Poincare representation

The positions of the positive a film and the negative a film shown in Figure 8.17(a) are exchangeable. Simulation results indicate that the required film thicknesses remain the same. The Poincar representation is still similar, except that the intermediate polarization state (point E) is on the lower hemisphere. The required film thicknesses da+ and da- are stUl the same as obtained in Equations (8.28) and (8.29). And finally, the viewing angle performance is almost identical to those shown in Figure 8.18. 

The function P can be computed from either an analytical or a numerical representation of the flow field. In such a way, a 3-D convection problem is essentially reduced to a mapping between two-dimensional Poincare sections. In order to analyze the growth of interfacial area in a spatially periodic mixer, the initial distri-... 

Thus, to completely solve the problem of symmetry reduction within the framework of the formulated algorithm above, we need to be able to perform steps 3-5 listed above. However, solving these problems for a system of partial differential equations requires enormous amount of computations moreover, these computations cannot be fully automatized with the aid of symbolic computation routines. On the other hand, it is possible to simplify drastically the computations, if one notes that for the majority of physically important realizations of the Euclid, Galileo, and Poincare groups and their extensions, the corresponding invariant solutions admit linear representation. It was this very idea that enabled us to construct broad classes of invariant solutions of a number of nonlinear spinor equations [31-33]. 

Representations of the Poincare group and their relation to mass and spin. 

Figure 8. Schematic representation of a chaotic repeller and its stable Ws and unstable IV manifolds in some Poincare surface of section (q,p) together with one-dimensional slices along the line L of typical escape time function T+1...

D. F. Roscoe, Maxwell s Equations as a Consequence of the Orthogonality between Irreducible Two Index Representation of the Poincare, preprint, 1997. 

Figure 1. (a) Poincare sphere representation of wave polarization and rotation (b) a Poincare... 

The representation of the sampling by a unipolar, single-rotation-axis, U(l) sampler of a SU(2) continuous wave that is polarization/rotation-modulated is shown in Fig. 2, which shows the correspondence between the output space sphere and an Argand plane [28]. The Argand plane, S, is drawn in two dimensions, x and v, with z = 0, and for a set snapshot in time. A point on the Poincare sphere is represented as P(t,x,y,z), and as in this representation t = 1 (or one step in the future), specifically as P(l,x,y,z). The Poincare sphere is also identified as a 3-sphere, S 1, which is defined in Euclidean space as follows ... 

The consequence of these relations is that every proper 2n rotation on S + — in the present instance the Poincare sphere—corresponds to precisely two unitary spin rotations. As every rotation on the Poincare sphere corresponds to a polarization/rotation modulation, then every proper 2n polarization/rotation modulation corresponds to precisely two unitary spin rotations. The vector K in Fig. lb corresponds to two vectorial components one is the negative of the other. As every unitary spin transformation corresponds to a unique proper rotation of S +, then any static (unipolarized, e.g., linearly, circularly or ellipti-cally polarized, as opposed to polarization-modulated) representation on S + (Poincare sphere) corresponds to a trisphere representation (Fig. 3a). Therefore... 

A /l = 7, where 7 is the identity matrix. Thus, a spin transformation is defined uniquely up to sign by its effect on a static instantaneous snapshot representation on the S+ (Poincare) sphere ... 

K thus defines a static polarization/rotation—whether linear, circular or elliptical—on the Poincare sphere. The 2, r representation of the vector K gives no indication of the future position of K that is, the representation does not address the indicated hatched trajectory of the vector K around the Poincare sphere. But it is precisely this trajectory which defines the particular polarization modulation for a specific wave. Stated differently a particular position of the vector K on the Poincare sphere gives no indication of its next position at a later time, because the vector can depart (be joined) in any direction from that position when only the static 2, r coordinates are given. 

Figure 6. Spin frame representation of a spin-vector by flagpole normalized pair representation a,b over the Poincare sphere in Minkowski tetrad (l,x,y,z) form (n representation) and for three timeframes or sampling intervals providing overall (t]. r ) a Cartan-Weyl form representation. The sampling intervals reset the clock after every sampling of instantaneous polarization. Thus polarization modulation is represented by the collection of samplings over time. Minkowski form after Penrose and Rindler [28]. This is an SU(2) Gd hx) m C over it representation, not an SO(3) Q(to, 8) in C representation over 2it. This can be seen by noting that an b or bt-z a over it, not 2n, while the polarization modulation in SO(3) repeats at a period of 2it.

While in some analogous works, it was possible to devise surfaces of section or even full representations of phase space this is hardly thinkable here. Let us recall that an on-shell (or constant energy H = E = 0.001 atomic units) Poincare section would be of dimension = D(phasespace) — 1 — 1=6. Instead we... 

Another of Poincare s new methods was the reduction of the continuous phase-space flow of a classical dynamical system to a discrete mapping. This is certainly one of the most useful techniques ever introduced into the theory of dynamical systems. Modem journals on nonlinear dynamics abound with graphical representations of Poincare mappings. A quick glance into any one of these journals will attest to this fact. Because of the usefulness and the formal simplicity of mappings, this topic is introduced and discussed in Section 2.2. 

The Poincare-Bendixson theorem can be difficult to apply in practice. One convenient case occurs when the system has a simple representation in polar coordinates, as in the following example. 

Since the time of Poincare elimination t from the presentation and focusing on the variables only has been accepted as an alternative for representation. In two dimensional systems this is called the phase plane, Fig. IV.2. 

The symmetry group of relativity theory tells the story. For the irreducible representations of the Poincare group (of special relativity) or the Einstein group (of general relativity) obey the algebra of quaternions. The basis functions of the quaternions, in turn, are two-component spinor variables [17]. 

The basis functions of this operator are the two-component spinor variables. Guided by the two-dimensional Hermitian structure of the representations of the Poincare group, we may make the following identification between the spinor basis functions 4>a(a = 1,2) of this operator and the components ( , H )(k = 1,2, 3) of the electric and magnetic fields, in any particular Lorentz frame ... 

How the correspondence principle should be applied to an atomic system thus depends critically on whether or not there exists a multiperiodic representation of the classical trajectories - the question first raised by Einstein. If the system possesses multiperiodic orbits, then its motion becomes separable, i.e. it becomes equivalent to as many independent modes as there are degrees of freedom. Dynamical separability is assumed in all independent particle and perturbative models of the many-electron atom. It is, however, not strictly applicable and the successes of simple quantum theory for many-electron systems are, to say the least, surprising. It was pointed out by Einstein, who based his arguments on the work of Poincare [519], that there exists no true separation of the three-body problem. 

The last expression gives the potential matrix in the standard representation. The transformation law (89) gives precisely the Poincare transformation of the electromagnetic field strengths E and B, which can be combined into a tensor field on Minkowski space. 

We know already that Hq commutes with each of the angular momentum operators Jfc because Hq and are generators of a representation of the Poincare group. 

In this representation, simple periodic behaviour corresponds to a fixed point located on the bisectrix, given that such a behaviour is characterized by the equality a +i = a for all values of n. A complex periodic solution will correspond to a finite number of points, none of which will be located on the bisectrix. Thus, the pattern of bursting tt(4), which contains four successive peaks of )8, corresponds to four distinct points in the one-dimensional equivalent (fig. 4.23b). The construction of the Poincare section for the similar situation of a pattern tt(3) is... 

Figure 26 A schematic representation of the Poincare integral invariants for a system with three degrees of freedom. (A) The invariant is the sum of oriented areas projected onto the three possible (qi, pj) planes. (B) The invariant 2 is rhe sum of oriented volumes projected onto the three possible pj, q, pl) hyperplanes. Not shown (Liouville s theorem). Reprinted with permission from f. 119.

The main difference between the Hamiltonian and dissipative systems arises from the conservation condition that applies to the former. In Hamiltonian systems, the total energy is fixed. A trajectory with a given initial condition and energy will continue with that same energy for the remainder of the trajectory. In the phase space representation, this will result in a stable trajectory that does not pull in toward an attractor. A periodic trajectory in a Hamiltonian system will have an amplitude and position in the phase space that is determined by the initial conditions. In fact, the phase space representation of a Hamiltonian system often includes many choices of initial conditions in the same phase space portrait. The Poincare section, to be described below, likewise contains many choices of initial conditions in one diagram. 

Fig. 7.8 Stroboscopic representation of perturbed variable, (a) Quasistable butterfly shape, (b) Decay of the quasistable shape and start to tend to a new shape, (c) Tending to a new attractor, (d) Arriving at a fixed point of the Poincare plot.

Figure 8.9 Schematic diagram of Poincare sphere representation and the effect of uniaxial medium on the polarization state change of a polarized incident hght.

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