Monodromy in the resonant swing spring

Abstract

This paper shows that an integrable approximation of the spring pendulum, when tuned to be in $1:1:2$ resonance, has monodromy. The stepwise precession angle of the swing plane of the resonant spring pendulum is shown to be a rotation number of the integrable approximation. Due to the monodromy, this rotation number is not a globally defined function of the integrals. In fact at lowest order it is given by $\arg(a+ib)$ where $a$ and $b$ are functions of the integrals. The resonant swing spring is therefore a system where monodromy has easily observed physical consequences.