Deformations of Poisson structures on fibered manifolds and adiabatic slow-fast systems. (English) Zbl 1379.53097

The authors of this short note study adiabatic perturbations of slow-fast Hamitonian systems and of some of their generalizations. At the core of the study is the fact that an adiabatic Hamiltonian system can be seen as a perturbed system on a fibre bundle with a “slow” base and a deformation of a “fast” fibrewise Poisson bracket. For the reader’s sake the authors recall the relevent definitions and facts. As they study objects on fibre bundles, Ehresmann connections play an important role in the study. Another ingredient of the study are actions of compact groups on Poisson fibre bundles and their influence on 2-cocycles related to Poisson structures. The authors demonstrate that every horizontal 2-cocycle is cohomologous to its average (cf. Corollary 10). The final section of the paper is dedicated to G-invariant normalizations. They show, among other things, that a generalized slow-fast Hamiltonian system can be transformed into a particular normal form (cf. Proposition 13). General considerations are supplemented by examples.


53D17 Poisson manifolds; Poisson groupoids and algebroids
70G45 Differential geometric methods (tensors, connections, symplectic, Poisson, contact, Riemannian, nonholonomic, etc.) for problems in mechanics
70H09 Perturbation theories for problems in Hamiltonian and Lagrangian mechanics
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