New coordinates for three interacting vortices

What is it about?

Point vortices are simple solutions to the equations of fluid motion, which we may think of as tiny whirlpools. Mathematicians in the late 1800s wrote down a simple system of equations that describes the motion of point vortices as a finite number of interacting particles. This is much simpler to study than the full fluid equations, which require keeping track of the fluid velocity at every point in space. The equations governing a system of of one or two vortices is simple enough to solve on a napkin, but the motion of three vortices is surprisingly complicated. Even this system, which consists of six equations (with two coordinates, x and y, for each vortex), is too complex to understand without first performing a mathematical reduction. The standard reduction, which was introduced by Gröbli in 1877, results in equations that break down whenever the three vortices lie in a straight line. This makes it difficult to understand the motion of the vortices and to understand the solutions by drawing graphs. In this work, we introduce a new reduction method that avoids these problems. We apply this to several well-known examples. In certain cases, the motion is described by a phase plane on a sphere as in the image below.

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Photo by Susan Wilkinson on Unsplash

Photo by Susan Wilkinson on Unsplash

Why is it important?

We found a new way to solve a well-known problem. Although people had solved this problem before, their solutions were clunky and difficult to interpret. Complex problems are often built from simple pieces, but the old reduction methods gave solutions that were difficult to combine to understand more complex problems. With these new pieces in hand, we are ready to begin work on more complex problems.

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