I have never completely understood the justification of this step in the derivation of the E-L equation:
$\delta L = L(q + \delta q, q' + \delta q', x) – L(q, q', x) = \partial_q L \delta q + \partial_{q'} L \delta q'$
This is only valid when both $\delta q$ and $\delta q'$ are small. Now $\delta q$ is small by assumption, but $\delta q'$ is completely determined by
$\delta q$, so the smallness of $\delta q'$ needs to be implied by the smallness of $\delta q$. However, if $\delta q$ has high frequency components (eq $\delta q = \epsilon sin(x/\epsilon)$ for a small $\epsilon$) then $\delta q'$ will not be small even though $\delta q$ is. I imagine this can be fixed by having additional assumtions on $\delta q$ like that it is of the form $\epsilon f$ where $\epsilon$ is small and $f$ is just a suitably nice function,
but the explanations I have seen do not make such assumptions explicit.
Also I have never seen a good explanation of the properties $\delta$ operator.
In the usual E-L derivation it also seems to be assumed that $\delta (q') = (\delta q)'$. I can convince myself of that informally, but it is something that would seem worth explaining. Is there a good reference that goes into the $\delta$ operator in some depth.
EDIT
When I first asked my question I was forgetting about Lanczos Variational Principles of Mechanics which goes into the $\delta$ notation in some length.
Now that I have gone back to Lanczos, I know a little bit more, but it is still far from clear, despite Lanczos's heroic efforts.
The responses seem to agree on the idea that $\delta$ is not an independent entity, but more of a naming convention, making $\delta L$ and $\delta q$ atomic names. This makes some sense to me, but it contradicts Lanczos in Variational Principles of Mechanics, where he uses the term $\delta$-process. He even writes things like $\delta^2 F$ and $\delta \int F$ and even has the equivalent of
$\delta (q') = (\delta q)'$ above as equation 29.3, together with a derivation.
Lanczos stresses the notion of virtual displacement . At one point he says that $\delta u$ is a virtual change while $du$ is an actual change. Unfortunately I have no clue what he is getting at, but virtual displacement seems to be the concept I am missing. I have found
http://en.wikipedia.org/wiki/Virtual_displacement, but I am not really parsing that either. However, the abstract of Subhankar Ray, J. Shamanna, Virtual Displacement in Lagrangian Dynamics sounds promising:
… However, the definition
of virtual displacement is rarely made precise and often
seems vague and ambiguous to students. In this article
we attempt a more systematic and precise definition,
which not only gives one a qualitative idea of virtual
displacement, but also allows one to quantitatively
express the same for any given constrained system. …
Sounds like what the doctor ordered.
Best Answer
Hint: the basic assumption is that the Lagrangian $L(q(x),q'(x),x)$ is differentiable at $y:=(q(x),q'(x),x)$ and the variation $\delta L$ expresses the differentiability, in a suitable limit.
In other words, we are considering the increment ( $h$ is just a vector in $\mathbb R^3$, for all $x$, but the components $\delta q$ and $\delta q'$ are functions of $x$)
$$h:=(\delta q,\delta q',0)$$
and looking at
$$L(y+h)-L(y)=\langle \nabla L(y),h\rangle+O(\|h\|);~~(*)$$
the gradient $\nabla L(y)$ of $L$ at $y$ is
$$\nabla L(y)=\left(\frac{\partial L}{\partial q},\frac{\partial L}{\partial q'},\frac{\partial L}{\partial x}\right) $$
and, by definition,
$$y+h=(q(x)+\delta q,q'(x)+\delta q',x).$$
In variational problems, the vector $h$ is not completely arbitrary: in fact, if we consider small variations _(i.e. $\delta q$) of the function $q:x\mapsto q(x)$ we need to impose boundary conditions on the components of $h$: such conditions are depending on the specific variational problem under exam. They are motivated by the variational approach and usually "kill" unfriendly boundary terms (at least in classical variational problems: in quantum field theory one can accept and subsequently work with boundary terms).
For example, in presence of the variational problem
$$S[q]:=\int_a^b L(q(x),q'(x),x)dx $$
we would be interested in all small variations $\delta q$ of $q=q(x)$ s.t. $\delta q(a)=\delta q(b)=0$.
If we expand $(*)$ using the definition of gradient and scalar product, we arrive at
$$L(q(x)+\delta q,q'(x)+\delta q',x)-L(q(x),q'(x),x)=\frac{\partial L}{\partial q}\delta q+\frac{\partial L}{\partial q'}\delta q'+O(\|h\|). $$
Remark: parameters
In some books and papers it is customary to define the small variations $\delta q$ as
$$\delta q:=\epsilon\varphi, $$ $$\delta q':=\epsilon\varphi' $$
where $\varphi$ is any function satisfying suitable boundary conditions, and $\epsilon$ is a parameter. In this setting we are considering the increment
$$h=(\epsilon\varphi,\epsilon\varphi',0)$$
around $y=(q(x),q'(x),x)$ and the first variation $\delta L$ of $L$ is defined as
$$\delta L:=\frac{dL}{d\epsilon}|_{\epsilon=0}:=\lim_{\epsilon\rightarrow 0} \frac{L(q(x)+\epsilon\varphi,q'(x)+\epsilon\varphi',x)-L(q(x),q'(x),x)-\epsilon\left(\frac{\partial L}{\partial q}\varphi+\frac{\partial L}{\partial q'}\varphi'\right)+O(\epsilon^2)}{\epsilon}. $$
I personally find this notation quite good.