Hamiltonian vector fields of homogeneous polynomials in two variables

Details Hamiltonian vector fields of homogeneous polynomials in two variables Hamiltonian vector fields of homogeneous polynomials in two variables

2007-09-16

arxiv.org/abs/0709.2511v2

Proceedings of the Institute of Mathematics NAS of Ukraine, (2006) vol 3, no 3, 269-308 (translation from Ukrainian)

Mathematics

Dynamical Systems

26 pages, 3 figures. In version 2 the latter section is removed, since it is not included into original article

Scientific paper

Let $g:R^2\to R$ be a homogeneous polynomial of degree $p>1$, $G=(-g'_{y}, g'_{x})$ be its Hamiltonian vector field, and $G_t$ be the local flow generated by $G$. Denote by $E(G,O)$ the space of germs of $C^{\infty}$ diffeomorphisms $(R^2,O)\to (R^2,O)$ that preserve orbits of $G$. Let also $E_{id}(G,O)$ be the identity component of $E(G,O)$ with respect to $C^1$-topology. Suppose that $g$ has no multiple prime factors. Then we prove that for every $h\in E_{id}(G,O)$ there exists a germ of a smooth function $\alpha:R^2\to R$ at $O$ such that $h(z)=G_{\alpha(z)}(z).$

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