Geometry of slow-fast Hamiltonian systems and Painlevé equations

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 bundle p:M→B where (M,ω) is a C∞-smooth presymplectic manifold with a closed constant rank 2-form ω and (B,λ) is a smooth symplectic manifold. The 2-form ω is supposed to be compatible with the structure of the bundle, that is the bundle fibers are symplectic manifolds with respect to the 2-form ω and the distribution on M generated by kernels of ω is transverse to the tangent spaces of the leaves and the dimensions of the kernels and of the leaves are supplementary. This allows one to define a symplectic structure Ωε=ω+ε−1p∗λ on M for any positive small ε, where p∗λ is the lift of the 2-form λ to M. Given a smooth Hamiltonian H on M one gets a slow-fast Hamiltonian system with respect to Ωε. We define a slow manifold SM for this system. Assuming SM is 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 to SM. We show that if dimM=4,dimB=2 and Hamilton function H is generic, then the behavior of the system near a singularity of fold type is described, to the main order, by the equation Painlevé-I, and if this singularity is a cusp, then the related equation is Painlevé-II.