Classical resonances, Fermi resonances, and canonical transformations for three nonlinearly coupled oscillators

摘要

Classical resonances arising from the interaction of three nonlinearly coupled oscillators are studied from both a theoretical and numerical perspective. In particular, our study focuses on ternary classical resonances defined byn1ohgr;1+n2ohgr;2minus;n3ohgr;3=0. We discuss some of the experimental and quantum mechanical consequences of binary and ternary classical resonances (e.g., Fermi resonances and vibrationndash;rotation coupling). Numerically we show that it is possible to construct a threehyphen;dimensional map such that ternary classical resonances can be systematically found. Theoretically, we show that canonical transformations exist between resonant and nonresonant motion. These transformations predict various structural features of the threehyphen;dimensional numerical maps which are subsequently observed in a model numerical calculation. Finally we argue that the methods and ideas presented in this paper are generic and can be used for more general systems.