Regular motion

However, the reader may be wondering, what is the connection of all of these classical notions—stable nonnal modes, regular motion on an invariant toms—to the quantum spectmm of a molecule observed in a spectroscopic experiment Recall that in the hannonic nonnal modes approximation, the quantum levels are defined by the set of quantum numbers (Up. . Uyy) giving the number of quanta in each of the nonnal modes. 

For regular motion, T> t) grows only linearly with time, so that the exponents are all zero. On the other hand, because chaotic flows are characterized by exponential divergences of initial nearby trajectories, a characteristic signature of such flows is the existence of at least one positive Lyapunov exponent. 

When the regular motion is simply uniform rotation of the absorption and emission dipoles with angular velocity to around the helix axis, one has p(t) - p(0) = cot. For the corresponding random motion, one might have m)2> = 2Dt, where D is the effective diffusion coefficient for Brownian rotation of the transition dipole around the helix axis. When these expressions are incorporated in Eqs. (4.31) and (4.24), the latter becomes a generalization of a relation recently derived using a more cumbersome approach/104-1... 

It measures the average loss of information. For regular motion, K = 0, and for random motion K = oo. 

We assume Newtonian mechanics to be valid and consider a mass point M bouncing in a two-dimensional square box of side length 1 (see Fig. 1.2). The box shown in Fig. 1.2 is used to illustrate regular motion. Therefore, we call it i . Another box is shown in Fig. 1.3. It is equipped with a hard stationary disk of radius r = 1/4 at its centre. It serves to illustrate chaotic motion. Therefore, we call it C7 . We assume that inside the boxes the mass point M travels on straight fine trajectories subject only to specular reflection whenever it hits the walls of R or C, or the central disk of box C. Since the motion of M is free between bounces, the mass of M is irrelevant for the kinematics of M. Therefore, M s velocity can be normalized to 1. 

Fig. 1.2. Regular motion of a mass point M inside an empty square box labelled R. A trajectory launched at L with p — 0.69 returns to wall d after three bounces.

Equation (6) shows that the dynamics is expected to be a regular motion when 5 is small, such as a lattice-like vibration in a solid phase. As the 8 value increases,... 

The spectrum of Lyapunov exponents provides fundamental and quantitative characterization of a dynamical system. Lyapunov exponents of a reference trajectory measure the exponential rates of principal divergences of the initially neighboring trajectories [1], Motion with at least one positive Lyapunov exponent has strong sensitivity to small perturbations of the initial conditions, and is said to be chaotic. In contrast, the principal divergences in regular motion, such as quasi-periodic motion, are at most linear in time, and then all the Lyapunov exponents are vanishing. The Lyapunov exponents have been studied both theoretically and experimentally in a wide range of systems [2-5], to elucidate the connections to the physical phenomena of importance, such as transports in phase spaces and nonequilibrium relaxation [6,7]. 

REGULAR MOTIONS IN EXTRA-SOLAR PLANETARY SYSTEMS... 

Regular motions in extra-solar planetary systems... 

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