dorsal/arxiv
View SchemaIntegrable systems and symmetric products of curves
| Authors | Pol Vanhaecke |
|---|---|
| Categories | |
| ArXiv ID | solv-int/9402002 |
| URL | https://arxiv.org/abs/solv-int/9402002 |
Abstract
We show how there is associated to each non-constant polynomial $F(x,y)$ a completely integrable system with polynomial invariants on $\Rd$ and on $\C{2d}$ for each $d\geq1$; in fact the invariants are not only in involution for one Poisson bracket, but for a large class of polynomial Poisson brackets, indexed by the family of polynomials in two variables. We show that the complex invariant manifolds are isomorphic to affine parts of $d$-fold symmetric products of a deformation of the algebraic curve $F(x,y)=0$, and derive the structure of the real invariant manifolds from it. We also exhibit Lax equations for the hyperelliptic case (i.e., when $F(x,y)$ is of the form $y^2+f(x)$) and we show that in this case the invariant manifolds are affine parts of distinguished (non-linear) subvarieties of the Jacobians of the curves. As an application the geometry of the H\'enon-Heiles hierarchy --- a family of superimposable integrable polynomial potentials on the plane --- is revealed and Lax equations for the hierarchy are given.
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"abstract": "We show how there is associated to each non-constant polynomial $F(x,y)$ a\ncompletely integrable system with polynomial invariants on $\\Rd$ and on\n$\\C{2d}$ for each $d\\geq1$; in fact the invariants are not only in involution\nfor one Poisson bracket, but for a large class of polynomial Poisson brackets,\nindexed by the family of polynomials in two variables. We show that the complex\ninvariant manifolds are isomorphic to affine parts of $d$-fold symmetric\nproducts of a deformation of the algebraic curve $F(x,y)=0$, and derive the\nstructure of the real invariant manifolds from it. We also exhibit Lax\nequations for the hyperelliptic case (i.e., when $F(x,y)$ is of the form\n$y^2+f(x)$) and we show that in this case the invariant manifolds are affine\nparts of distinguished (non-linear) subvarieties of the Jacobians of the\ncurves. As an application the geometry of the H\\\u0027enon-Heiles hierarchy --- a\nfamily of superimposable integrable polynomial potentials on the plane --- is\nrevealed and Lax equations for the hierarchy are given.",
"arxiv_id": "solv-int/9402002",
"authors": [
"Pol Vanhaecke"
],
"categories": [
"solv-int",
"nlin.SI"
],
"title": "Integrable systems and symmetric products of curves",
"url": "https://arxiv.org/abs/solv-int/9402002"
},
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