报告题目：Birkhoff normal forms, Dirac brackets and symplectic reduction

报 告 人：Jose Lamas Rodriguez (PostDoc @ DUT - Dalian University of Technology)

报告时间：2026年3月19日（星期四）9:00—9:45

报告地点：数学科学学院114（小报告厅）      

校内联系人：廖娴 教授      联系方式：84708351-8510

Abstract: The Birkhoff normal form provides a systematic way to simplify Hamiltonian dynamics near an equilibrium or relative equilibrium and remains one of the basic tools in local dynamical analysis. For Hamiltonian systems with symmetry, however, the relevant dynamics is the reduced dynamics, and the reduced phase space may be singular. This creates a serious obstacle for local normal form methods, which are usually most effective in smooth canonical coordinates. In this talk I will present a local approach that avoids performing the normal form construction directly on the reduced space. The idea is to work instead on a smooth momentum level set and to model the reduced dynamics locally by means of a transverse system of second-class constraints. The associated Dirac bracket then provides an effective Poisson structure on the slice. I will explain a condition on the quadratic part of the Hamiltonian ensuring that the dynamics on the momentum level, the Dirac-bracket dynamics on the slice, and the reduced dynamics on the corresponding symplectic stratum agree locally. This gives a practical framework for constructing Birkhoff normal forms in symmetric Hamiltonian systems without having to normalize directly on a possibly singular quotient.