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I have a vector field described by $$\mathbf{F}(x,y) = y \hat{x} -x\hat y$$

I am trying to find the field lines for it, ideally by choosing a starting point and then having a parameter t that will lead me through all points that are on the field line, starting with the starting point.

Now, I do know that the field lines are circles, but that's only from drawing them, which wouldn't work with every field.

I have also tried saying that $$ \frac {dy}{dx}=\frac{-x}{y}$$ $$y \cdot dy=-x \cdot dx$$ $$y^2 = - x^2 + C $$ $$C=x^2+y^2$$ Which is true, but now I still don't have a way to go through all points. Is this the only way to do this? I know I can still use polar coordinates, but if I ever end up with a weird vector field where a field line looks more like this? Or is that impossible?

Asfangen
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1 Answers1

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Well, the idea is that every field line/trajectory is uniquely defined in your system from the equation relating $x$ and $y$ above, through the choice of the constant $C$ and some orientation that tells you which way the "flow" is going.

As a result, all you need to do to determine the trajectory/field line associated with a given initial condition is to identify the $C$ associated with the initial condition, and then obtain the flow direction by evaluating $\mathbf{F}(x,y)$ at the initial condition.

For example, consider the initial condition $(x,y) = (1,1)$. In this case, substitution into the equation you found above tells me $$C = 1^2 + 1^2 = 2$$ so the trajectory will occur on a circle centered at the origin of radius $2$. For the flow direction, we find $$\mathbf{F}(1,1) = \hat{x} - \hat y$$ which indicates clockwise motion/rotation.

All trajectories in your system can be uniquely determined with this procedure.

aghostinthefigures
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