Relative equilibrium

By the argument in Section IIB, the presence of a locally quadratic cylindrically symmetric barrier leads one to expect a characteristic distortion to the quantum lattice, similar to that in Fig. 1, which is confirmed in Fig. 7. The heavy lower lines show the relative equilibria and the point (0,1) is the critical point. The small points indicate the eigenvalues. The lower part of the diagram differs from that in Fig. 1, because the small amplitude oscillations of a spherical pendulum approximate those of a degenerate harmonic oscillator, rather than the fl-axis rotations of a bent molecule. Hence the good quantum number is... 

Figure 10. Divisions of the Em map for the quadratic Hamiltonian in Eq. (39). Va, Vb, and Vx are the energies of the two isomers and that of the saddle point between them. The solid lines are relative equilibria, and the cusp points, a , indicate points at which the secondary minimum and the saddle point in Fig. 9 merge at a point of inflection.

Figure 11. Quantum monodromy in the spectrum of the quadratic Hamiltonian of Eq. (38). The solid lines indicate relative equilibria. Filled circles mark the eigenvalues of the most stable isomer and those above the relevant effective potential barrier in Fig. 8. Open circles indicate interpenetrating eigenvalues of the secondary isomer. The transported unit cell moves over the hlled circle lattice, around the curved fold line connecting the two spectra.

In order to establish conditions for the isolation of the image of point D under the EM map, the projection is performed by first taking a section through the surface ct = 0 at fixed J, an example of which is shown in the right-hand panel of Fig. 15, for the critical value = S — N. The shaded area of the (K, plane defines the classically allowed range for the specified value. The lines indicate energy contours for y = 0.5. Those that touch the section correspond to relative equilibria of the Hamiltonian, whose values... 

Tt2 are even functions of K, contours of touch any constant section of 712 at = 0, to give relative equilibria of the form... 

The dotted contour in Fig. 19 shows that a second form of touching condition can also occur, for sufficiently large values of the parameter ratio b/a. Simple analytical manipulations show that touching conditions of this second type, with real K, give rise to relative equilibria described by... 

It is evident that the projections of the fixed points A and B, in the EJ map, always lie on relative equilibria of type I, but that the position of the overlapping projections of C and D depends on the sign of b — a. If > b the double point is isolated between the two type I equilibria, and quantum monodromy is expected, for a sufficiently dense quantum lattice. If, on the other hand, the critical point lies on the type II relative equilibrium line and... 

Figure 20. Quantum eigenvalue lattices forji =22 = 5 and (a) hja = 0.2, (b) bja = 2. Solid and dashed lines are type / and type U relative equilibria, respectively. The large point at 7 = 0 is the overlapping projection of points C and D in Table 1. Those at = N are projections of points A and B.

In addition, one finds, by the methods outlined next, that the other critical points Y and Z always lie on relative equilibria. 

It was shown in the earlier sections that the existence or nonexistence of quantum monodromy in two-dimensional maps depends on the relative dispositions of the critical points and relative equilibria of the Hamiltonian, which involves a search for the stationary points of with respect to K and Qk- For L = 0 there is a root at the critical point K = J, and other possible roots given by... 

Cooper and Child [14] have given an extensive description of the effects of nonzero angular momentum on the nature of the catastrophe map and the quantum eigenvalue distributions for polyads in its different regions. Here we note that the fixed points and relative equilibria, for nonzero L = L/2J, are given by physical roots of the equation... 

Y-chelate [Pd(C(0)CH7CH C(0)Me)(phen)] were determined through a combination of competitive and relative equilibria studies. Based on this multiform investigation, a complete mechanistic cycle of chain propagation was proposed (Scheme 7.11) [26]. 

As Bell (1959d) has pointed out, the absolute values of AF° and AS° for equilibria involving the solvent are of somewhat uncertain significance because these functions depend on the concentration units employed. For consistency the concentration of solvent might be expressed in units of moles liter-1, but the activity of pure water on this basis is an open question. Therefore it is generally more satisfactory to consider relative equilibria of the type... 

As is apparent from Eq. (32), besides the total energy E = H, the dynamical system has another conserved quantity, the total angular momentum A. The angular momentum A plays definitely an important role here, but we do not deal yet with relative equilibria, as in Section V. While derivation of the equation of... 

There have been various studies concerning relative equilibria [58,65,66, 70-72], especially so for bound systems. Rotating scattering systems have been much less studied except when the rotating frequency is imposed and constant. Many gravitational A -body problems belong to this class. 

While relative equilibria and relative TS might occur in bound motion (isomerization with nonzero J, we restrict ourselves to scattering situations in all that follows. A generalization of the isomerization for the three-body system, for instance, is still lacking. Also, the very important case of three-body (and four-body) reactive scattering, with angular momentum, is only treated in the literature without explicitly resorting to a TS concept [73-75]. 

Relative equilibria arise when the shape of the system does not change in time while the object as a whole is rotating. A relative equilibrium (RE) point means that ... 

Figure 19. For the value % = 0, tt/2, n, energy of the relative equilibria as a function of J. Note that the three families undergo a saddle-node bifurcation at different energy each.

It is quite ambitious for a scientist to describe a natural phenomenon in terms of a specific reaction. The situation in the atmospheric environment is however more complicated as a variety of reactions are occurring simultaneously and a certain species may take part in different reactions affecting the relative equilibria. Most data are coming from laboratory work and experimental conditions are definitely different from the ones observed in the troposphere. As an example the mechanism of oxidation of sulphur dioxide, in gas phase is usually reported occur to a large extent through free radicals. If the presence of humidity and of particulated matter is considered, specifically in the lower part of the troposphere, definitely also heterogeneous reactions play a very important role. I feel that experiments carried on in the atmosphere yield more consistent results to elucidate the chemistry of the atmospheric environment. 

Birds living near lakes are also affected because the content of metals, mainly Al, increases in fishes and plankton. Another important aspect of the effect of acid deposition is on soil microorganisms, especially in poorly buffered soils, as reported by R.J. Buck both affecting bacteria and fungi and their relative equilibria. 

This assumption implies that 4>t u ) is a time-periodic solution of (not necessarily minimal) period 27t/cu. An element of our phase space is called a relative equilibrium if the time evolution 4>t u ) lies inside the group orbit pgiu, ). Hence the periodic solution of u, consists of relative equilibria. 

B. Fiedler, B. Sandstede, A. Scheel, and C. Wulff. Bifurcation from relative equilibria of noncompact group actions Skew products, meanders, and drifts. Doc. Math., J. DMV, 1 479-505, 1996. 

B. Fiedler and D. Turaev. Normal forms, resonances, and meandering tip motions near relative equilibria of Euclidean group actions. Arch. Ration. Mech. Anal., 145(2) 129-159, 1998. 

The group motion associated to the relative equilibria of the discrete equations (16)-(17) were obtained in Armero Romero (2001a) and are given by... 

The standard choice in the governing equation (16) does not lead, however, to a conserving approximation. The conditions (21) can be easily seen to be satisfied, but the conditions (26) and (29) for the conservation of the relative equilibria and energy are not. This situation is to be traced to the spatial gradients i in (39). First, we... 

The different terms in the expression (46) can be seen to be second-order approximation of the variations of the assumed C. Some long algebraic manipulations show that this new B-bar operator satisfies the desired conditions (21), (26) and (29). In particular, the relation (26) for the relative equilibria is satisfied for the assumed B-bar operator... 

Simo, J.C., Posbergh, T.A. Marsden, J.E. 1991. Stability of relative equilibria. Part II Application to nonlinear elasticity , Arch. Rational Mech. Anal., 115 61-100. 

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