I have trouble understanding this problem of Arnold's ODE book. It goes as follows:

Prove that a diffeomorphism taking the vector field $v$ to the vector field $w$ takes the phase curves of the field $v$ to the phase curves of the field $w$. Is the converse true?

I'am confused about what type of diffeomorphism is supposed to be taken. Either a diffeomorphism between vector fields (for which I don't know if it is possible to define such mapping) or the diffeomorphism between the domains of the vector fields. In the latter case I think the answer is yes, in fact it is a corollarly of the theorem proved just below that problem. But he gives a negative answer by giving two vector fields, namely $v=x\frac{\partial}{\partial x}$ and $w=2x\frac{\partial}{\partial x}$. Hope you can help me to understand this.

  • $\begingroup$ A diffeomorphism induces a map of vector fields. So he means: If $f \colon U \to V$ is a diffeomorphism, $v$ is a vector field on $U$, and $w = f_* v$, then $f$ takes the phase curves of $v$ to those of $w$. $\endgroup$ – Matthew Leingang Mar 6 '17 at 4:02
  • $\begingroup$ Thanks! But then the converse is that if $f$ takes the phase curves of $v$ to those of $w$ that does not convert f into a diffeomorphism, right ? Or the converse means that is not always possible to take the phase curves of $w$ to those of $v$ under a diffeomorphism? Sorry if this seems stupid, but as I got it $f$ makes a bijective map between the phase curves of $v$ and $w$, since it is a diffeomorphism $\endgroup$ – Rogelio Mar 6 '17 at 4:20
  • $\begingroup$ I think the converse (which you are supposed to show is not true) is that if $f \colon U \to V$ is a diffeomorphism that takes phase curves of $v$ to phase curves of $w$, then $w = f_* v$. $\endgroup$ – Matthew Leingang Mar 6 '17 at 4:23

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