[Paper Review] Renormalization group method and canonical perturbation theory

Yoshiyuki Y. Yamaguchi, Yasusada Nambu|arXiv (Cornell University)|Feb 16, 1999

Quantum chaos and dynamical systems7 references3 citations

TL;DR

This paper demonstrates that the renormalization group (RG) method yields identical approximate solutions to canonical perturbation theory for Hamiltonian systems with integrable parts linear in action variables, up to second order in a small parameter. By applying RG to action-angle coordinates, it establishes equivalence between the two approaches in this context, offering a new perspective on perturbative methods in Hamiltonian dynamics.

ABSTRACT

Renormalization group method is one of the most powerful tool to obtain approximate solutions to differential equations. We apply the renormalization group method to Hamiltonian systems whose integrable parts linearly depend on action variables. We show that the renormalization group method gives the same approximate solutions as canonical perturbation theory up to the second order of a small parameter with action-angle coordinates.

Motivation & Objective

• To investigate the applicability of the renormalization group (RG) method to Hamiltonian systems with integrable parts linear in action variables.
• To compare the RG method with canonical perturbation theory in terms of accuracy and solution structure.
• To determine whether the RG method reproduces the same approximate solutions as canonical perturbation theory up to second order in a small parameter.
• To establish a formal connection between RG and canonical perturbation theory in the context of action-angle variables.

Proposed method

• The RG method is applied to Hamiltonian systems where the integrable part is linear in action variables.
• The analysis is conducted in action-angle coordinates, which are standard in canonical perturbation theory.
• The small parameter is introduced to expand the solution perturbatively up to second order.
• The RG method systematically eliminates secular terms through a renormalization procedure in the perturbative expansion.
• Solutions from the RG method are compared directly with those from canonical perturbation theory.
• The equivalence of solutions is verified order-by-order in the small parameter expansion.

Experimental results

Research questions

• RQ1 Does the renormalization group method produce the same approximate solutions as canonical perturbation theory for Hamiltonian systems with action-linear integrable parts?
• RQ2 How do the solutions from the RG method compare to those from canonical perturbation theory up to second order in the small parameter?
• RQ3 Can the RG method effectively handle secular terms in perturbative expansions of Hamiltonian systems?
• RQ4 What is the formal relationship between the RG method and canonical perturbation theory in action-angle coordinates?

Key findings

• The renormalization group method produces approximate solutions that are identical to those from canonical perturbation theory up to second order in the small parameter.
• The equivalence holds specifically for Hamiltonian systems where the integrable part is linear in action variables.
• The RG method successfully removes secular terms through renormalization, mirroring the structure of canonical perturbation theory.
• The use of action-angle coordinates enables a direct comparison between the two methods.
• The results confirm the consistency and validity of the RG method as an alternative to canonical perturbation theory in this class of systems.