Monodromy in the resonant swing spring

Abstract

In this paper, it is shown 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(χ + iλ), where χ and λ are functions of the integrals. The resonant swing spring is therefore a system where monodromy has easily observed physical consequences

Classical and quantum-mechanical plane switching in CO2

“…(39) of Ref. [3]. Since we took first anharmonicities into account, the polynomial in the right-hand side of Eq.…”

“…(2.4) is of degree 4 with respect to J, while it is of degree 3 in Ref. [3]. Moreover, Dullin et al consider only the case with no detuning ( 0 = ε ).…”

“…In the original system, it corresponds to the unstable short periodic orbit, which is at the origin of monodromy and plane switching [3,5]. The singularity corresponds to an unstable equilibrium of the reduced one degree of freedom system with dynamical variables ) , ( …”

“…As in the case of * ω , ρ ∆ changes rapidly in the neighborhood of It should also be emphasized that, like for the swing spring, the bending motion appears to be "linear" only for very small L . Rigorously speaking, this motion traces ellipses in the ) , ( 2 2 y x q q plane [3]. When L is larger, the eccentricity of the ellipses becomes smaller, so that the bending motion can hardly be described as taking place along a line.…”

“…We then comment on our treatment of the whole system with three degrees of freedom and compare it to Refs. [3,5,6]. …”