Phase space structures

The quote is from the third volume of Henri Poincare s New Methods of Celestial Mechanics, and is a description of his discovery of homoclinic orbits (see below) in the restricted three-body problem. It is also one of the earliest recorded formal observations that very complicated behavior may be found even in seemingly simple classical Hamiltonian systems. Although Hamiltonian (or conservative) chaos often involves fractal-like phase-space structures, the fractal character is of an altogether different kind from that arising in dissipative systems. An important common thread in the analysis of motion in either kind of dynamical system, however, is that of the stability of orbits. 

The geometric version of TST laid out in Section II is centered around the NHIM that defines the dividing surface and its stable and unstable manifolds that act as separatrices. The NHIMs at different energies are in turn organized by the saddle point. It forms a fixed point of the dynamics—that is it is itself an invariant object—and it provides the Archimedean point in which the geometric phase-space structure is anchored. 

Hence, for small perturbations the system is nearly integrable Most classical trajectories are restricted to two-dimensional phase-space structures that are often called KAM tori. Since two classical trajectories cannot cross each other, a toms such as shown in Fig. 1 is in fact an impenetrable phase-space stmcture, dividing the three-dimensional energy shell into disjoint regions. With stronger perturbations, more KAM tori are expected to be destroyed and therefore more trajectories become chaotic. 

The KAM theorem demonstrates the existence of KAM tori when the perturbations to the motion are small. What happens when a nearly integrable Hamiltonian is strongly perturbed For example, with increasing perturbation strength, what is the last KAM torus to be destroyed and how should we characterize the phase space structures when all KAM tori are destroyed Using simple dynamical mapping systems, which can be regarded as Poincare maps in Hamiltonian systems with two DOFs, MacKay, Meiss, and Percival [8,9] and Bensimon and Kadanoff [10] showed that the most robust KAM curve... 

The Davis-Gray theory teaches us that by retaining the most important elements of the nonhnear reaction dynamics it is possible to accurately locate the intramolecular bottlenecks and to have an exact phase space separatrix as the transition state. Unfortunately, even for systems with only two DOFs, there may be considerable technical difficulties associated with locating the exact bottlenecks and the separatrix. Exact calculations of the fluxes across these phase space structures present more problems. For these reasons, further development of unimolecular reaction rate theory requires useful approximations. 

A beautiful classical theory of unimolecular isomerization called the reactive island theory (RIT) has been developed by DeLeon and Marston [23] and by DeLeon and co-workers [24,25]. In RIT the classical phase-space structures are analyzed in great detail. Indeed, the key observation in RIT is that different cylindrical manifolds in phase space can act as mediators of unimolecular conformational isomerization. Figure 23 illustrates homoclinic tangling of motion near an unstable periodic orbit in a system of two DOFs with a fixed point T, and it applies to a wide class of isomerization reaction with two stable isomer... 

The phase space structure of classical molecular dynamics is extensively used in developing classical reaction rate theory. If the quanmm reaction dynamics can also be viewed from a phase-space perspective, then a quantum reaction rate theory can use a significant amount of the classical language and the quantum-classical correspondence in reaction rate theory can be closely examined. This is indeed possible by use of, for example, the Wigner function approach. For simplicity let us consider a Hamiltonian system with only one DOF. Generalization to many-dimensional systems is straightforward. The Wigner function associated with a density operator /)( / is defined by... 

The fact that classical unstable periodic trajectories can manifest themselves in the Wigner function implies that nonstatistical behavior in the quanmm dynamics can be intimately related to the phase-space structure of the classical molecular dynamics. Consider, for example, the bottlenecks to intramolecular energy flow. Since the intramolecular bottlenecks are caused by remnants of the most robust tori, they are presumably related to the least unstable periodic trajectories. Hence quantum scars, being most significant in the case of the least unstable periodic trajectories, are expected to be more or less connected with intramolecular bottlenecks. Indeed, this observation motivated a recent proposal [75] to semiclassically locate quantum intramolecular bottlenecks. Specifically, the most robust intramolecular bottlenecks are associated with the least unstable periodic trajectories for which Eq. (332) holds, that is,... 

Figure 51. Classical phase-space structures of a modified kicked rotor system. Note that the regular islands are transporting islands. [From J. B. Gong, H. J. Worner, and P. Brumer, Phys. Rev. E. 68, 026209 (2003).]...

IV. Phase-Space Structure Around a Simple (Rank-One) Saddle... 

E. Searching for the Transition State and Other Phase-Space Structures... 

IV. PHASE-SPACE STRUCTURE AROUND A SIMPLE (RANK-ONE) SADDLE... 

In this section we will develop the phase-space structure for a broad class of n-DOF Hamiltonian systems that are appropriate for the study of reaction dynamics through a rank-one saddle. For this class of systems we will show that on the energy surface there is always a higher-dimensional version of a saddle (an NHIM [22]) with codimension one (i.e., with dimensionality one less than the energy surface) stable and unstable manifolds. Within a region bounded by the stable and unstable manifolds of the NHIM, we can construct the TS, which is a dynamical surface of no return for the trajectories. Our approach is algorithmic in nature in the sense that we provide a series of steps that can be carried out to locate the NHIM, its stable and unstable manifolds, and the TS, as well as describe all possible trajectories near it. 

Most realistic Hamiltonians with simple saddles do not appear in this form. In what follows, we show how to transform such a Hamiltonian into this form using Normal-Form theory [13]. The phase-space structures that form the subject of this review will then be expressed in terms of the normal-form coordinates (qi,..., q ,pi,..., p ). Therefore, before analyzing Eq. (2) we show that any Hamiltonian vector field in the neighborhood of an equilibrium point of saddle (g) center (8> center type can be transformed to the form of Eq. (2). 

It is well known (see, e.g.. Ref. 13) that the normal form transformations do not converge in the sense that normalization to all orders generally does not yield a meaningful result. However, this is of no consequence for our purposes. We view the technique more as the input to a numerical method for realizing the NHIM, its stable and unstable manifolds, and the TS. In this sense the limitations of machine precision make normalization beyond a certain finite order meaningless. This is a local result valid in the neighborhood of the equilibrium point of center center saddle type. However, once the phase-space structure is established locally, it can be numerically continued outside of the local region. 

As mentioned above, we can obtain all these phase-space structures in explicit form following these steps ... 

All these properties are more or less obvious in the linear case. However, because of the essentially four-dimensional nature of phase space, it is somewhat difficult to get an intuitive nature of the phase-space structure. This is... 

These conditions can be understood as follows. Since the system is time-dependent, we consider the phase-space structure at a fixed time t. Then, the stable and unstable manifolds are two-dimensional manifolds, and they are parametrically represented using (x, a). Their intersections, if any, are points in the four-dimensional phase space 0). (Note that the equi-energy surface... 

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