Symmetric relative equilibria with one dominant and four infinitesimal point vortices

Abstract

We investigate the symmetry of point vortices with one dominant vortex and four vortices with infinitesimal circulations in the (1+4)-vortex problem, a subcase of the five-vortex problem. The four infinitesimal vortices inscribe quadrilaterals in the unit circle with the dominant vortex at the origin. We consider symmetric configurations which have one degree of spacial freedom, namely the (1+N)-gon, kites, rectangles, and trapezoids with three equal sides. We show there is only one possible rectangular configuration (up to rotation and ordering of the vortices) and one possible trapezoid with three equal sides (up to rotation and ordering), while there are parametrically defined families of kites. Additionally we consider the (1+4)-gon and show that the infinitesimal vortices must have equal circulations on opposite corners of the square. The proofs are heavily dependent on techniques from algebraic geometry and require the use of a computer to calculate Grobner bases.

Figures (10)

• Figure 1: Examples of symmetric relative equilibria in the $(1+4)$-vortex problem
• Figure 2: Pictured are the surfaces in $\mathcal{P}$ for Example \ref{['GBelimination example']}, the variety $Var(\mathcal{P})$ and its projection onto the $xy$-plane, i.e. the variety of the elimination ideal which eliminates the variable $z$.
• Figure 3: The three possible configurations satisfying $\theta_1=0, \theta_3=2\theta_2, \theta_4=3\theta_2$ and $\theta_2 \in (0,\pi)\setminus\{2\pi/3\}$ . The configuration in Figure \ref{['threeequal_ex1']} is the only configuration with $\theta_2\in (0,2\pi/3)$ and the only true inscribed trapezoid with three equal sides.
• Figure 4: Regions corresponding to the linear stability of relative equilibria forming a trapezoid with three equal sides. Critical points for parameter values on the boundaries of the region are degenerate.
• Figure 5: The partition of the $\mu_1=1$ plane in $\mu_1\mu_2\mu_3$-parameter space for kite configurations. There is one kite configuration in the blue shaded regions, along the blue curves and at the blue dots. There are two kite configurations in the gray regions and along the gray curves. There are three kite configurations in the rest of the plane.

Theorems & Definitions (25)

• lemma 1: Lemma 2 in barry2016existence
• theorem 1: Theorem 1 in barry2016existence
• definition 1
• theorem 2: Theorem 2 in barry2016existence
• theorem 3: Theorem 3 in barry2016existence
• theorem 4: Sturm
• theorem 5
• proof
• theorem 6
• proof