Stepwise Precession of the Resonant Swinging Spring

Darryl D. Holm ( and Peter Lynch (

Submitted April, 2001, to SIAM Journal on Applied Dynamical Systems


The swinging spring, or elastic pendulum, has a 2:1:1 resonance arising at cubic order in its approximate Lagrangian. The corresponding modulation equations are the well-known three-wave equations that also apply, for example, in laser-matter interaction in a cavity. We use Hamiltonian reduction and pattern evocation techniques to derive a formula that describes the characteristic feature of this system's dynamics, namely, the stepwise precession of its azimuthal angle.

PACS numbers: 02.40.-k, 05.45.-a, 45.10.Db, 45.20.Jj

Keywords: Classical mechanics, Variational principles, Averaged Lagrangian, Elastic Pendulum, Nonlinear Resonance.