Canonical structure of renormalization group equations and separability of Hamiltonian systems

1998/10/07 by Yoshiyuki Y. Yamaguchi, Yasusada Nambu, Yamaguchi, Yoshiyuki Y. +1
Computer Science · Physics and Astronomy · #Chaotic Dynamics (nlin.CD) #FOS: Physical sciences #Nonlinear Dynamics and Pattern Formation #Nonlinear Waves and Solitons #Quantum chaos and dynamical systems

paper · pdf · doi:10.48550/arxiv.chao-dyn/9810010

openalex publication_date 1998/10/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We investigate perturbed Hamiltonian systems with two degrees of freedom by renormalization group method, which derives a reduced equation called renormalization group equation (RGE) by handling secular terms. We found that RGE is not always a Hamiltonian system. The necessary and sufficient condition that RGE becomes a Hamiltonian system up to the second leading order of a small parameter is that the original system is separable by Cartesian coordinates. Moreover, RGE is integrable when it is a Hamiltonian system. These statements are partial generalizations of our previous paper.

Citations

• Long-time correlations in the stochastic regime
• The Renormalization Group and Singular Perturbations: Multiple-Scales, Boundary Layers and Reductive Perturbation Theory

Related

• Renormalization group method and canonical perturbation theory
• Hamiltonian Renormalization Groups
• KAM--renormalization-group for Hamiltonian systems with two degrees of freedom
• The Canonical Symmetry and Hamiltonian Formalism. II. Hamiltonian Operators
• Canonical equations of Hamilton with beautiful symmetry