# 廣義速度

${\displaystyle {\dot {q}}_{i}={dq_{i} \over dt}\,\!}$

## 與動能的關係

${\displaystyle T={\frac {1}{2}}mv^{2}\,\!}$

${\displaystyle \mathbf {v} ={\frac {d\mathbf {r} }{dt}}=\sum _{i}\ {\frac {\partial \mathbf {r} }{\partial q_{i}}}{\dot {q}}_{i}+{\frac {\partial \mathbf {r} }{\partial t}}\,\!}$

${\displaystyle T={\frac {1}{2}}m\left(\sum _{i}\ {\frac {\partial \mathbf {r} }{\partial q_{i}}}{\dot {q}}_{i}+{\frac {\partial \mathbf {r} }{\partial t}}\right)^{2}\,\!}$

${\displaystyle T=T_{0}+T_{1}+T_{2}\,\!}$

${\displaystyle T_{0}={\frac {1}{2}}m\left({\frac {\partial \mathbf {r} }{\partial t}}\right)^{2}\,\!}$
${\displaystyle T_{1}=\sum _{i}\ m{\frac {\partial \mathbf {r} }{\partial t}}\cdot {\frac {\partial \mathbf {r} }{\partial q_{i}}}{\dot {q}}_{i}\,\!}$
${\displaystyle T_{2}=\sum _{i,j}\ {\frac {1}{2}}m{\frac {\partial \mathbf {r} }{\partial q_{i}}}\cdot {\frac {\partial \mathbf {r} }{\partial q_{j}}}{\dot {q}}_{i}{\dot {q}}_{j},\!}$

${\displaystyle T_{0}\,\!}$ ${\displaystyle T_{1}\,\!}$ ${\displaystyle T_{2}\,\!}$ 分別為廣義速度${\displaystyle {\dot {q}}_{i}\,\!}$ 的0次、1次、2次齊次函數。如果這系統是定常系統，位置不顯性地含時間，${\displaystyle {\frac {\partial \mathbf {r} }{\partial t}}=0\,\!}$ ，則只有${\displaystyle T_{2}\,\!}$ 不等於零。所以，${\displaystyle T=T_{2}\,\!}$ ，動能是廣義速度的2次齊次函數。

## 參考文獻

1. ^ Goldstein, Herbert. Classical Mechanics 3rd. United States of America: Addison Wesley. 1980: pp. 25. ISBN 0201657023 （英语）.