An area preserving map

This choice of G X) is designed to mimic some aspects of a molecular process. The classical phase space is two-dimensional. Let (X ,P ) be the position and momentum of the particle just before the rath kick. Then the kicking field induces an area-preserving map... 

To conclude this section we discuss the baker s map (Farmer et al. (1983)) as an example for an area preserving mapping in two dimensions. Area preservation is of utmost importance for Hamiltonian systems, since Liouville s theorem (Landau and Lifechitz (1970), Goldstein (1976)) guarantees the preservation of phase-space volume in the course of the time evolution of a Hamiltonian system. The baker s map is a transformation of the unit square onto itself. It is constructed in the following four steps illustrated in Fig. 2.5. 

Because of the impulsive nature of the <5-kick drive, the time evolution of a point 9, y) over one cycle of the external perturbation can be written in the form of an area-preserving mapping... 

This is an area-preserving map. The multipliers of the fixed point 0(x = y = 0) satisfy the relations... 

The distinction is crucial because area-preserving maps cannot have attractors (strange or otherwise). As defined in Section 9.3, an attractor should attract all orbits starting in a sufficiently small open set containing it that requirement is incompatible with area-preservation. 

Moser, J. (1962). On invariant curves of area-preserving mappings of an annulus. Nachr. Akad. Wiss. Gott,. II Math. Phys. Kl 1962, 1-20. 

A mapping is said to be symplectic or canonical if it preserves the differential form dp A dq which defines the symplectic structure in the phase space. Differential forms provide a geometric interpretation of symplectic-ness in terms of conservation of areas which follows from Liouville s theorem [14]. In one-degree-of-freedom example symplecticness is the preservation of oriented area. An example is the harmonic oscillator where the t-flow is just a rigid rotation and the area is preserved. The area-preserving character of the solution operator holds only for Hamiltonian systems. In more then one-degree-of-freedom examples the preservation of area is symplecticness rather than preservation of volume [5]. 

The key to effective mixing lies in producing stretching and folding, an operation that is referred to in the mathematics literature as a horseshoe map. Horseshoe maps, in turn, imply chaos. The 2D case is the simplest. The equations of motion for a two-dimensional area preserving flow can be written as... 

The classical dynamics of a system can also be analyzed on the so-caUed Poincare surface of section (PSS). Hamiltonian flow in the entire phase space then reduces to a Poincare map on a surface of section. One important property of the Poincare map is that it is area-preserving for time-independent systems with two DOFs. In such systems Poincare showed that all dynamical information can be inferred from the properties of trajectories when they cross a PSS. For example, if a classical trajectory is restricted to a simple two-dimensional toms, then the associated Poincare map will generate closed KAM curves, an evident result considering the intersection between the toms and the surface of section. If a Poincare map generates highly erratic points on a surface of section, the trajectory under study should be chaotic. The Poincare map has been a powerful tool for understanding chemical reaction dynamics in few-dimensional systems. 

Figure 5.71. Geographical distribution of Januctry average space heating requirements in the year 2050, based on satellite measurements of temperature (on a gross scale of an approximately 50-km grid) in combination with the scenario s assumed building standards. The variations thus reflect both dimate differences across Denmark and differences in heated space per unit cell. The map uses the Mollweide area-preserving projection, in contrast to the straight longitude-latitude co-ordinate system of, e.g.. Fig. 5.70 (Sorensen ei al., 2001).

Some of the above methods can be miniaturized, and the instrumentation can be adapted for use in field situations (see Section 4 of this book (Potential Use of Screening Methods and Performance Evaluation)), and some can be used only in the laboratory. In the former case a spot sample can be taken, processed and analysed in the field without the need for sample preservation, transport, or storage. Since many of these methods are rapid, they can provide either quantitative (concentration) or qualitative (above or below a threshold) data on water quality in a time-scale that enables a timely and appropriate response (for instance in the case of an accidental spillage) or the rapid mapping of water quality in a wide area. Table 1.3.3 shows the main classes of chemical priority substances and the different methods that can be used for their analysis. 

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