Singular reduction of resonant Hamiltonians

• Published in: Nonlinearity Vol. 31; no. 6; pp. 2854 - 2894
• Main Authors: Meyer, Kenneth R, Palacián, Jesús F, Yanguas, Patricia
• Format: Journal Article
• Language: English
• Published: IOP Publishing 01.06.2018
• Subjects: cross section; normal form and resonant Hamiltonian; orbit space and orbifold; peak and ridge; plateau; singular reduction; symplectic coordinates and symplectic smoothing
• ISSN: 0951-7715; 1361-6544
• DOI: 10.1088/1361-6544/aab591

We investigate the dynamics of resonant Hamiltonians with n degrees of freedom to which we attach a small perturbation. Our study is based on the geometric interpretation of singular reduction theory. The flow of the Hamiltonian vector field is reconstructed from the cross sections corresponding to an approximation of this vector field in an energy surface. This approximate system is also built using normal forms and applying reduction theory obtaining the reduced Hamiltonian that is defined on the orbit space. Generically, the reduction is of singular character and we classify the singularities in the orbit space, getting three different types of singular points. A critical point of the reduced Hamiltonian corresponds to a family of periodic solutions in the full system whose characteristic multipliers are approximated accordingly to the nature of the critical point.