Is the flow of a vector field independent of coordinates?
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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?
differential-geometry manifolds vector-fields
asked Dec 13 '16 at 21:05
user136592user136592
818511
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add a comment |
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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?
differential-geometry manifolds vector-fields
asked Dec 13 '16 at 21:05
user136592user136592
818511
$endgroup$
1
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Sure. This follows from uniqueness of solutions of ODE. Think about the overlap of your two coordinate charts.
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– Ted Shifrin
Dec 13 '16 at 22:31
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@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?
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– user136592
Dec 14 '16 at 15:59
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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 $X=y_2frac{partial}{partial y_1}-y_1frac{partial}{partial y_2}$. What did you mean by saying the $a_i$ and $b_i$ should be exactly the same?
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– Ted Shifrin
Dec 14 '16 at 21:00
add a comment |
$begingroup$
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?
differential-geometry manifolds vector-fields
asked Dec 13 '16 at 21:05
user136592user136592
818511
$endgroup$
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?
differential-geometry manifolds vector-fields
differential-geometry manifolds vector-fields
asked Dec 13 '16 at 21:05
user136592user136592
818511
asked Dec 13 '16 at 21:05
user136592user136592
818511
asked Dec 13 '16 at 21:05
user136592user136592
818511
asked Dec 13 '16 at 21:05
user136592user136592
818511
asked Dec 13 '16 at 21:05
user136592user136592
818511
818511
1
$begingroup$
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 $X=y_2frac{partial}{partial y_1}-y_1frac{partial}{partial y_2}$. 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
add a comment |
1
$begingroup$
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 $X=y_2frac{partial}{partial y_1}-y_1frac{partial}{partial y_2}$. 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
1
1
$begingroup$
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
$begingroup$
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
$begingroup$
@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 $X=y_2frac{partial}{partial y_1}-y_1frac{partial}{partial y_2}$. 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
$begingroup$
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 $X=y_2frac{partial}{partial y_1}-y_1frac{partial}{partial y_2}$. 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
add a comment |
1 Answer
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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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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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add a comment |
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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, $X = y_2 frac{partial}{partial y_1} - y_1 frac{partial}{partial y_2}$.
$endgroup$
add a comment |
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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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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, $X = y_2 frac{partial}{partial y_1} - y_1 frac{partial}{partial y_2}$.
answered Dec 13 '18 at 13:02
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Brahadeesh
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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
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@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 $X=y_2frac{partial}{partial y_1}-y_1frac{partial}{partial y_2}$. 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