Suppose that a particle is free to move in all directions. Three coordinates are needed to specify its location.
The presence of constrains mean that some coordinates might be less than three positions. A constraint that reduces the number of position of a particle is called a holonomic constraint. The minimal set of required independent coordinates are called generalized coordinates, denoted by $q_k$.
The Lagrangian is defined as
$$ \begin{equation} L=T-U \end{equation} $$
where $T$ is kinetic energy and $U$ is potential energy. Having chosen a set of generalized coordinates, the Lagrangian can be written as:
$$ L=T-U=L(t, q_1, q_1, \cdots, \dot{q_1}, \dot{q_2}, \cdots) $$
We define its action $S[q_k(t)]$ as the functional of the time integrand over the Lagrange $L$, from a starting time $t_a$ to an ending time $t_b$.
$$ S[q_k(t)]=\int^{t_b}{t_a} dt L(t, q_1, q_1, \cdots, \dot{q_1}, \dot{q_2}, \cdots)=\int^{t_b}{t_a} dt L(t, q_k, \dot{q_k}) $$
When $S$ is stationary, $\delta S$(the change in position) is zero, giving us
$$ 0=\delta\int^{t_b}_{t_a} dt L(t, q_k, \dot{q_k}) $$
Using knowledge from the previous chapter, the Lagrange is equal to (The integral represents the area under curve, where the curve/function is defined by the integrand. In this case, the integrand is $L$. When $S$ is stationary, it has reached a minimum or a maximum. As we derived in chapter, when the value of a function is extremeized, by Euler’s equation $\color{red}0={\partial L}/{\partial q_k}-({d}/{dt})({\partial L}/{\partial \dot{q_k}})$):