Phase space invariant tori

Figure Al.2.7. Trajectory of two coupled stretches, obtained by integrating Hamilton s equations for motion on a PES for the two modes. The system has stable anhamionic synmretric and antisyimnetric stretch modes, like those illustrated in figrne Al.2.6. In this trajectory, semiclassically there is one quantum of energy in each mode, so the trajectory corresponds to a combination state with quantum numbers nj = [1, 1]. The woven pattern shows that the trajectory is regular rather than chaotic, corresponding to motion in phase space on an invariant torus.

Thus, the orbits in the domain Q x Tn of phase space lie on invariant tori parameterized by the action variables h,..., /n, and the motion on each torus is a Kronecker flow with frequencies wi(/),..., ojn(I). 

The dynamics of such systems is described by the Kolmogorov-Arnold-Moser theory of nearly integrable conservative dynamical systems (see e.g. Ott (1993)). For e = 0 the fluid elements move along the streamlines and the trajectories in the phase space form tubes parallel to the time axis. Due to the periodicity in the temporal direction these tubes form tori that fill the whole phase space and are invariant surfaces for the motion of the fluid elements. Each torus... 

According to the Poincare-Birkhoff fixed point theorem all resonant tori break up for arbitrarily small perturbations. If the rotation number is p/q the perturbation leaves q pairs of hyperbolic and elliptic periodic orbits. The unstable hyperbolic orbits are embedded in a layer filled by aperiodic, chaotic orbits that do not stay on an invariant surface, but cover a finite non-zero volume of the phase space in a chaotic layer around the original resonant torus. The elliptic points, however, are wrapped around by new concentric tori that form islands of regular orbits within the chaotic band (Fig. 2.5). 

Although in a weakly time-dependent flow all resonant tori disappear together with some of the nearly resonant tori around them, the Kolmogorov-Arnold-Moser theorem ensures that infinitely many invariant surfaces survive a small perturbation. For sufficiently small e the remaining invariant surfaces formed by quasiperiodic orbits, so called KAM tori, still occupy a non-zero volume of the phase space. The condition for a torus to survive a given perturbation is that its rotation number should be sufficiently far from any rational number so that the inequality... 

Although the phase space trajectories appear as simple curves on the two-dimensional Iz,ip phase space diagram (the 0 coordinate is suppressed) most trajectories are actually quasiperiodic. The actual trajectories he on the 2-dimensional surface of a 3-dimensional invariant torus in 4-dimensional phase space. Fig. 9.14 shows such a torus. Any point on the surface of the torus is specified by two angles, 0 and. The 0 and circuits about the torus are shown, respectively, as large and small diameter circles. The diameter of the 0... 

Fig. 3.5. Motion of a phase space point for an integrable Hamiltonian system with two degrees of freedom, (a) Invariant tori in a 3D constant energy space E and (b) the flow on a 2D torus.

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