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Global continua of periodic solutions of singular first-order Hamiltonian systems of N-vortex type

Abstract

The paper deals with singular first order Hamiltonian systems of the form

$$\begin{aligned} \Gamma _k\dot{z}_k(t)=J\nabla _{z_k} H(z(t)),\quad z_k(t)\in \Omega \subset \mathbb {R}^2,\;\; k=1,\dots ,N, \end{aligned}$$

where \(J\in \mathbb {R}^{2\times 2}\) defines the standard symplectic structure in \(\mathbb {R}^2\), \(\Gamma _1,\dots ,\Gamma _N\in \mathbb {R}\setminus \{0\}\) are given, and the Hamiltonian H is of N-vortex type:

$$\begin{aligned} H(z_1,\dots ,z_N) = -\frac{1}{2\pi }\sum _{\genfrac{}{}{0.0pt}{}{j,k=1}{j\ne k}}^N \Gamma _j\Gamma _k\log \left| z_j-z_k\right| - F(z). \end{aligned}$$

This is defined on the configuration space \(\{(z_1,\ldots ,z_N)\in \Omega ^N:z_j\ne z_k\text { for }j\ne k\}\) of N different points in the domain \(\Omega \subset \mathbb {R}^2\). The function \(F:\Omega ^N\rightarrow \mathbb {R}\) may have additional singularities near the boundary of \(\Omega ^N\). We prove the existence of a global continuum of periodic solutions \(z(t)=(z_1(t),\dots ,z_N(t))\in \Omega ^N\) that emanates, after introducing a suitable singular limit scaling, from a relative equilibrium \(Z(t)\in \mathbb {R}^{2N}\) of the N-vortex problem in the whole plane (where \(F=0\)). Examples for Z include Thomson’s vortex configurations, or equilateral triangle solutions. The domain \(\Omega \) need not be simply connected. A special feature is that the associated action integral is not defined on an open subset of the space of \(2\pi \)-periodic \(H^{1/2}\) functions, the natural form domain for first order Hamiltonian systems. This is a consequence of the singular character of the Hamiltonian. Our main tool in the proof is a degree for \(S^1\)-equivariant gradient maps that we adapt to this class of potential operators.

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Acknowledgements

The authors thank the reviewer for several remarks that helped to improve the presentation.

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Correspondence to Thomas Bartsch.

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Communicated by Nalini Anantharaman.

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Bartsch, T., Gebhard, B. Global continua of periodic solutions of singular first-order Hamiltonian systems of N-vortex type. Math. Ann. 369, 627–651 (2017). https://doi.org/10.1007/s00208-016-1505-z

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Mathematics Subject Classification

  • Primary 37J45
  • Secondary 37N10
  • 76B47