Hamiltonian systems resonance

W, g potential functions, k 1, has been discussed in various papers (see, for example, [6, 11, 9, 16, 3]). It has been pointed out that, for step-sizes /j > e = 1/ /k, the midpoint method can become unstable due to resonances [9, 16], i.e., for specific values of k. However, generic instabilities arise if the step-size k is chosen such that is not small [3, 6, 18], For systems with a rotational symmetry this has been shown rigorously in [6j. This effect is generic for highly oscillatory Hamiltonian systems, as argued for in [3] in terms of decoupling transformations and proved for a linear time varying system without symmetry. 

Benettin, G., Galgani, L., Giorgilli, A. (1985). A proof of Nekhoroshev s theorem for the stability times in nearly integrable Hamiltonian systems. Cel. Mech., 37 1-25. Benettin, G., Gallavotti, G. (1986). Stability of motions near resonances in quasi-integrable Hamiltonian systems. Joum. Stat. Phys., 44 293-338. 

In previous papers (Froeschle et al. 2000, Guzzo et al. 2002, Lega et al. 2002) we used the FLI to describe the geometry of the resonances, integrating orbits of the Hamiltonian system of equation 6 and of the following 4-dimensional symplectic map ... 

Both for the Hamiltonian system (equation 6) and for the 4 dimensional symplectic mapping (equation 7) we have computed the FLI charts for different values of the perturbing parameter and we have selected a low order resonance. In the case of the Hamiltonian system we have considered the unperturbed resonance l = 2/2, while for the mapping we have chosen 2 = —x/2. 

In this article we discuss the problem of understanding the long-term stability properties of a solution of a quasi-integrable Hamiltonian system by means of a Fourier analysis on a short observation time. Precisely, even for resonant chaotic motions, we will show how the combined use of Fourier analysis and Nekhoroshev theorem allows to understand the stability properties on a time T exp(T), where T is a suitable observation time, of the order of the resonant period. To be definite, we will refer to quasi-integrable Hamiltonian systems with Hamiltonian of the form ... 

Figure 4 Fast Fourier transform of g(t) = (t)Q(I(t), tp(t)), where (7(t),

The transition of spectra from structured to unstructured ones is not described by a theorem, but has been studied numerically in Guzzo, Lega and Froeschle (2002) by comparing the geometry of resonances of a given system computed with the Fast Lyapunov Indicator with the structure of the spectra of an observable computed on well selected chaotic solutions. More precisely, in Froeschle et al. (2000) we estimated with the FLI method that the transition between Littlewood and Chirikov regime for the Hamiltonian system ... 

Benettin, G. and Gallavotti,G. (1986) Stability of motions near resonances in quasi-integrable Hamiltonian systems J. Stat. Phys., vol. 44, 293-338. 

In the recent past, analytical research in Celestial Mechanics has centred on KAM theory and its applications to the dynamics of low dimensional Hamiltonian systems. Results were used to interpret observed solutions to three body problems. Order was expected and chaos or disorder the exception. Researchers turned to the curious exception, designing analytical models to study the chaotic behaviour at resonances and the effects of resonant overlaps. Numerical simulations were completed with ever longer integration times, in attempts to explore the manifestations of chaos. These methods improved our understanding but left much unexplained phenomena. 

In the resonant case, integral trajectories are everywhere dense on tori of smaller dimension. We recall that precisely such a situation characterizes those Hamiltonian systems which admit noncommutative integration (see above). 

To treat real molecules the theory must be extended to typical unbounded Hamiltonian systems, for which we expect there to be long-lived resonances associated with the quasi periodic islands in the classical description. 

The relevant Hamiltonian for the gas-phase solute molecules can be treated by the same three-orbitals four-electron model used in Chapter 2. Since the energy of 3 is much higher than that of , and d>2 (see Table 2.4), we represent the system by its two lowest energy resonance structures, using now the notation fa and fa as is done in eq. (2.40). The energies of these two effective configurations are now written as... 

OH ion is denoted iff%. The atoms depicted in the figure are considered as our solute system (5) while the rest of the protein-water environment constitutes the solvent (s) for the enzyme reaction. Although the Ca2+ ion does not actually react, it is included in the reacting system for convenience. As before, we describe the diagonal elements of the EVB Hamiltonian associated with the three resonance structures (t/rf,, t/ff) by... 

A wide variety of ID and wD NMR techniques are available. In many applications of ID NMR spectroscopy, the modification of the spin Hamiltonian plays an essential role. Standard techniques are double resonance for spin decoupling, multipulse techniques, pulsed-field gradients, selective pulsing, sample spinning, etc. Manipulation of the Hamiltonian requires an external perturbation of the system, which may either be time-independent or time-dependent. Time-independent... 

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