Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
5778884 | Indagationes Mathematicae | 2016 | 26 Pages |
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 bundle p:MâB where (M,Ï) is a Câ-smooth presymplectic manifold with a closed constant rank 2-form Ï and (B,λ) is a smooth symplectic manifold. The 2-form Ï 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 Ï and the distribution on M generated by kernels of Ï 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=Ï+εâ1pâλ on M for any positive small ε, where pâλ 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.
Related Topics
Physical Sciences and Engineering
Mathematics
Mathematics (General)
Authors
L.M. Lerman, E.I. Yakovlev,