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I was reading John Lee's wonderful book Introduction to Smooth Manifolds and believe I found a small mistake or omission in the proof of Theorem D.6 (Fundamental Theorem for Nonautonomous ODEs) on page 672. Non-typo mistakes in this book seem to be exceedingly rare, so I wanted to check that my understanding of this proof is correct. I want to focus on just the existence part of the statement.

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The equations (D.1) and (D.2) are the following.

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Stated informally, $V$ is a vector field dependent on time and defined on a space-time cylinder $J\times U$. For any point $(s_0,x_0)$ within that cylinder, there is a cylindrical neighborhood $J_0\times U_0$ containing it so that we can pick initial conditions $(t_0,c)$ from $J_0\times U_0$ and find a function $y$ satisfying the ODE and defined for all time in $J_0.$ The proof of the fundamental theorem for nonautonomous ODEs relies on the fundamental theorem for autonomous ODEs. We only need the existence part.

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Notice that the initial time $t_0$ is fixed before we get the domain of definition $J_0,$ in contrast to the fundamental theorem of nonautonomous ODEs where we can choose an initial time $t_0$ after fixing a domain of definition $J_0.$ The basic idea of the proof is that every non-autonomous ODE corresponds to an autonomous ODE when we make $t$ a variable. Given a non-autonomous ODE, we replace it with this autonomous ODE and apply the fundamental theorem for autonomous ODEs. In the proof, this fundamental theorem allows us to fix a time variable $s_0$ that we'll use for the initial condition of the autonomous ODE. On the other hand, we need to be able to choose the initial condition $t_0$ of the corresponding non-autonomous ODE freely in a neighborhood of $s_0$ to fulfill the existence part of Theorem D.6. This is permitted by the existence portion of the fundamental theorem on autonomous ODEs because the variable $t$ is a dependent "space" variable in the autonomous ODE.

My concrete objection to Lee's version of the proof is that the statement "Theorem D.1 guarantees that... there exists a unique solution to (D.20)" is not correct because, after fixing $s_0$, Theorem D.1 can only make statements about $y^i(s_0)$ rather than $y^i(t_0).$ Of course we could apply Theorem D.1 for $t_0$ but then the domain of definition $J_0$ on which the solution existed would be dependent on $t_0$, not fulfilling the claim of Theorem D.6.

At this point, I believe that the last two equations of (D.20) should written in terms of the variable $s$ as $\dot{y}^0(s)=1,\dot{y}^i(s)=..., y^0(s_0)=t_0$ and $y^i(s_0)=c^i$ so that we can define $t=y^0(s).$ The fundamental theorem of autonomous ODE gives a solution to the system (D.20), and since $\dot{y}_0(s)=1$ we'll get $t=y^0(s)=s+(t_0-s_0).$ At this point, to get a path satisfying the non-autonomous ODE, I think we need to do a change of variables to express the path in terms of $t$, letting $Y(t)=y(t-(t_0-s_0))$ to find a function satisfying (D.1-D.2). Then $Y(t_0)=y(s_0)=c$ as required.

Am I right?

Edit: I think a problem with my solution is that the domain on which $Y$ is defined would be $J+(t_0-s_0)$ which depends on $t_0$.

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  • $\begingroup$ D.6 (a) does not follow from D.1 (a) as far as I can see. $\endgroup$
    – copper.hat
    Feb 2, 2022 at 4:45
  • $\begingroup$ @copper.hat Why is that? $\endgroup$
    – subrosar
    Feb 2, 2022 at 5:01
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    $\begingroup$ The time interval $J_0$ in D.1 is a "red herring". The solution of an autonomous DE does not depend on the location of the time interval, one could as well replace $t_0$ with $0$. If $y$ is a solution centered at $0$, then $y(t-t_0)$ is a solution centered at $t_0$. $\endgroup$ Feb 2, 2022 at 7:13
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    $\begingroup$ Yes, that would be the trivial construction to recover that. The formulation of D.1 makes it similar to the general non-autonomous case without having a deeper reason to do so. In D.6 the variability of the initial time is shifted into the initial value of the autonomized system, it does not seem necessary to keep the time-lines of the system time and the zeroth variable synchronized in all aspects. $\endgroup$ Feb 2, 2022 at 11:53
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    $\begingroup$ Sedimentated mathematics (with decades or centuries of lectures and text books on top) tends to be presented in a telegram style, with stringency, there is a pressure to be as short and straightforward as possible. The nature of the subject however is not as linear as one would often like. So there are some puzzle pieces in the web of statements that can be shifted around in the linear order of the text. In my opinion the raised points more organically belong close to or in the proof of D.6, simplifying the statement of D.1. But I might be missing the big picture. $\endgroup$ Feb 2, 2022 at 18:38

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I think the key to explaining my question is that in the proof of Theorem D.6, Lee actually uses a result slightly stronger than Theorem D.1 which can be seen as a corollary of Theorem D.1. This slightly stronger result would replace the existence portion of D.1 with

EXISTENCE: For any $s_0\in \mathbb{R}$ and $x_0\in U$, there exist an open interval $J_0$ containing $s_0$ and an open subset $U_0\subseteq U$ containing $x_0$ such that for each $t_0\in J_0$ and $c\in U_0$, there is a $C^1$ map $y:J_0\to U$ that solves $(D.3)-(D.4)$.

This "corollary" can be proven by restricting to a smaller $J_0$ and using the fact that the system is autonomous.

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