Resonance motion

The two species of 202 amu are taken to be the initially formed Franck-Condon structure and the parent species giving coherent resonance motion loss of a bromine atom gives BrCH2CH2CH2, which is detected after ionization by the probe pulse as C3H5, at 41 amu. The decay of BrCH2CH2CH2 leads to cyclopropane, a product not ionized by the probe pulse, and hence not seen through mass spectrometry. 

Different methods have been developed either for a rapid computation of the LCIs (Cincotta and Simo 2000) or for detecting the structure of the phase space (chaotic zones, weak chaos, regular resonant motion, invariant tori). Especially for this last purpose we quote the frequency map analysis (Laskar 1990, Laskar et al. 1992, Laskar 1993, Lega and Froeschle 1996), the sup-map method (Laskar 1994, Froeschle and Lega 1996), and more recently the fast Lyapunov indicator (hereafter FLI, Froeschle et al. 1997, Froeschle et al. 2000) and the Relative Lyapunov Indicator (Sandor et al. 2000). The definitions and comparisons between different methods including a preliminary version of the FLI have been discussed in Froeschle and Lega (1998, 1999). 

In quasi-integrable systems we do not find only KAM tori, but also resonant motions, and among resonant motions we find the chaotic ones. If e is small and h satisfies a suitable geometric condition (convexity of h is sufficient) the Nekhoroshev theorem proves the exponential stability of the actions for all initial conditions, including the resonant ones. More precisely, there exist positive constants eo, a,b,Io,to such that if < so, for any (I(0),ip(0)) e B x T it is I(t) — /(0) < Io a for any time t satisfying the exponential estimate ... 

C. The truncated normal form approximation. Consider a resonant motion with initial condition in the resonant domain of a lattice A and replace the true solution with... 

Hadjidemetriou, J. D. and Psychoyos, D. (2003), Dynamics of extrasolar planetary systems 2/1 resonant motion. In Galaxies and chaos (G. Contopoulos and N. Voglis, eds.). Lectures Notes in Physics, Springer, 412-432. 

Vella-Coleiro, G. P., Smith, D. H. Van Uitert, L. G. (1972). Resonant motion of domain walls in yttrium gadolinium iron garnets. Journal of Applied Physics, 43, 2428-30. 

Unlike the rf fields of the Paul trap, the Penning trap uses only dc fields a dc electric field between a ring electrode and the pole caps (Fig. 9.42a) and a dc magnetic field in the z-direction. Two different designs are shown in Fig. 9.42a and b. The ionic motion is complex. It consists of three components. There is cyclotron resonance motion in circles around a point that moves with the motion of the magnetron in a circle around the magnetic field lines (Fig. 9.42c). Superimposed on this is an axial oscillation in the z-direction. The superposition of all three components leads to the ion path depicted in Fig. 9.42d. 

It is important to emphasize the physical diflerence between relaxational and resonance motions because confusion frequently arises due to the use of similar words to describe physically very dissimilar phenomena. 

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