Assume we have a vector field $X=a_i(x)\frac{\partial}{\partial x_i}$ in local coordinates of a manifold M. Then we can solve an ODE to get the flow of the vector field. If we have in some other coordinates $y_i$, $X=b_i(y)\frac{\partial}{\partial y_i}$, will we end up with the same flow by solving an ODE as in the previous coordinates?

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    $\begingroup$ Sure. This follows from uniqueness of solutions of ODE. Think about the overlap of your two coordinate charts. $\endgroup$ – Ted Shifrin Dec 13 '16 at 22:31
  • $\begingroup$ @TedShifrin If the flows are always the same, wouldn't it imply the same ODE is satisfied in the two coordinates? Then $a_i(x)$ and $b_i(y)$ should be exactly the same, but is that true for all local coordinates? $\endgroup$ – user136592 Dec 14 '16 at 15:59
  • $\begingroup$ Just to convince yourself that even the simplest change of coordinates changes the differential equation, consider $X = -x_2 \frac{\partial}{\partial x_1} + x_1 \frac{\partial}{\partial x_2}$ in the plane. Consider the linear change of coordinates: $y_1=x_2$, $y_2=x_1$. Now $X=y_2\frac{\partial}{\partial y_1}-y_1\frac{\partial}{\partial y_2}$. What did you mean by saying the $a_i$ and $b_i$ should be exactly the same? $\endgroup$ – Ted Shifrin Dec 14 '16 at 21:00

From the comments above.

Yes, we will end up with the same flow. This follows from the uniqueness of solutions of ODEs. Note that this does not mean that the equation of the vector field looks the same regardless of the coordinates. Consider the differential equation $X = -x_2 \frac{\partial}{\partial x_1} + x_1 \frac{\partial}{\partial x_2}$ and consider the change of coordinates $y_1 = x_2$, $y_2 = x_1$. Now, $X = y_2 \frac{\partial}{\partial y_1} - y_1 \frac{\partial}{\partial y_2}$.


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