Secular terms

In celestial mechanics difficulties were experienced in connection with the so-called secular terms, and at the end of the last century there was a tendency to get rid of these secular terms by a proper determination of the available constants (Lindstedt). 

Liapounov theorems for stability and instability, 346, 347, 348 Lienard, A., 334 Ufskitz, E. M., 726,759,768 Light quantum hypothesis, 485 Limit cycles, 328 coalescence of, 339 stable, 329 unstable, 329 Lindsay, R. ., 4,47 Lindstedt method of eliminating secular terms, 349... 

The solvability condition for (6.2.29b) (elimination of secular terms in P2) implies... 

The simplest interaction to be considered is the dipole-dipole interaction which can be divided in two parts a secular term which has a 3 cos2 — 1 L variation and a non-secular term which, when associated with the previous one, results in a variation of the form ... 

Several formalisms have been applied to relaxation in exchanging radicals. Principal among these are modifications of the classical Bloch equations (8, l ) and the more rigorous quantum mechanical theory of Redfield al. (8 - IJ ). When applied in their simplest form, as in the present case for K3, both approaches lead to the same result. Since the theory has been elegantly described by many authors (8 - 12 ), only those details which pertain to the particular example of K3 will be presented here. Secular terms contribute to the ESR linewidth (r) and transverse relaxation time (T ) by an amount... 

Figure 14 illustrates the spectra of a stearic spin probe for two greatly different values of the ratio D /D. For very different diffusion limits in the presence of an orienting potential, this shows how sensitive to changes of the order parameter the dynamics of the spin probe are. The slightly different features of the two series of spectra can be ascribed to the modulation of secular terms of the Hamiltonian of Eq. (3.3). The two series of spectra exhibit virtually the same behavior, except that the senes of curves reproduced in Fig. 14 by the continuous lines seem to be more rigid. 

As argued earlier in the text, 72 (t) serves as an appropriate spatially independent measure of the impact of the dynamical fluctuations on the kinetics of the reaction. The absence of secular terms in 73(1) [cf. Eq. (36)] enables us to treat it as a purely perturbative correction to Jo(t). In such a case, we can discern the impact of the dynamical fluctuations by analyzing... 

If 72(z) had. in contrast, possessed secular terms, then as t oo it would be impossible to choose an e small enough to maintain the perturbative nature of the correction—necessitating a renormalization procedure. 

The true solution (7) exhibits two time scales a fast time t 0(1) for the sinusoidal oscillations and a slow time t - l/ over which the amplitude decays. Equation (17) completely misrepresents the slow time scale behavior. In particular, because of the secular term tsint, (17) falsely suggests that the solution grows with time whereas we know from (7) that the amplitude A = (l - ) c decays exponentially. 

Poincare-Lindstedt method) This exercise guides you through an improved version of perturbation theory known as the Poincare-Lindstedt method. Consider the Duffing equation x + x + e.v = 0, where 0phase plane analysis that the true solution x(r, e) is periodic our goal is to find an approximate formula for x(z, ) that is valid for all t. The key idea is to regard the frequency co as unknown in advance, and to solve for it by demanding that x(z,e) contains no secular terms. 

For example, the form of the secular terms in the two-body homonuclear dipolar Hamiltonian in the frame of the Zeeman interaction is... 

If we reexamine the governing equation, (4-230), for Re x = 0, we see that the mathematical difficulty for co = a>0 is that the forcing function is actually a solution of the homogeneous equation. The forcing function in this case is known as a secular term. Obviously, when a secular forcing term is present, a particular solution cannot exist in the form C sin 7 as is possible for co >0 - see, for example (4-232) - but must instead take the form Cl cos 7. Indeed, substituting this form into the governing equation, (4-230), with co = two, we find... 

Now, Eq. (4-246) is nothing more than the homogeneous equation obtained in our earlier analysis, and it might appear that nothing has been accomplished because the secular term will now appear in (4-248). However, it is at this point that the analysis departs from the preceding work. For in this case, we consider gi to depend not only on the independent variable 7, which appears explicitly in (4-246), but also on the slow time scale r. Hence a convenient form for the general solution of (4-246) is... 

Thus, to avoid secular terms and therefore ensure bounded solutions for g3, we must choose A(t) and right-hand side of (4-252) are equal to zero. After some manipulation, this condition leads to the coupled pair of equations... 

Thus, to eliminate secular terms, we require that... 

The problem of small divisors is related to another well known problem that shows up in classical perturbation theory, namely the problem of secular terms. Let us illustrate the problem with a very simple example. We consider the Duffing s equation... 

The elimination of secular terms from the power series expansion of the solution is achieved by the method of Lindstedt. The underlying idea is to pick a fixed frequency p, and to look for a quasi-periodic solution with basic frequencies /i and v. This is actually the same thing as looking for a quasi-periodic orbit on an invariant 2-dimensional torus. The process of solution is the following. Write the Duffing s equation as... 

For an isoenergetic displacement, no secular term appears in the general solution, as is readily obtained from Equations(47),(48) and (49). 

The stability that we mentioned before refers to the evolution of the deviation vector ( ) = x (t) — x(l) between the perturbed solution x (t) and the periodic orbit x(t) at the same time t. If ( ) is bounded, then the periodic orbit is stable. In this case two particles, one on the periodic orbit x t) and the other on the perturbed orbit x (t), that start close to each other at t = 0, would always stay close. A necessary condition is that all the eigenvalues of the monodromy matrix be on the unit circle in the complex plane. However, in a Hamiltonian system this condition is not enough for stability, because there is only one eigenvector corresponding to the double unit eigenvalue and consequently a secular term always appears in the general solution, as can be seen from Equation (49). We remark that this secular term appears if the vector of initial deviation (0) = a (0) — s(0) has a component along the direction /2(C)). 

In order to understand the meaning of the secular term, we consider the initial conditions, for e small,... 

Thus we come to the conclusion that the secular term introduces a phase shift only along the orbit. This means that the two orbits, x(t) and x (t), considered as geometrical curves, are close to each other (Figure 7). In this case we say that we have orbital stability, provided that the eigenvalues A3, A4 are on the unit circle and consequently the corresponding solution is bounded. 

As with the Zeeman interaction, the secular term HJ,fc of the hyperfine interaction cannot alter the z component of the electron spin state (Eq. 3). However, the nonsecular terms can, with concomitant opposite alteration of the z projection of the nuclear spin state. Intersystem crossing, on the other hand, can be effected by both secular and nonsecular terms, c. The exchange interaction Hex of two electrons... 

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