For the Hamilton's principle: $$\delta s =\int_{t_1}^{t_2}L(\mathbf {q+\delta q},\mathbf {\dot q+\delta \dot q},t) dt-\int_{t_1}^{t_2}L(\mathbf {q},\mathbf {\dot q},t) dt=0.\\$$
In the textbooks, using the Taylor's series:
$$L(\mathbf {q+\delta q},\mathbf {\dot q+\delta \dot q},t) =L(\mathbf {q},\mathbf {\dot q},t)+\frac{\partial L(\mathbf {q},\mathbf {\dot q},t)}{\partial \mathbf q} \bar \delta \mathbf q(t) +\frac{\partial L(\mathbf {q},\mathbf {\dot q},t)}{\partial \mathbf {\dot q}} \bar \delta \mathbf {\dot q(t)}+\frac{\partial}{\partial \mathbf q} (\frac{\partial L(\mathbf {q},\mathbf {\dot q},t)}{\partial \mathbf {\dot q}}\bar \delta \dot q)\bar \delta q + …$$
But why is omitting the rest of the Taylor's series?
I think it will propagate a lot of error in the derivation…
Best Answer
The Euler Lagrange equation is a differential equation resulting from the search for the extremum of a functional: this extremum is given by the first variation only.
This is similar to the condition for finding a point where a function $f$ is extremum: the condition $df/dx=0$ is on the first derivative only.
In both cases, one does not seek to approximate $L$ or $f$ by a series, but rather extract from the first term in the series either a differential equation to be satisfied by a function or an algebraic equation to be satisfied by a point.
Edit:
In addition, the order of the resulting Euler-Lagrange equation can be greater than 1 even if one considers only the first variation. For instance, in 1d, with a Lagrangian of the type $L=L(q,\dot{q},\ddot{q},t)$, the resulting EL equation is of second order: $$ \frac{\partial L}{\partial q}-\frac{d}{dt}\frac{\partial L}{\partial \dot{q}}+ \frac{d^2}{dt^2}\frac{\partial L}{\partial \ddot{q}}=0. $$
Finally, the first variation (like the first derivative) only guarantees an extremum. To verify if this is a local max or min, one ought to use the second variation (like the second derivative) but it is often technically complicated and unnecessary: one can simply verify if any other construction gives a greater or smaller action or shorter or longer path.