非完整系統

$f(x_1,\ x_2,\ x_3,\ \dots,\ x_N,\ t)=0$

廣義座標的轉換

$x_d=g_k(x_1,\ x_2,\ x_3,\ \dots,\ x_{d-1},\ x_{d+1},\ \dots,\ x_N,\ t)$

$x_i=x_i(q_1,\ q_2,\ \dots,\ q_m,\ t)\ ,\qquad\qquad\qquad i=1,\ 2,\ 3,\ \dots N$

微分形式表示

$\sum_j\ c_{ij} dq_j+c_i dt=0$

$df_i=\sum_j\ c_{ij} dq_j+c_i dt=0$

半完整系統

$f_i(q_1,\ q_2,\ \dots,\ q_N,\ \dot{q}_1,\ \dot{q}_2,\ \dots,\ \dot{q}_N)=0\ ,\qquad\qquad\qquad i=1,\ 2,\ 3,\ \dots n$

$\sum_{i=1}^n\ \lambda_i f_i=0$

$\delta\int_{t_1}^{t_2}\ L\ dt=0$

$\int_{t_1}^{t_2}\ \sum_j\ \left(\frac{\partial L}{\partial q_j} - \frac{d}{dt}\left(\frac{\partial L}{\partial \dot{q}_j}\right)\right)\delta q_j \ dt=0$

$\delta\int_{t_1}^{t_2}\ \left(L+\sum_{i=1}^n\ \lambda_i f_i\right)\ dt=0$

$\int_{t_1}^{t_2}\ \sum_j\ \left(\frac{\partial L}{\partial q_j} - \frac{d}{dt}\left(\frac{\partial L}{\partial \dot{q}_j}\right) + \mathcal{F}_j\right)\delta q_j \ dt=0$ ;

$\mathcal{F}_j=\sum_i\ \left[\frac{\partial (\lambda_i f_i)}{\partial q_j} - \frac{d}{dt}\left(\frac{\partial (\lambda_i f_i)}{\partial \dot{q}_j}\right)\right]$

$\frac{d}{dt}\left(\frac{\partial L}{\partial \dot{q}_j}\right) - \frac{\partial L}{\partial q_j}=\mathcal{F}_j$

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

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。

參考文獻

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