Poincare variables

Having performed the reduction of the centre of mass and of the angular momentum, the Hamiltonian of the problem of three bodies can be written in heliocentric coordinates and using the Poincare variables as... 

Using the classical methods of Celestial Mechanics, we can expand the distance A in the Hamiltonian (19) as a function of the Poincare variables, and we can calculate the so called secular system at order two in the masses (used, for instance, in Laskar 1988 in a model with 8 planets, to study the long term evolution of the solar system). In the secular system the dependency on the angles Ai, A2 (which evolve much faster than the other Poincare variables) is dropped out by simply averaging the Hamiltonian over the angles themselves. Thus, the actions Ai, A2 are first integrals for the secular system, which are replaced with their numerical values corresponding to the data for the real system Sun-Jupiter-Saturn at a fixed initial time. Therefore, we can actually expand the secular Hamiltonian as a power series in the form... 

The angle AO] is called the rotation angle. It depends on the action J only, for a given parameter h (or J2). For this reason the map (73) is called a twist map. The map (73) is usually presented in the Poincare variables... 

To get exact solutions that are symmetric in all the variables, we exploit the formulas for generating solutions by Lorentz transformations (see the previous subsection) and thus come to the following general form of the Poincare-invariant ansatz ... 

In the previous section we gave a complete list of P( 1,3)-inequivalent ansatzes for the Yang-Mills held, which are invariant under the three-parameter subgroups of the Poincare group P(l, 3). These ansatzes can be represented in the unified form (53), where Bv(co) are new unknown vector functions, a> - co(x) is the new independent variable, and the functions (x) are given by (54). 

Another useful rule which can frequently guide us to situations where oscillatory solutions will be found is the Poincare-Bendixson theorem. This states that if we have a unique stationary state which is unstable, or multiple stationary states all of which are unstable, but we also know that the concentrations etc. cannot run away to infinity or become negative, then there must be some other non-stationary atractor to which the solutions will tend. Basically this theorem says that the concentrations cannot just wander around for an infinite time in the finite region to which they are restricted they must end up somewhere. For two-variable systems, the only other type of attractor is a stable limit cycle. In the present case, therefore, we can say that the system must approach a stable limit cycle and its corresponding stable oscillatory solution for any value of fi for which the stationary state is unstable. 

Concerning the Poincare surface of section, it should be noticed that a sort of quantum surface of section can be constructed by intersection of the Wigner or Husimi transform of the eigenfunctions expressed in the quantum action-angle variables of the effective Hamiltonian, which can provide a comparison with the classical Poincare surfaces of section (e.g., in acetylene). 

Figure 7.33(C) is the return point histogram with the Poincare surface drawn at Cx = 1.55 kg/m3 for variable Cs values at D = 0.04584 hr 1.

Stroboscopic and Poincare maps are different from phase plane plots in that they plot the variables on the trajectory at specifically chosen and repeated time intervals. For... 

The stability of transport and rate systems is studied either by nonequilibrium thermodynamics or by conventional rate theory. In the latter, the analysis is based on Poincare s variational equations and Lyapunov functions. We may investigate the stability of a steady state by analyzing the response of a reaction system to small disturbances around the stationary state variables. The disturbed quantities are replaced by linear combinations of their undisturbed stationary values. In nonequilibrium thermodynamics theory, the stability of stationary states is associated with Progogine s principle of minimum entropy production. Stable states are characterized by the lowest value of the entropy production in irreversible processes. The applicability of Prigogine s principle of minimum entropy production is restricted to stationary states close to global thermodynamic equilibrium. It is not applicable to the stability of continuous reaction systems involving stable and unstable steady states far from global equilibrium. The steady-state deviation of entropy production serves as a Lyapunov function. 

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 ... 

Poincare map of the unperturbed problem in action-angle variables... 

In Poincare s reduction, the variables are the components of the heliocentric position vectors X — X0 and the momenta are the same linear momenta IT of the barycentric formulation. Hence,... 

It may be demonstrated (the Poincare-Bendixon theorem, see Appendix A2) that for the autonomous system in two variables (5.2) a limit set can only be a point or a closed curve. Linear systems, gradient and Hamiltonian systems, will be shown in Appendix A1 to be uncapable of having a limit cycle. 

The advection problem is thus described by a periodically driven non-autonomous Hamiltonian dynamical system. In such case, besides the two spatial dimensions an additional variable is needed to complete the phase space description, which is conveniently taken to be the cyclic temporal coordinate, r = t mod T, representing the phase of the periodic time-dependence of the flow. In time-dependent flows ip is not conserved along the trajectories, hence trajectories are no longer restricted to the streamlines. The structure of the trajectories in the phase space can be visualized on a Poincare section that contains the intersection points of the trajectories with a plane corresponding to a specified fixed phase of the flow, tq. On this stroboscopic section the advection dynamics can be defined by the stroboscopic Lagrangian map... 

In the phase space formed by the concentrations of the chemical variables involved in the reaction, sustained oscillations correspond to the evolution towards a closed curve called a limit cycle by Poincare,... 

In writing eqns (9.2), provision was made that these equations can in principle generate periodic behaviour. The question is indeed how the external signal induces not just a rise in cytosolic Ca, but also the oscillations that are observed in a large variety of cells. The usefulness of two-variable models is that they are amenable to phase plane analysis. In particular, there exists a powerful criterion due to Poincare and Bendixson (see Minorsky, 1962), which allows us to rule out the occurrence of sustained oscillations in two-variable systems. Although... 

The second approach, successfully followed in the analysis of complex oscillations observed in the model of the multiply regulated biochemical system, relies on a further reduction that permits the description of the dynamics of the three-variable system in terms of a single variable only, by means of a Poincare section of the original system. Based on the one-dimensional map thus obtained from the differential system, a piecewise linear map can be constructed for bursting. The fit between the predictions of this map and the numerical observations on the three-variable differential system is quite remarkable. This approach allows us to understand the mechanism by which a pattern of bursting with n peaks per period transforms into a pattern with (n + 1) peaks. 

The phase space structures near equilibria of this type exist independently of a specific coordinate system. However, in order to carry out specific calculations we will need to be able to express these phase space structures in coordinates. This is where Poincare-Birkhoff normal form theory is used. This is a well-known theory and has been the subject of many review papers and books, see, e.g.. Refs. [34-AOj. For our purposes it provides an algorithm whereby the phase space structures described in the previous section can be realized for a particular system by means of the normal form transformation which involves making a nonlinear symplectic change of variables. 

The traditional Poincare-Von Zeipel approach [53] of CPT is based on mixed-variable generating functions F ... 

For nonautonomous systems, an additional term involving the time derivatives of W(p, q s) must be included in Eq. (A.66) [45, 46, 53]. In this Appendix, we have described how Lie transforms provide us with an important breakthrough in the CPT free from any cumbersome mixed-variable generating function as one encounters in the traditional Poincare-Von Zeipel approach. After the breakthrough in CPT by the introduction of the Lie transforms, a few modifications have been established in the late 1970s by Dewar [56] and Drag and Finn [47]. Dewar established the general formulation of Lie canonical perturbation theories for systems in which the transformation is not... 

We now have a set of four new differential equations, but the dependent variables are (pi, t, p2, q-f), and q plays the role of the independent variable. Thus, the Poincare map of a trajectory is constructed by integrating Hamilton s equations of motion (Eq. [37]) until the surface-of-section function Siq-f) - q f... 

Figure 37 shows a bifurcation diagram formed by plotting the value of the variable A in this Poincare section as a function of the bifurcation parameter kx- The 51151 state can be seen to go through a series of period-doubling bifurcations before the chaotic state appears. At the other end of the chaotic region, the state abruptly appears, with no evidence of a reverse... 

The idea that time has no objective existence but depends on events led some scientists to abandon the assumption that it is a continuous variable. Following the establishment of the Planck-Einstein quantum theory, it was suggested by Poincare and others that time is quantized one calculation gave the value 10 " s to the chronon . 

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