A new method to improve validity range of Lie canonical perturbation theory: with a central focus on a concept of non-blow-up region

Abstract

Validity ranges of Lie Canonical Perturbation Theory (LCPT) are investigated in terms of non blow-up regions. We investigate how the validity ranges depend on the perturbation order in two systems, one of which is a simple Hamiltonian system with one degree of freedom and the other is a HCN molecule. Our analysis of the former system indicates that non blow-up regions become reduced in size as the perturbation order increases. In case of LCPT by Dragt and Finn and that by Deprit, the non blow-up regions enclose the region inside the separatrix of the Hamiltonian but it may not be the case for LCPT by Hori. We also analyze how well the actions constructed by these LCPTs approximate the true action of the Hamiltonian in the non blow-up regions and find that the conventional truncated LCPT does not work over the whole region inside the separatrix whereas LCPT by Dragt and Finn without truncation does. Our analysis of the latter system indicates that non blow-up regions do not necessarily cover the whole regions inside the HCN well.We propose a new perturbation method to improve non blow-up regions and validity ranges inside them. Our method is free from blowing up and retains the same normal form as the conventional LCPT. We demonstrate our method in the two systems and show that the actions constructed by our method have larger validity ranges than those by the conventional and our previous methods proposed in [1,2].