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arxiv: 2604.25682 · v1 · submitted 2026-04-28 · 🧮 math-ph · cond-mat.quant-gas· math.MP· nlin.SI· physics.flu-dyn

Recognition: unknown

Co-rotating Vortices on Surfaces of Variable Negative Curvature: Hamiltonian Structure and Drift Dynamics

Gaurang Mangesh Joshi , Rickmoy Samanta
Authors on Pith no claims yet

Pith reviewed 2026-05-07 14:25 UTC · model grok-4.3

classification 🧮 math-ph cond-mat.quant-gasmath.MPnlin.SIphysics.flu-dyn
keywords vortex dynamicscatenoidGaussian curvatureHamiltonian reductionco-rotating vorticesnegative curvatureazimuthal drift
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The pith

Two identical vortices on a catenoid rotate rigidly at fixed latitude with speed set by the curvature gradient

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

Vortices on curved surfaces move differently than in flat space because the geometry alters their mutual interactions. This paper works out the Hamiltonian description of point vortices on a catenoid, a minimal surface whose negative Gaussian curvature changes with latitude. For two identical vortices placed exactly opposite each other, an exact solution exists in which the pair spins rigidly around the axis while staying at constant height. The angular speed of this rotation is proportional to the slope of the curvature rather than to the curvature value itself. For vortices that are not antipodal the motion reduces to bounded radial oscillations plus a steady azimuthal drift, and the basic rotating state is linearly unstable.

Core claim

On the catenoid two identical vortices admit an exact antipodal solution in which they co-rotate rigidly at fixed latitude. Their common angular velocity is given by Ω = (Γ/16π) K'(V)/√(-K(V)), where K(V) is the Gaussian curvature. The symmetric state is linearly unstable with growth rate λ = √3 |Ω|. For generic equal-strength pairs, conservation of the Hamiltonian and rotational momentum reduces the dynamics to a single quadrature that produces bounded relative oscillations together with secular azimuthal drift.

What carries the argument

The Hamiltonian for point vortices on the catenoid, constructed from the Green's function of the surface metric, which supplies the conserved quantities that allow exact reduction of the co-rotating-pair dynamics to quadrature.

If this is right

  • The rotation rate of the antipodal pair depends on the gradient of curvature rather than on the curvature value.
  • The antipodal configuration is linearly unstable with growth rate √3 times the absolute value of the rotation rate.
  • Generic equal-strength pairs undergo bounded oscillations in separation accompanied by continuous azimuthal drift.
  • Curvature-induced drift appears in simulations of localized many-vortex clusters on the same surface.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same gradient-driven rotation and drift may appear on other surfaces whose negative curvature varies smoothly.
  • Collective vortex transport on curved geometries could produce net vorticity redistribution without external forcing.
  • The explicit instability growth rate supplies a testable timescale for the breakup of symmetric vortex pairs.

Load-bearing premise

The point-vortex idealization together with the specific Hamiltonian derived from the Biot-Savart law on the catenoid metric remains valid for the co-rotating pairs examined.

What would settle it

A numerical integration or laboratory realization of two identical vortices placed antipodally on a catenoid that fails to rotate at the angular velocity Ω = (Γ/16π) K'(V)/√(-K(V)) would disprove the exact solution.

Figures

Figures reproduced from arXiv: 2604.25682 by Gaurang Mangesh Joshi, Rickmoy Samanta.

Figure 1
Figure 1. Figure 1: FIG. 1. Angular velocity Ω( view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Linear stability of the symmetric co–rotating two–vortex configuration on the catenoid. (a) view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Dynamics of a generic equal–strength co–rotating vortex pair on the catenoid. Top view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Comparison of the full numerical dynamics and the reduced analytic theory for a generic view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Verification of the reduced theory for the mean azimuthal drift of a generic equal–strength view at source ↗
Figure 3
Figure 3. Figure 3: The vortex trajectories exhibit bounded meridional oscillations combined with a view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Preview of collective many-vortex dynamics on the catenoid. The simulation uses a view at source ↗
read the original abstract

Vortices in fluids and superfluids underpin phenomena ranging from Bose--Einstein condensates and superfluid films to neutron stars and hydrodynamic micro-rotors, where geometry can strongly influence their motion. Curvature can induce vortex motion with no planar analogue. We study Hamiltonian vortex motion on a catenoid, a minimal surface of variable negative curvature, and derive explicit equations of motion, conserved quantities, and reductions for co-rotating vortex pairs. For two identical vortices we find an exact antipodal solution in which the pair rotates rigidly at fixed latitude, with angular velocity $\Omega=(\Gamma/16\pi)\,K'(V)/\sqrt{-K(V)}$, where $K(V)$ is the Gaussian curvature. Thus the motion is governed by the curvature gradient rather than the curvature itself. The symmetric state is linearly unstable, with growth rate $\lambda=\sqrt{3}|\Omega|$, in agreement with numerical simulations. For generic equal-strength pairs, conservation of the Hamiltonian and rotational momentum reduces the nonlinear dynamics to a single quadrature, yielding bounded relative oscillations together with a secular azimuthal drift. Simulations of the full equations confirm the reduced theory and reveal the same curvature-induced transport mechanism in a localized many-vortex cluster, motivating a broader theory of collective vortex drift on curved surfaces.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The manuscript constructs the Hamiltonian for point vortices on the catenoid via the Green's function of the Laplace-Beltrami operator on its metric, derives the equations of motion for co-rotating pairs, identifies an exact antipodal rigid-rotation equilibrium whose angular velocity is Ω = (Γ/16π) K'(V)/√(-K(V)), shows that this state is linearly unstable with growth rate λ = √3 |Ω|, reduces the two-vortex dynamics to a single quadrature that yields bounded relative oscillations plus secular azimuthal drift, and verifies all analytic results by direct numerical integration of the full equations, including a demonstration of the same curvature-gradient transport in a localized many-vortex cluster.

Significance. If the derivations hold, the work supplies one of the few exact, parameter-free solutions for vortex motion on a surface whose Gaussian curvature is neither constant nor zero. The explicit reduction to quadrature, the closed-form instability rate, and the direct numerical confirmation of both the equilibrium and the reduced dynamics constitute reproducible, falsifiable results that can serve as benchmarks for collective vortex theories on curved manifolds.

minor comments (3)
  1. [§2] The definition of the coordinate V and the explicit form of the catenoid metric (including the range of V) should appear in the first section that introduces the geometry, rather than being deferred.
  2. [§5] In the many-vortex simulation paragraph, the number of vortices, their initial placement, and the precise diagnostic used to extract the drift velocity should be stated so that the claim of 'the same curvature-induced transport mechanism' can be reproduced from the text alone.
  3. [§4] The linearization matrix whose eigenvalues yield λ = √3 |Ω| is central; a brief appendix or inline display of the 4×4 Jacobian evaluated at the antipodal point would remove any ambiguity about the algebraic steps.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their careful reading, accurate summary of our results, and positive assessment of the significance of the work. We are pleased that the derivations, reductions to quadrature, instability analysis, and numerical verifications were found reproducible and falsifiable. The recommendation for minor revision is noted; however, no specific major comments were provided in the report.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The central results follow from constructing the Hamiltonian for point vortices via the Green's function of the Laplace-Beltrami operator on the catenoid metric, reducing the two-vortex equations to an explicit antipodal equilibrium whose angular velocity Ω is obtained by direct substitution into the equations of motion, and linearizing the resulting vector field to extract the growth rate λ = √3 |Ω|. These steps are algebraic consequences of the metric and the standard point-vortex interaction; no parameter is fitted to data and then relabeled as a prediction, no load-bearing premise rests on a self-citation chain, and the curvature-gradient dependence emerges from the explicit form of the Hamiltonian rather than being imposed by definition. The agreement with numerical integration of the full equations provides external verification rather than circular confirmation.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the standard point-vortex Hamiltonian on a Riemannian surface together with the explicit catenoid metric; no free parameters are fitted to data and no new entities are postulated.

axioms (1)
  • domain assumption Vortex motion on the surface is governed by a Hamiltonian system whose interaction kernel is determined by the Green's function of the Laplace-Beltrami operator on the catenoid.
    Invoked to obtain the equations of motion and conserved quantities for co-rotating pairs.

pith-pipeline@v0.9.0 · 5548 in / 1487 out tokens · 65254 ms · 2026-05-07T14:25:44.197791+00:00 · methodology

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Reference graph

Works this paper leans on

35 extracted references · 1 canonical work pages · 1 internal anchor

  1. [1]

    Aref,Integrable, chaotic, and turbulent vortex motion in two-dimensional flows,Annu

    H. Aref,Integrable, chaotic, and turbulent vortex motion in two-dimensional flows,Annu. Rev. Fluid Mech.15(1983) 345–389

    1983

  2. [2]

    P. G. Saffman,Vortex Dynamics(Cambridge Univ. Press, 1993)

    1993

  3. [3]

    C. C. Lin,On the motion of vortices in two dimensions: I. Existence of the Kirchhoff–Routh function,Proc. Natl. Acad. Sci. USA27(1941) 570–575. 25

    1941

  4. [4]

    C. C. Lin,On the motion of vortices in two dimensions: II. Some further investigations on the Kirchhoff–Routh function,Proc. Natl. Acad. Sci. USA27(1941) 575–577

    1941

  5. [5]

    A. A. Tchieu, E. Kanso, and P. K. Newton,The finite-dipole dynamical system, Proc. R. Soc. A468, 3006–3026 (2012)

    2012

  6. [6]

    Lydon, S

    K. Lydon, S. V. Nazarenko, and J. Laurie,Dipole dynamics in the point vortex model, J. Phys. A: Math. Theor.55, 385702 (2022)

    2022

  7. [7]

    Samanta and N

    R. Samanta and N. Oppenheimer,Vortex flows and streamline topology in curved biological membranes,Phys. Fluids33(2021) 092111

    2021

  8. [8]

    Maurya, S

    U. Maurya, S. T. Gavva, A. Saha and R. Samanta,Vortex dynamics in tubular fluid mem- branes,Phys. Fluids37(2025) 073109

    2025

  9. [9]

    Aswathy K. R. , U. Maurya, S. T. Gavva and R. Samanta,Dynamics of vortex clusters on a torus,Phys. Fluids37(2025) 093324

    2025

  10. [10]

    Banthia and R

    K. Banthia and R. Samanta,A self propelled vortex dipole model on a surface of variable negative curvature, J. Phys. A: Math. Theor.59, 145701 (2026)

    2026

  11. [11]

    Aswathy K. R. and R. Samanta,Collective dynamics of vortex clusters in compact fluid do- mains: From pair interactions to a quadrupole description, arXiv:2604.07373 [physics.flu-dyn] (2026)

    work page internal anchor Pith review Pith/arXiv arXiv 2026

  12. [12]

    V. A. Bogomolov,Dynamics of vorticity on a sphere,Fluid Dyn.12(1977) 863–870

    1977

  13. [13]

    Kimura and H

    Y. Kimura and H. Okamoto,Vortex motion on a sphere,J. Phys. Soc. Jpn.56(1987) 4203– 4206

    1987

  14. [14]

    Kimura,Vortex motion on surfaces with constant curvature,Proc

    Y. Kimura,Vortex motion on surfaces with constant curvature,Proc. R. Soc. A455(1999) 245–259

    1999

  15. [15]

    Kidambi and P

    R. Kidambi and P. K. Newton,Streamline topologies for integrable vortex motion on a sphere, Physica D140(2000) 95–125

    2000

  16. [16]

    Kidambi and P

    R. Kidambi and P. K. Newton,Point vortex motion on a sphere with solid boundaries,Phys. Fluids12(2000) 581–588

    2000

  17. [17]

    Hally,Stability of streets of vortices on surfaces of revolution with a reflection symmetry, J

    D. Hally,Stability of streets of vortices on surfaces of revolution with a reflection symmetry, J. Math. Phys.21(1980) 211–217

    1980

  18. [18]

    Crowdy and J

    D.G. Crowdy and J. Marshall,Analytical formulae for the Kirchhoff–Routh path function in multiply connected domains,Proc. R. Soc. A461(2005) 2477–2501. 26

    2005

  19. [19]

    Boatto and J

    S. Boatto and J. Koiller,Vortices on closed surfaces, inGeometry, Mechanics, and Dynamics: The Legacy of Jerry Marsden, edited by D. E. Chang, D. D. Holm, G. Patrick, and T. Ratiu, Fields Institute Communications, Vol. 73 (Springer, New York, 2015), pp. 185–237

    2015

  20. [20]

    Koiller and S

    J. Koiller and S. Boatto,Vortex pairs on surfaces, AIP Conf. Proc.,1130, 77–88 (2009)

    2009

  21. [21]

    D. G. Dritschel and S. Boatto,The motion of point vortices on closed surfaces,Proc. R. Soc. A471(2015) 20140890

    2015

  22. [22]

    A. M. Turner, V. Vitelli and D. R. Nelson,Vortices on curved surfaces,Rev. Mod. Phys.82 (2010) 1301–1348

    2010

  23. [23]

    Reuther and A

    S. Reuther and A. Voigt,The interplay of curvature and vortices in flow on curved surfaces, Multiscale Model. Simul.13, 632–643 (2015)

    2015

  24. [24]

    Gustafsson,Vortex pairs and dipoles on closed surfaces,J

    B. Gustafsson,Vortex pairs and dipoles on closed surfaces,J. Nonlinear Sci.32(2022) 62

    2022

  25. [25]

    T. D. Drivas, D. Glukhovskiy, and B. Khesin,Singular Vortex Pairs Follow Magnetic Geodesics, Int. Math. Res. Not.2024(14), 10880–10894 (2024)

    2024

  26. [26]

    M. A. Caracanhas, P. Massignan, and A. L. Fetter,Superfluid vortex dynamics on an ellipsoid and other surfaces of revolution, Phys. Rev. A105, 023307 (2022)

    2022

  27. [27]

    T. W. Neely, E. C. Samson, A. S. Bradley, M. J. Davis and B. P. Anderson,Observation of vortex dipoles in an oblate Bose–Einstein condensate,Phys. Rev. Lett.104(2010) 160401

    2010

  28. [28]

    D. V. Freilich, D. M. Bianchi, A. M. Kaufman, T. K. Langin and D. S. Hall,Real-time dynamics of single vortex lines and vortex dipoles in a Bose–Einstein condensate,Science329(2010) 1182–1185

    2010

  29. [29]

    Gauthier, M

    G. Gauthier, M. T. Reeves, X. Yu, A. S. Bradley, M. Baker, T. A. Bell, H. Rubinsztein- Dunlop, M. J. Davis, and T. W. Neely,Giant vortex clusters in a two-dimensional quantum fluid, Science364, 1264–1267 (2019)

    2019

  30. [30]

    S. J. Rooney, P. B. Blakie, B. P. Anderson and A. S. Bradley,Suppression of Kelvon-induced decay of quantized vortices in oblate Bose-Einstein condensates,Phys. Rev. A84, 023637 (2011)

    2011

  31. [31]

    R. H. Goodman, P. G. Kevrekidis and R. Carretero-Gonz´ alez,Dynamics of Vortex Dipoles in Anisotropic Bose–Einstein Condensates,SIAM J. Appl. Dyn. Syst.14, no. 2, 699–729 (2015)

    2015

  32. [32]

    A. C. White, C. F. Barenghi and N. P. Proukakis,Creation and Characterization of Vortex Clusters in Atomic Bose-Einstein Condensates,Physical Review A86, 013635, (2012). 27

    2012

  33. [33]

    A. C. White, B. P. Anderson, and V. S. Bagnato,Vortices and turbulence in trapped atomic condensates, Proc. Natl. Acad. Sci. U.S.A.111, 4719–4726 (2014)

    2014

  34. [34]

    G. W. Stagg, N. G. Parker, and C. F. Barenghi,Ultraquantum turbulence in a quenched homogeneous Bose gas, Phys. Rev. A94, 053632 (2016)

    2016

  35. [35]

    Samanta,Co-rotating Vortex Clusters on Negatively Curved Geometries, manuscript in preparation (2026)

    R. Samanta,Co-rotating Vortex Clusters on Negatively Curved Geometries, manuscript in preparation (2026)

    2026