Pendulum phase space

Thus, the pendulum phase space is four-dimensional. For the special case considered here, the equations of motion (3.2.6) agree with those derived by Shinbrot et al. (1992). 

The pendulum phase space is not periodic in Z as was the case for the standard mapping (see Fig. 5.4). But in the vicinity of Z = 0 it qualitatively reproduces the phase-space features of the kicked rotor very... 

Fig. 5.7. Qualitative sketch of the pendulum phase space indicating libration and rotation regions. The separatrix, in this case a one-dimensional curve, separates the two regions.

Peculiar particle velocity, 19 Pendulum problem, 382 Periodicity conditions, 377 Perturbed solution, 344 Pessimism-optimism rule, 316 Petermann, A., 723 Peterson, W., 212 Phase plane, 323 "Phase portrait, 336 Phase space, 13 Photons, 547... 

For a given set of initial conditions Zh t = 0), k = 1,. ..,4, the solution of (3.2.6) defines a system trajectory. Since the energy is conserved, this trajectory winds in a three-dimensional sub-space of the four-dimensional phase space of the pendulum. It is not easy to imagine the motion in such a high-dimensional space, and we need to devise a visualization method that gives some clues as to the qualitative nature of the system trajectories. One particularly useful method, the method of the surface of... 

Fig. 3.2. Projection of a phase-space trajectory of the double pendulum on the plane.

Besides the one little regular island at 0 and — 2 there are undoubtedly more regular islands in the phase space of the double pendulum at E = 2. We missed them by our rather coarse choice of initial conditions. As indicated in Fig. 3.3(b), their total area in phase space is probably very small. Nevertheless, Fig. 3.3(b) illustrates an important feature of the phase space of most physical systems the phase space contains an intricate mixture of regular and chaotic regions. The system is said to exhibit a mixed phase space. 

As illustrated by Figs. 3.3(a) and (b), Poincare sections are a very powerful tool for the visual inspection and classification of the dynamics of a given Hamiltonian. The double pendulum illustrates that for autonomous systems with two degrees of fireedom a Poincare section can immediately suggest whether a given Hamiltonian allows for the existence of chaos or not. Moreover, it tells us the locations of chaotic and regular regions in phase space. 

The helium atom is an atomic physics example of a three-body problem. On the basis of Poincare s result we have to expect that the helium atom is classically chaotic. Richter and Wintgen (1990b) showed that this is indeed the case the helium atom exhibits a mixed phase space with intermingled regular and chaotic regions (see also Wintgen et al. (1993)). Thus, conceptually, the helium atom is a close relative of the double pendulum studied in Section 3.2. Given the classical chaoticity of the helium atom we are confronted with an important question How does chaos manifest itself in the helium atom ... 

The phase portrait for the pendulum is more illuminating when wrapped onto the surface of a cylinder (Figure 6.7.4). In fact, a cylinder is the natural phase space for the pendulum, because it incor-... 

In research similar to that described in section 4.3.1, phase space structures and phase space bottlenecks have been used to analyze unimolecular reaction dynamics (Davis and Gray, 1986 Gray et al., 1986b Gray and Rice, 1987 Zhao and Rice, 1992 Jain et al., 1993 DeLeon, 1992a,b Davis and Skodje, 1992). Important phase space structural properties are illustrated in figure 8.11, for the one-dimensional pendulum Hamiltonian (Lichtenberg and Lieberman, 1991) ... 

Figure 8.11 Correspondence between (a) an energy diagram and (b) a phase space diagram for the pendulum Hamiltonian (Lichtenberg and Lieberman, 1992).

This process in a free-electron laser is described by a nonlinear pendulum equation. The ponderomotive phase if[= k + kyj)z -cot] is a measure of the position of an electron in both space and time with respect to the ponderomotive wave. The ponderomotive phase satisfies the circular pendulum equation... 

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