Abstract
In this study, we geometrically analyze the relation between a point vortex system and deviation curvatures on the Jacobi field. First, eigenvalues of deviation curvatures are calculated from relative distances of point vortices in a three point vortex system. Afterward, based on the assumption of self-similarity, time evolutions of eigenvalues of deviation curvatures are shown. The self-similar motions of three point vortices are classified into two types, expansion and collapse, when the relative distances vary monotonously. Then, we find that the eigenvalues of self-similarity are proportional to the inverse fourth power of relative distances. The eigenvalues of the deviation curvatures monotonically convergent to zero for expansion, whereas they monotonically diverge for collapse, which indicates that the strengths of interactions between point vortices related to the time evolution of spatial geometric structure in terms of the deviation curvatures. In particular, for collapse, the collision point becomes a geometric singularity because the eigenvalues of the deviation curvature diverge. These results show that the self-similar motions of point vortices are classified by eigenvalues of the deviation curvature. Further, nonself-similar expansion is numerically analyzed. In this case, the eigenvalues of the deviation curvature are nonmonotonous but converge to zero, suggesting that the motion of the nonself-similar three point vortex system is also classified by eigenvalues of the deviation curvature.
Highlights
Geometric theories are often applied to analyze dynamical system behavior
The KCC theory has been applied to a three point vortex system, and eigenvalues of a deviation curvature have been calculated using relative distances
For self-similar motions, the eigenvalues of deviation curvature are proportional to the inverse fourth power of relative distances
Summary
Geometric theories are often applied to analyze dynamical system behavior. The relationship between a dynamical system and geometry is such that the Euler-Lagrange equation corresponds to the geodesic equation as a second-order differential equation. The theory of the Jacobi field has been applied to various dynamical systems, and concrete phenomena in biology, oscillating system, geophysics, astrophysics, and chaos models have been analyzed using the KCC invariants [4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28]. Translation and rigid rotation are classified by circulation conditions, whereas expansion and collapse are determined by circulation conditions and initial positions [30, 33, 35] In such a point vortex system, equations of motion are expressed by a system of first-order differential equations, which means that the theory of the Jacobi field can be applied to the point vortex system by differentiating the equations of motion.
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