Action
$$S=\int_{t_1}^{t_2}dt\quad L\left(q_i,{\dot{q}}_i,t\right)$$
Lagrangian
$$L\left(q_i,{\dot{q}}_i,t\right)=E_{kin}-E_{pot}$$
Generalized coordinates and velocities \(q_i,{\dot{q}}_i\)
Lagrange's equations
$$\frac{d}{dt}\left(\frac{\partial L}{\partial{\dot{q}}_i}\right)-\frac{\partial L}{\partial{q}_i}=0$$
Generalized momenta
$$p_i=\frac{\partial L}{\partial{\dot{q}}_i}$$
\(\delta q\left(t\right)=q'\left(t\right)-q\left(t\right)\)
\(\delta\dot{q}\left(t\right)=\frac{d}{dt}\delta q\left(t\right)\)
\(\delta q\left(t_1\right)=\delta q\left(t_2\right)=0\)

$$\delta S=\int_{t_1}^{t_2}dt\left(\frac{\partial L}{\partial{\dot{q}}_i}{\delta\dot{q}}_i+\frac{\partial L}{\partial{q}_i}\delta{q}_i\right)=\left.\left(\frac{\partial L}{\partial{\dot{q}}_i}\right)\delta{q}_i\right|_{t_1}^{t_2}+\int_{t_1}^{t_2}dt\left(\frac{\partial L}{\partial{q}_i}-\frac{d}{dt}\frac{\partial L}{\partial{\dot{q}}_i}\right)\delta{q}_i$$

$$\delta L=\frac{\partial L}{\partial {\dot{q}}_i}{\delta\dot{q}}_i+\frac{\partial L}{\partial{q}_i}\delta{q}_i=\frac{d}{dt}F$$
$$\frac{d}{dt}\left(\frac{\partial L}{\partial{\dot{q}}_i}\delta{q}_i-F\right)=0$$
Conserved quantity:
$$Q=\frac{\partial L}{\partial{\dot{q}}_i}\delta{q}_i-F$$
$$\frac{d}{dt}Q=0$$