Phase space action/angle

We begin with the simple case of one-dimensional problems described algebraically by U(2). The coset space for this case is just a single complex variable, which we call We denote the complex conjugate by 2,. These variables can be interpreted in terms of the position (q) and momentum (p) variables in phase space. Equivalently the t, variables can be related to the action-angle variables /,0 introduced in Section 3.4. To be more precise... 

As has been discussed in Section 111, the initial phase-space distribution pyj, for the nuclear DoF xj and pj may be chosen from the action-angle (18) or the Wigner (17) distribution of the initial state of the nuclear DoF. To specify the electronic phase-space distribution pgj, let us assume that the system is initially in the electronic state v i ). According to Eq. (80b), the electronic state vl/ ) is mapped onto Ne harmonic oscillators, whereby the nth oscillator is in its first excited state while the remaining Nei — 1 oscillators are in their ground state. The initial density operator is thus given by... 

They would become the stars of Prigoginian statistical mechanics. Their importance lies in the fact that, whenever it is possible to determine these variables by a canonical transformation of the initial phase space variables, one obtains a description with the following properties. The action variables / ( = 1,2,..., N, where N is the number of degrees of freedom of the system) are invariants of motion, whereas the angles a increase linearly in time, with frequencies generally action-dependent. The integration of the equations... 

Let us now lift this disk D2 into phase space. To do so, one must go back to the sphere equation, Eq. (37). There are several ways of depicting a 3-sphere one is particularly appropriate here [24]. The sphere is dynamically composed of two identical harmonic oscillators without explicit coupling, but whose total energy is a constant, hs3 > 0. Let us thus transform the Hamiltonian (37) in action angle variables, where N,Iy are the actions of the two oscillators and Q, 0, are the two associated angles. Since... 

Fig. 7. Doubly logarithmic plot of the survival probabilities N(n) in kicked hydrogen (Eq. (11)) as functions of the number n of cks for scaled field strength s = 1. The various curves are obtained with initial conditions on different cuts through phase space corresponding to fixed values of the scaled action f = no and uniformly distributed angles. (From [45])...

However, the effect of a small perturbation in action-action-angle type flows is quite different. The two-parameter family of invariant cycles coalesce into invariant tori that are connected by resonant sheets defined by the u(h,l2) = 0 condition. The consequence of this is that contrary to action-angle-angle flows in this case a trajectory can cover the whole phase space and no transport barriers exist. Thus, in this type of flows global uniform mixing can be achieved for arbitrarily small perturbations. This type of resonance induced dispersion has been demonstrated numerically in a low-Reynolds number Couette flow between two rotating spheres by Cartwright et al. 

Second, calculations can be performed in a semiclassical regime, and the results plotted on a Poincare section in action-angle (I, 0) coordinates. Such diagrams may seem complicated (see figs. 10.16 and 10.17), but are at least in principle readily understood a near-horizontal line across the (1,0) plot corresponds to a torus in ordinary phase space. When periodically extended in the time coordinate, each line corresponds to a vortex tube embedded in the extended phase space of the periodically... 

Trajectories in action/angle polyad phase space convey all of the most important qualitative relationships between a quantum spectrum and classical intramolecular dynamics. However, coordinate space trajectories are both more easily visualized and more directly comparable to quantum probability densities, ip(Qi,Q2,... ( 3jv-6) 2 Xiao and Kellman (1989) describe how the action/angle phase space trajectories for each eigenstate may be converted into a coordinate space trajectory. The key to this is the exact relationship between Morse oscillator displacement coordinates, rit and the action, angle variables, Ii,4>i (Rankin and Miller, 1971). Figure 9.17 shows, for the 6 eigenstates in the I = 3 (N = vs + va = 5) polyad of H20, the correspondences between the phase space trajectories, the coordinate space trajectories, and the probability densities. The resemblance between the classical coordinate space trajectories and the quantum probability densities is striking ... 

The 8-dimensional phase space of two 2-dimensional oscillators is reduced by the existence of two conserved actions, Ka and Kb, and by the absence of the conjugate angles, classical mechanical polyad 7feff. The conserved actions appear parametrically in 7feff, thus the phase space accessible at specified values of Ka and Kb is four dimensional. Since energy is conserved, in addition to Ka and Kb, all trajectories lie on the surface of a 3-dimensional energy shell. 

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