Suppose that $\dot{ x }= f(x)$ is a differential equation defined on an open subset $U$ of $\mathbb{R}^n$. I want to show that the differential equation has a complete reparametrization.
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What do you mean by complete reparametrization? – Chee Han Jun 12 '18 at 05:49
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Possibly a special case of “Completing” a vector field on a non-compact manifold M. – user539887 Jun 12 '18 at 06:28
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@CheeHan By complete representation, I mean the solution exists for all $t \in \mathbb{R}$. – Arthur Jun 13 '18 at 04:34
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Thanks for the link, but I do not understand it. I appreciate it if you explain it in a more simple language @user539887 – Arthur Jun 13 '18 at 04:40
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You know what is "complete Riemannian metric"? – user539887 Jun 13 '18 at 07:25
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Unfortunately, I am not familiar with that (I do not have any background in differential geometry). – Arthur Jun 13 '18 at 12:51
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Perhaps that can be proved in a more straightforward way, without using such a heavy machinery of differential geometry as in the references I have given. But I don't know of any such proof. – user539887 Jun 13 '18 at 19:40
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@Artour See Evgeny's comment in Techniques to prove that solutions of ODE are defined for all time; this is just what you need. – user539887 Jul 05 '18 at 21:18