Abstract
We are concerned with the dynamics of N point vortices \({z_1,\dots,z_N\in\Omega\subset\mathbb{R}^2}\) in a planar domain. This is described by a Hamiltonian system
where \({\Gamma_1,\dots,\Gamma_N\in\mathbb{R}\setminus\{0\}}\) are the vorticities, \({J\in\mathbb{R}^{2\times2}}\) is the standard symplectic \({2\times2}\) matrix, and the Hamiltonian H is of N-vortex type:
Here \({g:\Omega\times\Omega\to\mathbb{R}}\) is an arbitrary symmetric function of class \({\mathcal{C}^2}\), e.g., the regular part of a hydrodynamic Green function. Given a non-degenerate critical point \({a_0\in\Omega}\) of \({h(z)=g(z,z)}\) and a non-degenerate relative equilibrium \({Z(t)\in\mathbb{R}^{2N}}\) of the Hamiltonian system in the plane with \({g=0}\), we prove the existence of a smooth path of periodic solutions \({z^{(r)}(t)=(z^{(r)}_1(t),\dots,z^{(r)}_N(t))\in\Omega^N}\), \({0<r<r_0}\), with \({z^{(r)}_k(t)\to a_0}\) as \({r\to0}\). In the limit \({r\to0}\), and after a suitable rescaling, the solutions look like \({Z(t)}\).
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Bartsch, T. Periodic solutions of singular first-order Hamiltonian systems of N-vortex type. Arch. Math. 107, 413–422 (2016). https://doi.org/10.1007/s00013-016-0928-9
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DOI: https://doi.org/10.1007/s00013-016-0928-9