Consider the Euler-Lagrange equation in multiple dimensions (which is actually a system of equations): $$ \frac{\partial L}{\partial q_i}(t, \mathbf{q}(t), \dot{\mathbf{q}}(t)) - \frac{d}{dt}\frac{\partial L}{\partial \dot{q}_i}(t, \mathbf{q}(t),\dot{\mathbf{q}}(t)) =0. $$ My question is the following:
Is the relationship $$\dot{\mathbf{q}}(t)=\frac{d}{dt}\mathbf{q}(t)$$ an assumption of the Euler-Lagrange equation, or is it a conclusion?
My argument -- where are the mistakes? I have argued that it is a conclusion before, see in this answer, I don't know how to prove or disprove that this is correct.
Specifically, when evaluating $\displaystyle\frac{\partial L}{\partial q_i}$ no terms of the form $\displaystyle\frac{\partial \dot{q}_i}{\partial q_i}$ show up, and when evaluating $\displaystyle\frac{\partial L}{\partial \dot{q}_i}$ no terms of the form $\displaystyle\frac{\partial q_i}{\partial \dot q_i}$ show up, which wouldn't make sense if $\dot{q}_i$ was being treated as a function, for example, if $q_i = \frac{1}{2}t^2$ then $\dot{q}_i = t$ so then $\displaystyle\frac{\partial q_i}{\partial \dot{q}_i} = t \not=0$.
However, neglecting such terms would/does make sense if the $\dot{q}_i$ are being treated as independent variables, rather than functions. The only problem then is that one can't say that one independent variable is the derivative of another independent variable.
BUT, along the solution curves, the independent variable denoted by an abuse of notation $\dot{q}_i$ could/would be/is an implicit function of $t$, call it $f_i(t)$, as would the independent variable $q_i$ be an implicit function of $t$ along the solution curve call it $g_i(t)$, and then a consequence of the Euler-Lagrange equation might be then that $f_i(t) = \frac{d}{dt}g_i(t)$.
This seems to be a natural way to think of the situation/setup if one thinks about differential equations as "phase flows" in "phase space", e.g. as in the first section of the first chapter of Vladimir Arnold's excellent book on ODE's.
The answers to this question seem to suggest this might be a possibility: Derivative with respect to $y'$ in the Euler-Lagrange differential equation; however they don't state whether or not the relation $\dot{\mathbf{q}}(t)=\frac{d}{dt}\mathbf{q}(t)$ is a consequence or not, although that should be implied since it could not be an assumption if $y'$ were an independent variable, as the answers suggest.
Note: Before dismissing this possibility out of hand, I would ask one to at least consider that partial derivative notation is notorious for obfuscating and making more confusing issues surrounding implicit differentiation and implicit functions, see for example here, here, here, here, here, here, here, here, here: "A problem with Jacobi's notation is that... substitution of variables can lead to absurd formulae or, at least, to formulae that require careful interpretation", and here.
Let $L: \Bbb R^3 \to \Bbb R: (x_1, x_2, x_3) \mapsto L(x_1, x_2, x_3)$ and $q: \Bbb R\to \Bbb R: t \mapsto q(t)$. Then denote by $\dot q$ the derivative of $q$. Let $\gamma: \Bbb R \to \Bbb R^3: t\mapsto (t,q(t),\dot q(t))$. Now consider the restriction of $L$ to $\gamma$:
$$(L\circ \gamma)(t):\Bbb R\to \Bbb R \\ t\mapsto L(t,q(t),\dot q(t))$$
This is the usual way of writing the Lagrangian.
Now the expressions $\frac{\partial L}{\partial q}$ and $\frac{\partial L}{\partial {\dot q}}$ in the EL equation are really just the partial derivatives of the above function $L$ wrt its second and third arguments: $$\frac{\partial L}{\partial q}:= \frac{\partial L}{\partial x_2} \qquad \frac{\partial L}{\partial \dot q}:= \frac{\partial L}{\partial x_3}$$
Thus the value of these partials does not depend on the fact that $\dot q = \frac{dq}{dt}$.
Now let's look at $\frac{d}{dt}\frac{\partial L}{\partial \dot q}$. This is really the derivative of the restriction of $\frac{\partial L}{\partial x_3}$ to the path $\gamma$
$$\frac{d}{dt}\frac{\partial L}{\partial \dot q} := \left[\frac{d}{dt}\left(\frac{\partial L}{\partial x_3}\circ \gamma\right)\right](t)$$
So the EL equation, written in this new notation is:
$$\left(\frac{\partial L}{\partial x_2}\circ \gamma\right)(t) - \left[\frac{d}{dt}\left(\frac{\partial L}{\partial x_3}\circ \gamma\right)\right](t) =0$$
or written slightly nicer (but also slightly less clear):
$$\left.\frac{\partial L}{\partial x_2}\right|_{\gamma(t)} - \frac{d}{dt}\left(\left.\frac{\partial L}{\partial x_3}\right|_{\gamma(t)}\right) = 0$$
Or equivalently, one could interpret the EL equation as saying that this particular function is the null function on $\Bbb R$:
$$\frac{\partial L}{\partial x_2}\circ \gamma - \frac{d}{dt}\left(\frac{\partial L}{\partial x_3}\circ \gamma\right) = 0$$