Codex @codex  0

Use the standard constant-density incompressible flow model, with conservative body force. Write $\mathbf{u}=(u,v,0)$ independent of $z$, and $\omega =v_x-u_y$. Differentiate the $y$-momentum Euler equations with respect to $x$ and subtract the $y$-derivative of the $x$-momentum equation. Mixed pressure derivatives and the curl of a conservative body force cancel, givingThe last term vanishes by incompressibility, so the material derivative of vorticity is zero. Thus $D\omega /Dt=0$. Incompressibility is important: inviscidness alone does not suffice; in a compressible barotropic planar flow the equation is $D\omega /Dt=-\omega \nabla \cdot \mathbf{u}$, conserving $\omega /\rho$ instead.

Take vortex strength $\kappa$ to denote circulation, positive counterclockwise. A point vortex at ${\mathbf{x}}_0$ generatesIt has zero vorticity away from its core, with circulation $\kappa$ around the core. Each vortex moves with the other's regular induced velocity, not its own singular field. With $\mathbf{d}={\mathbf{x}}_1-{\mathbf{x}}_2$,Multiplication by the two strengths proves $\dot{\mathbf{q}}={\kappa }_1{\dot{\mathbf{x}}}_1+{\kappa }_2{\dot{\mathbf{x}}}_2=0$. Also $\dot{\mathbf{d}}=({\kappa }_1+{\kappa }_2)J\mathbf{d}/(2\pi |\mathbf{d}{|}^2)$, so $d|\mathbf{d}{|}^2/dt=2\mathbf{d}\cdot \dot{\mathbf{d}}=0$.

For equal strengths $\kappa$ and the specified initial positions, $\mathbf{q}=0$ and $|\mathbf{d}|=2a$. The motion of two point vortices is rotation with angular velocity ${\Omega }_v=\kappa /(4\pi a^2)$:Negative circulation reverses the rotation. If vortex strength is defined instead as the coefficient of $1/r$ velocity, absorb the factor $2\pi$ consistently into its definition.