Periodic solutions of singular first-order Hamiltonian systems of N-vortex type

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 \[ Γ_k\dot{z}_k(t)=J\nabla_{z_k} H\big(z(t)\big),\quad k=1,\dots,N, \] where ${\Gamma }_1,\dots ,{\Gamma }_N\in \mathbb{R}\setminus \{0\}$ are the vorticities, $J\in {\mathbb{R}}^{2\times 2}$ is the standard symplectic $2\times 2$ matrix, and the Hamiltonian $H$ is of $N$-vortex type: \[ H(z_1,\dots,z_N) = -\frac1{2π}\sum_{j\ne k}^N Γ_jΓ_k\log|z_j-z_k| - \sum_{j,k=1}^NΓ_jΓ_kg(z_j,z_k). \] Here $g:\Omega \times \Omega \to \mathbb{R}$ is an arbitrary symmetric function of class $C^2$, e.g.\ the regular part of a hydrodynamic Green function. Given a nondegenerate critical point $a_0\in \Omega$ of $h(z)=g(z,z)$ and a nondegenerate 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 $z^{(r)}(t)=(z_1^{(r)}(t),\dots ,z_N^{(r)}(t))\in {\Omega }^N$, $0<r<r_0$, with $z_k^{(r)}(t)\to a_0$ as $r\to 0$. In the limit $r\to 0$, and after a suitable rescaling, the solutions look like $Z(t)$.

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