$

$T=mv^2/2$

$\mathbf{F}=-\nabla V$

$\frac{\mathrm{d}E}{\mathrm{d}t}=m\mathbf{v}\cdot\frac{\mathrm{d}\mathbf{v}}{\mathrm{d}t}+\mathbf{v}\cdot \nabla V=\mathbf{v}\cdot(m\mathbf{a}-\mathbf{F})=0$

## 能量函數

$\mathcal{L}=T - V$

$\frac{d}{dt}\left(\frac{\partial \mathcal{L}}{\partial \dot{q}_i}\right) - \frac{\partial \mathcal{L}}{\partial q_i}=0$

$\frac{d\mathcal{L}}{dt}=\sum_i\frac{\partial \mathcal{L}}{\partial q_i}\dot{q}_i+\sum_i\frac{\partial \mathcal{L}}{\partial \dot{q}_i}\ddot{q}_i+\frac{\partial \mathcal{L}}{\partial t}$

\begin{align}\frac{d\mathcal{L}}{dt} & =\sum_i \frac{d}{dt}\left(\frac{\partial \mathcal{L}}{\partial \dot{q}_i}\right)\dot{q}_i+\sum_i\frac{\partial \mathcal{L}}{\partial \dot{q}_i}\ddot{q}_i+\frac{\partial \mathcal{L}}{\partial t} \\ &=\sum_i \frac{d}{dt}\left(\frac{\partial \mathcal{L}}{\partial \dot{q}_i}\dot{q}_i\right)+\frac{\partial \mathcal{L}}{\partial t} \\ \end{align}

$\mathit{h}\ \stackrel{def}{=}\ \sum_i \frac{\partial \mathcal{L}}{\partial \dot{q}_i}\dot{q}_i - \mathcal{L}$

$\frac{d\mathit{h}}{dt}= - \frac{\partial \mathcal{L}}{\partial t}$

## 參考文獻

1. ^ Morin, David. Introduction to classical mechanics: with problems and solutions. Cambridge University Press. 2008: 138. ISBN 9780521876223.
2. ^ 2.0 2.1 2.2 Goldstein, Herbert, Classical Mechanics 3rd, United States of America: Addison Wesley, pp. 2–5, 61, 312–324, 1980, ISBN 0201657023 （English）