Morse Index and Stability of the Planar N-vortex Problem

@article{Hu2020MorseIA,
  title={Morse Index and Stability of the Planar N-vortex Problem},
  author={Xijun Hu and Alessandro Portaluri and Qin Xing},
  journal={Qualitative Theory of Dynamical Systems},
  year={2020}
}
This paper concerns the investigation of the stability properties of relative equilibria which are rigidly rotating vortex configurations sometimes called vortex crystals, in the N-vortex problem. Such a configurations can be characterized as critical point of the Hamiltonian function restricted on the constant angular impulse hyper-surface in the phase space (topologically a pseudo-sphere whose coefficients are the circulation strengths of the vortices). Relative equilibria are generated by… 
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2 Citations

Citation Type

Citation Type

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References

SHOWING 1-10 OF 15 REFERENCES
Morse Theory and Relative Equilibria in the Planar n-Vortex Problem
Morse theoretical ideas are applied to the study of relative equilibria in the planar n-vortex problem. For the case of positive circulations, we prove that the Morse index of a critical point of the…
Stability of Relative Equilibria in the Planar n-Vortex Problem
TLDR
It is shown that for the case of positive circulations, a relative equilibrium is linearly stable if and only if it is a nondegenerate minimum of the Hamiltonian restricted to a level surface of the angular impulse (moment of inertia).
Relative equilibria of vortices in two dimensions.
  • J. Palmore
  • Physics
    Proceedings of the National Academy of Sciences of the United States of America
  • 1982
An old problem of the evolution of finitely many interacting point vortices in the plane is shown to be amenable to investigation by critical point theory in a way that is identical to the study of…
Morse Index and Linear Stability of the Lagrangian Circular Orbit in a Three-Body-Type Problem Via Index Theory
It is well known that the linear stability of the Lagrangian elliptic solutions in the classical planar three-body problem depends on a mass parameter β and on the eccentricity e of the orbit. We…
On The Motion of Three Vortices
  • J. Synge
  • Mathematics
    Canadian Journal of Mathematics
  • 1949
In a perfect incompressible fluid extending to infinity, the determination of the motion of N parallel rectilinear vortex filaments involves the solution of N non-linear differential equations, each…
The principle of symmetric criticality
It is frequently explicitly or implicitly assumed that if a variational principle is invariant under some symmetry groupG, then to test whether a symmetric field configuration ϕ is an extremal, it…
Linear Stability of Elliptic Lagrangian Solutions of the Planar Three-Body Problem via Index Theory
It is well known that the linear stability of Lagrangian elliptic equilateral triangle homographic solutions in the classical planar three-body problem depends on the mass parameter…
The N-Vortex Problem: Analytical Techniques
Introduction * N-Vortices in the Plane * Domains with Boundaries * Vortex Motion on a Sphere * Geometric Phases * Statistical Point Vortex Theories * Vortex Patch Models * Vortex Filament Models *…
The N-Vortex Problem
Über Integrale der hydrodynamischen Gleichungen, welche den Wirbelbewegungen entsprechen.
sind bisher Integrale der hydrodynamischen Gleichungen fast nur unter der Voraussetzung gesucht worden, dafs die rechtwinkligen Componenten der Geschwindigkeit jedes Wassertheilchens gleich gesetzt…
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