Geometry of slow-fast Hamiltonian systems and Painlev\'e equations

Abstract

In the first part of the paper we introduce some geometric tools needed to describe slow-fast Hamiltonian systems on smooth manifolds. We start with a smooth Poisson bundle $p: M\to B$ of a regular (i.e. of constant rank) Poisson manifold $(M,\omega)$ over a smooth symplectic manifold $(B,\lambda)$, the foliation into leaves of the bundle coincides with the symplectic foliation generated by the Poisson structure on $M$. This defines a singular symplectic structure $\Omega_{\varepsilon}=$ $\omega + \varepsilon^{-1}p^*\lambda$ on $M$ for any positive small $\varepsilon$, where $p^*\lambda$ is a lift of 2-form $\lambda$ on $M$. Given a smooth Hamiltonian $H$ on $M$ one gets a slow-fast Hamiltonian system w.r.t. $\Omega_{\varepsilon}$. We define a slow manifold $SM$ of this system. Assuming $SM$ to be a smooth submanifold, we define a slow Hamiltonian flow on $SM$. The second part of the paper deals with singularities of the restriction of $p$ on $SM$ and their relations with the description of the system near them. It appears, if $\dim M = 4,$ $\dim B = 2$ and Hamilton function $H$ is generic, then behavior of the system near singularities of the fold type is described in the principal approximation by the equation Painlev\'e-I, but if a singular point is a cusp, then the related equation is Painlev\'e-II. This fact for particular types of Hamiltonian systems with one and a half degrees of freedom was discovered earlier by R.Haberman.