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Deformation of Geometry and Bifurcations of Vortex Rings

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Recent Trends in Dynamical Systems

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 35))

Abstract

We construct a smooth family of Hamiltonian systems, together with a family of group symmetries and momentum maps, for the dynamics of point vortices on surfaces parametrized by the curvature of the surface. Equivariant bifurcations in this family are characterized, whence the stability of the Thomson heptagon is deduced without recourse to the Birkhoff normal form, which has hitherto been a necessary tool.

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Notes

  1. 1.

    Actually formulaic convenience leads us to take 4λ to be the Gaussian curvature.

  2. 2.

    Despite three formulae, a is a single analytic function of r, λ, with series expansion \(a = r -\frac{1} {3}{r}^{3}\lambda + \frac{1} {5}{r{}^{5}\lambda }^{2} -\frac{1} {7}{r{}^{7}\lambda }^{3} + \cdots \) convergent for | r 2 λ |  < 1.

  3. 3.

    The minus sign makes − Δ a positive operator. But we shall be casual about the sign and use + Δ as well as − Δ. All that the casualness causes is to reverse the direction of the flow.

  4. 4.

    S n is the group of permutations of the n vortices and D n has order 2n.

  5. 5.

    The vector field, not the Hessian.

  6. 6.

    Cf. ( 14.25 ).

  7. 7.

    We put \(\mathbf{D}_{1} = \mathbb{Z}_{2}\) acting by reflection.

  8. 8.

    “Nonagon” mixes Latin and Greek.

  9. 9.

    “Undecagon” is another Greco-Latin hybrid.

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Acknowledgements

JM thanks Tudor Ratiu and the staff of the Bernoulli Centre in Lausanne for their hospitality, as much of this paper was written during an extended visit there. TT thanks L. Mahadevan and the staff of SEAS at Harvard for their hospitality, as much of this paper was finished during an extended visit there.

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Authors and Affiliations

  1. School of Mathematics, University of Manchester, Oxford Road, Manchester, M13 9PL, UK

    James Montaldi

  2. Trinity Hall, Cambridge, CB2 1TJ, UK

    Tadashi Tokieda

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  1. James Montaldi
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  2. Tadashi Tokieda
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Correspondence to James Montaldi .

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Editors and Affiliations

  1. Zentrum Mathematik, Technische Universität München, Garching bei München, Germany

    Andreas Johann

  2. Zentrum Mathematik, Technische Universität München, Garching bei München, Germany

    Hans-Peter Kruse

  3. Zentrum Mathematik, Technische Universität München, Garching bei München, Germany

    Florian Rupp

  4. Zentrum Mathematik, Technische Universität München, Garching bei München, Germany

    Stephan Schmitz

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Montaldi, J., Tokieda, T. (2013). Deformation of Geometry and Bifurcations of Vortex Rings. In: Johann, A., Kruse, HP., Rupp, F., Schmitz, S. (eds) Recent Trends in Dynamical Systems. Springer Proceedings in Mathematics & Statistics, vol 35. Springer, Basel. https://doi.org/10.1007/978-3-0348-0451-6_14

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