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. •Geometric tools to describe slow–fast Hamiltonian systems on smooth manifolds.•Direct derivation of Painlevé-I equation near a fold point of a slow manifold.•Direct derivation of Painlevé-II equation near a cusp point of a slow manifold.