Renormalization group equations and integrability in Hamiltonian systems

We investigate Hamiltonian systems with two degrees of freedom by using renormalization group method. We show that the original Hamiltonian systems and the renormalization group equations are integrable if the renormalization group equations are Hamiltonian systems up to the second leading order of small parameter. To understand temporal evolutions of Hamiltonian systems, one useful method is to construct approximate solution to motion of equation. The approximate solution is usually obtained by perturbative methods. That is, we asymptotically expand variables as series of small parameter and solve equation of motion order by order. However, because of resonance, naive perturbation often produces secular terms which break the asymptotic expansion. Various tools have been developed to vanish the secular terms: averaging method, multiple scale method, matched asymptotic expansions, canonical perturbation theory [1, 2], and so on. Recently, renormalization group method [3, 4] is proposed as one of the most powerful tools to handle secular terms and it unifies many of the perturbation technique listed above. Roughly speaking, the method reduces equation of motion by ignoring fast motion, and the reduced equation is called renormalization group equation (RGE). Although RGE gives approximate solution to equation of motion, it is not obvious whether RGE keeps characteristic properties of the original Hamiltonian system. In this article, we investigate symplectic properties, E-mail address: [email protected] E-mail address: [email protected]