Lagrange Multipliers

Consider a Lagrangian of the form

$$\tilde{L}=L(x_i,{\dot{x}}_i,t)+{\lambda }_i(t)F_i(x_i,{\dot{x}}_i,t)$$

Then the Euler-Lagrange Equation would be

$$\frac{d}{dt}\frac{\partial L}{\partial {\dot{x}}_i}-\frac{\partial L}{\partial x_i}-\sum _{i=1}^n\left[{\lambda }_i(t)\frac{\partial F}{\partial x_i}-\frac{d}{dt}\left[\frac{\partial F}{\partial {\dot{x}}_i}{\lambda }_i(t)\right]\right]$$