A general solution for rotating lattices of identical point vortices

The rotating equilibrium solutions of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mi>N</mml:mi> </mml:math> identical point vortices are the stationary energy states of a higher-dimensional object projected to the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mrow> <mml:mi>x</mml:mi> <mml:mo>−</mml:mo> <mml:mi>y</mml:mi> </mml:mrow> </mml:math> plane, in this case, an <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mrow> <mml:mo form="prefix" stretchy="false">(</mml:mo> <mml:mi>N</mml:mi> <mml:mo>−</mml:mo> <mml:mn>1</mml:mn> <mml:mo form="postfix" stretchy="false">)</mml:mo> </mml:mrow> </mml:math> -dimensional regular simplex. Parameterizing the point vortex Hamiltonian with this fundamental geometrical object leads to a simple bivector condition, which contains the equilibrium states and resolves the origin of the asymmetric solutions. This novel approach uses the geometric product and the multivector derivative, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:msub> <mml:mi>∂</mml:mi> <mml:mi>ψ</mml:mi> </mml:msub> </mml:math> , of the Hamiltonian in the geometric algebra, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mrow> <mml:mi>C</mml:mi> <mml:mi>l</mml:mi> <mml:mo form="prefix" stretchy="false">(</mml:mo> <mml:mi>N</mml:mi> <mml:mo>−</mml:mo> <mml:mn>1</mml:mn> <mml:mo separator="true">,</mml:mo> <mml:mn>0</mml:mn> <mml:mo form="postfix" stretchy="false">)</mml:mo> </mml:mrow> </mml:math> to find a geometric condition that determines the equilibrium states. The resulting bivector equation is then used as the input to an optimizer, which rotates a simplex until the equilibrium condition is met, leading to a wealth of new solutions. If the vertices of the oriented simplex are projected to the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mi>x</mml:mi> </mml:math> -axis, the values form the roots of the Hermite polynomials, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mrow> <mml:msub> <mml:mi>H</mml:mi> <mml:mi>N</mml:mi> </mml:msub> <mml:mo form="prefix" stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo form="postfix" stretchy="false">)</mml:mo> </mml:mrow> </mml:math> , and obey the Stieltjes relations, capturing the collinear solutions. The vortex simplex exhibits a striking geometrical connection with the amplituhedron of quantum field theory and gives deep insight into the quantization of a classical system.