= Line-vortex trajectory in a quarter-plane
{title2=$r\sin2\theta=\text{constant}$}
In the quadrant $x,y>0$, a <line vortex> of <circulation> $\Gamma$ at $(x,y)$ has three images: circulations $-\Gamma$ at $(-x,y)$ and $(x,-y)$, and $+\Gamma$ at $(-x,-y)$. Their induced velocity is
$$
\dot x=\frac{\Gamma x^2}{4\pi y(x^2+y^2)},\qquad\dot y=-\frac{\Gamma y^2}{4\pi x(x^2+y^2)}.
$$
In <polar coordinates>, these equations give $\dot r=\Gamma\cot(2\theta)/(2\pi r)$ and $\dot\theta=-\Gamma/(4\pi r^2)$. Direct <differentiation> yields $\frac d{dt}(r\sin2\theta)=0$, so the trajectory obeys $r\sin2\theta=\text{constant}$.
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