Invariant tori, action-angle variables, and phase space structure of the Rajeev-Ranken model

Abstract

We study the classical Rajeev-Ranken model, a Hamiltonian system with three degrees of freedom describing nonlinear continuous waves in a 1+1-dimensional nilpotent scalar field theory pseudodual to the SU(2) principal chiral model. While it loosely resembles the Neumann and Kirchhoff models, its equations may be viewed as the Euler equations for a centrally extended Euclidean algebra. The model has a Lax pair and r -matrix leading to four generically independent conserved quantities in involution, two of which are Casimirs. Their level sets define four-dimensional symplectic leaves on which the system is Liouville integrable. On each of these leaves, the common level sets of the remaining conserved quantities are shown, in general, to be 2-tori. The non-generic level sets can only be horn tori, circles and points. They correspond to measure zero subsets where the conserved quantities develop relations and solutions degenerate from elliptic to hyperbolic, circular or constant functions. A geometric construction allows us to realize each common level set as a bundle with base determined by the roots of a cubic polynomial. A dynamics is defined on the union of each type of level set, with the corresponding phase manifolds expressed as bundles over spaces of conserved quantities. Interestingly, topological transitions in energy hypersurfaces are found to occur at energies corresponding to horn tori, which support purely homoclinic orbits. The dynamics on each horn torus is non-Hamiltonian, but expressed as a gradient flow. Finally, we discover a family of action-angle variables for the system that apply away from horn tori.

Quantum Rajeev-Ranken model as an anharmonic oscillator

“…Thus, the RR model can be viewed as an Euler top for the nilpotent Lie algebra n 3 . Similarly, the RR equations can also be viewed as Euler equations for a centrally extended Euclidean algebra as mentioned in [10]; without the central extension, one gets the Kirchhoff model.…”

“…The latter bears some resemblance to the Neumann (for a particle moving on a 2-sphere) and Kirchhoff models, but is different from them. In [9,10], we found a Lagrangian and a pair of Hamiltonian formulations for the RR model, based on compatible degenerate nilpotent and Euclidean Poisson algebras. Lax pairs and classical r -matrices were found.…”

“…In §3, we take a complementary approach by interpreting the RR model as a novel 3D cylindrically symmetric anharmonic oscillator. This interpretation follows from rewriting the Hamiltonian in terms of the Darboux coordinates introduced in [9,10] and identifying the coordinates and momenta as those of a nonrelativistic particle subject to a quartic potential and possessing an unusual rotational energy. The parameters of the RR model (the coupling constant λ and screwon wavenumber k ) enter the oscillator in an intricate way, which leads to an unconventional physical interpretation of solutions of the oscillator.…”