Experimental demonstration of classical Hamiltonian monodromy in the 1:1:2 resonant elastic pendulum.

The 1:1:2 resonant elastic pendulum is a simple classical system that displays the phenomenon known as Hamiltonian monodromy. With suitable initial conditions, the system oscillates between nearly pure springing and nearly pure elliptical-swinging motions, with sequential major axes displaying a stepwise precession. The physical consequence of monodromy is that this stepwise precession is given by a smooth but multivalued function of the constants of motion. We experimentally explore this multivalued behavior. To our knowledge, this is the first experimental demonstration of classical monodromy.

Stability of halo orbits.

We predict new populations of trapped nonequatorial ("halo") orbits of charged dust grains about an arbitrary axisymmetric planet. Simple equilibrium and stability conditions are derived, revealing dramatic differences between positively and negatively charged grains in prograde or retrograde orbits. Implications for the Cassini mission to Saturn are discussed.

An integrable shallow water equation with linear and nonlinear dispersion.

We use asymptotic analysis and a near-identity normal form transformation from water wave theory to derive a 1+1 unidirectional nonlinear wave equation that combines the linear dispersion of the Korteweg-deVries (KdV) equation with the nonlinear/nonlocal dispersion of the Camassa-Holm (CH) equation. This equation is one order more accurate in asymptotic approximation beyond KdV, yet it still preserves complete integrability via the inverse scattering transform method. Its traveling wave solutions contain both the KdV solitons and the CH peakons as limiting cases.

CO2 molecule as a quantum realization of the 1:1:2 resonant swing-spring with monodromy.

We consider the wide class of systems modeled by an integrable approximation to the 3 degrees of freedom elastic pendulum with 1:1:2 resonance, or the swing-spring. This approximation has monodromy which prohibits the existence of global action-angle variables and complicates the dynamics. We study the quantum swing-spring formed by bending and symmetric stretching vibrations of the CO2 molecule. We uncover quantum monodromy of CO2 as a nontrivial codimension 2 defect of the three dimensional energy-momentum lattice of its quantum states.