Why are there $2s -1$ independent integrals of motion?

Question

I was reading Mechanics by Landau and Lifshitz and I am confused when it is stated in chapter 2 section 6 that one of the integrals of motion is not independent and can be considered an additive constant of time. Hence I tried searching it up a few SE posts.

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Constants of motion vs. integrals of motion vs. first integrals

According to the OP in the link above (first paragraph second sentence), we need to specify $2N$ initial conditions, one of them is the initial time, the others the initial positions and velocity.

However, shouldn't it be the initial $N$ position and $N$ velocities? Can it be shown to be equivalent? Plus aren't we working with an autonomous system of equations? Is this why we need $t - t_0$ so that it invariant to translation in time?

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https://physics.stackexchange.com/a/592205/259297

The answer provided above is interesting. However, there are several points that I would like to verify...

Questions:

1. Because the Lagrangian, $\mathcal{L} (q, \dot q)$ is independent of the acceleration, Lagrange's equations, $$\frac{d}{dt} \frac{\partial \mathcal{L} }{\partial \dot q} - \frac{\partial \mathcal{L}}{\partial q}$$ which only involves one time derivative, only introduces terms linear in $\ddot q$, right?
2. According to the author (see a comment below the post as well), $$\ddot q_i =\frac{\text{d}\dot q_i}{\text{d} q_1} \frac{\text{d}q_1}{\text{d}t} =\dot q_1\frac{\text{d}\dot q_i}{\text{d}q_1}$$ How does a total time derivative become a total derivative in $q_1$? Are we performing a change of variables by inverting $q_1(t)$ to get time as a function of $q_1$ then all coordinates become $q_i(t(q_1))$?
3. Are there other proofs that there are only $2s - 1$ integrals of motion?

Answer 1

Questions:

1. Because the Lagrangian, $\mathcal{L} (q, \dot q)$ is independent of the acceleration, Lagrange's equations, $$\frac{d}{dt} \frac{\partial \mathcal{L} }{\partial \dot q} - \frac{\partial \mathcal{L}}{\partial q}$$ which only involves one time derivative, only introduces terms linear in $\ddot q$, right?

Yes.

2. According to the author (see a comment below the post as well), $$\ddot q_i =\frac{\text{d}\dot q_i}{\text{d} q_1} \frac{\text{d}q_1}{\text{d}t} =\dot q_1\frac{\text{d}\dot q_i}{\text{d}q_1}$$ How does a total time derivative become a total derivative in $q_1$? Are we performing a change of variables by inverting $q_1(t)$ to get time as a function of $q_1$ then all coordinates become $q_i(t(q_1))$?

This equation $\ddot q_i = \frac{d \dot q_i}{dq_1}\frac{dq_1}{dt}$ is nonsense. Or at least it needs a lot of explaining regarding what this notation is supposed to mean... I would not proceed this way if I were you.

3. Are there other proofs of this?

Proofs of what? Proofs regarding how many integrals of the motion there are? I'm sure you can find proofs related to counting integrals of the motion in almost every classical mechanics textbook ever written...