Stepwise Precession of the Resonant Swinging Spring

Abstract

Abstract. 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.

Asymptotic Method for Approximation of Resonant Interaction of Nonlinear Multidimensional Hyperbolic Waves

“…Similar ideas of asymptotical representation of unknown functions are used in many papers for description of different wave processes, and frequently the anzats of the solution is fixed even more strongly. For example, in [2,18] for (1.1) type problems the anzats of the asymptotical solution has (in our notation) slow variables τ , η, ζ and fast variables t, x. However, in these papers the main terms of the asymptotical expression have only one harmonic e iωt .…”

Classical and quantum-mechanical plane switching in CO2

“…In the presence of axial symmetry, many prefer using polar coordinates and conjugate because in these coordinates the Hamiltonian of an axially symmetric system (such as CO 2 ) does not depend on the cyclic variable ϕ, L can be immediately treated as a constant of motion with value l , and the system can thus be reduced to one degree of freedom with reduced phase space R 2 and coordinates ) , ( 2 2 p q [16]. In particular,…”

“…was used previously in the studies of the model 1:1:2 resonant swing-spring system [1][2][3][4]. Solution for the case 0 ≠ c is given in Ref.…”

“…One observes that the system periodically evolves into a planar swinging motion before returning back to its original springing motion. Most interestingly, the orientation of the swinging plane changes by a constant amount from one swinging phase to the next one, the size of the step depending on the initial conditions [2][3][4]. Under appropriate conditions, the orientation of the plane remains nearly constant for a time of several pendular oscillations and then changes abruptly.…”

Resonant Rossby Wave Triads and the Swinging Spring

“…These are the modulation equations, that is, the equations for the envelope amplitudes. They are in the canonical form of the system known as the three-wave equations (see, e.g., Holm and Lynch 2002).…”