I have been trying to review some parts of classical mechanics using Landau and Lifschitz and Arnolds Mathematical Methods of Classical Mechanics and have run into a question regarding how the Lagrangian is determined in the opening chapter of Landau and Lifschitz.

Let's take the following statement from Landau and Lifschitz,

The homogeneity of space and time implies that the Lagrangian cannot con­tain explicitly either the radius vector.

My understanding is that a motion, $ x(\cdot) $, should be an extremal of a given action, $ S[x(\cdot)] = \int_{0}^{T} \mathcal{L}(x(t), \dot{x}(t), t) dt $. From this we find that a motion should satisfy the Euler-Lagrange equations,

$$ -\frac{d}{dt} \left( \frac{d\mathcal{L}}{d\dot{x}}(x(t), \dot{x}(t), t)\right) + \frac{d\mathcal{L}}{dx}(x(t), \dot{x}(t), t) = 0.$$

The way I interpreted homogeneity of space is that all points are equivalent so mathematically, $y(t) = x(t) + r$, for fixed $r$, is another valid measurement which in turn means it should satisfy the Euler-Lagrange equations, or mathematically that,

$$ -\frac{d}{dt} \left( \frac{d\mathcal{L}}{d\dot{x}}(x(t) + r, \dot{x}(t), t)\right) + \frac{d\mathcal{L}}{dx}(x(t) + r, \dot{x}(t), t) = 0. $$

Writing the Euler-Lagrange equations as $F(x(t), \dot{x}(t),\ddot{x}(t), t) = 0 $ space is homogeneous tells us that $$F(x(t), \dot{x}(t),\ddot{x}(t), t) = F(x(t) + r, \dot{x}(t), \ddot{x}(t), t).$$ From here I think we can determine that $F$, or the Euler-Lagrange equations, is independent of the argument for $x(t)$ if there is only a single particle(Arnold uses this same argument to show forces must depend on only relative positions in his opening chapter). Analogously shifting time instead of spatial positions shows $F$ must be invariant of time

From this I have the following questions,

1. Was there any misunderstanding in what I have stated above?

2. How does one go from $F$ is independent of $x$, to deducing information about the Lagrangian being independent of $x$? To add some context to the second question I have found that some other sources such as this attempt to show that homogeneity and isotropy can lead to deductions about the Lagrangian although no source I have found attempts to show any detail about how to do this.

Edit: To address the comment about specificity, I am unclear about how the Euler-Lagrange equations being independent of position and time implies the Lagrangian must also be independent of those properties as implied by Landau. Is it possible to complete this argument? If not, can other concrete information be gained about the Lagrangian from the Euler-Lagrange equations being independent of position or time? If someone has specific feedback about why the current question scope is not specific enough or lacks enough detail I would appreciate it.