非完整系統

${\displaystyle f(x_{1},\ x_{2},\ x_{3},\ \dots ,\ x_{N},\ t)=0}$

廣義坐標的轉換

${\displaystyle x_{d}=g_{k}(x_{1},\ x_{2},\ x_{3},\ \dots ,\ x_{d-1},\ x_{d+1},\ \dots ,\ x_{N},\ t)}$

${\displaystyle x_{i}=x_{i}(q_{1},\ q_{2},\ \dots ,\ q_{m},\ t)\ ,\qquad \qquad \qquad i=1,\ 2,\ 3,\ \dots N}$

微分形式表示

${\displaystyle \sum _{j}\ c_{ij}dq_{j}+c_{i}dt=0}$

${\displaystyle df_{i}=\sum _{j}\ c_{ij}dq_{j}+c_{i}dt=0}$

半完整系統

${\displaystyle f_{i}(q_{1},\ q_{2},\ \dots ,\ q_{N},\ {\dot {q}}_{1},\ {\dot {q}}_{2},\ \dots ,\ {\dot {q}}_{N})=0\ ,\qquad \qquad \qquad i=1,\ 2,\ 3,\ \dots n}$

${\displaystyle \sum _{i=1}^{n}\ \lambda _{i}f_{i}=0}$

${\displaystyle \delta \int _{t_{1}}^{t_{2}}\ L\ dt=0}$

${\displaystyle \int _{t_{1}}^{t_{2}}\ \sum _{j}\ \left({\frac {\partial L}{\partial q_{j}}}-{\frac {d}{dt}}\left({\frac {\partial L}{\partial {\dot {q}}_{j}}}\right)\right)\delta q_{j}\ dt=0}$

${\displaystyle \delta \int _{t_{1}}^{t_{2}}\ \left(L+\sum _{i=1}^{n}\ \lambda _{i}f_{i}\right)\ dt=0}$

${\displaystyle \int _{t_{1}}^{t_{2}}\ \sum _{j}\ \left({\frac {\partial L}{\partial q_{j}}}-{\frac {d}{dt}}\left({\frac {\partial L}{\partial {\dot {q}}_{j}}}\right)+{\mathcal {F}}_{j}\right)\delta q_{j}\ dt=0}$ ;

${\displaystyle {\mathcal {F}}_{j}=\sum _{i}\ \left[{\frac {\partial (\lambda _{i}f_{i})}{\partial q_{j}}}-{\frac {d}{dt}}\left({\frac {\partial (\lambda _{i}f_{i})}{\partial {\dot {q}}_{j}}}\right)\right]}$

${\displaystyle {\frac {d}{dt}}\left({\frac {\partial L}{\partial {\dot {q}}_{j}}}\right)-{\frac {\partial L}{\partial q_{j}}}={\mathcal {F}}_{j}}$

${\displaystyle N}$個方程式加上${\displaystyle n}$個約束方程式，給予了${\displaystyle N+n}$個方程式來解${\displaystyle N}$個未知廣義座標與${\displaystyle n}$個拉格朗日乘子。

實例

1. 物體在做滾動運動。
2. 系統的約束包括不等式
3. 系統的約束與速度有關（例如普法夫約束）。

參考文獻

1. ^ Goldstein, Herbert. Classical Mechanics 3rd. United States of America: Addison Wesley. 1980: pp. 46–47. ISBN 0201657023 （英语）.