Nekhoroshev theorem Hamiltonian systems

In addition to computer simulations, what drives the research in this direction is elaborated perturbation theories developed almost simultaneously. In particular, the Kolmogorov-Arnold-Moser (KAM) theorem, which has shown the existence of invariant tori under a small perturbation to completely inte-grable systems, and the Nekhoroshev theorem, which has proved exponentially long-time stability of trajectories close to completely integrable ones, are landmarks in this field. Although a lot of works have been done, there still remain unsolved important questions, and the Hamiltonian system is being studied as one of important branches in the theory of dynamical systems [3-5]. 

The perturbation strength for which the Nekhoroshev s theorem holds is also so small that it cannot be applied to realistic physical and chemical situations. Indeed it was shown that the range of perturbation strength is much smaller than the situation where the power spectrum density of observables exhibits a continuous one [24]. This means that, in its rigorous sense, the Nekhoroshev s theorem can only be applied to sufficiently weak perturbed systems. For the same reason as mentioned above, Nekhoroshev s theorem is nevertheless a key guiding principle to sticky or stagnant motions in nearly integrable Hamiltonian systems. 

Keywords Hamiltonian systems, perturbation theory, normal forms, KAM theory, Nekhoroshev s theorem, planetary systems... 

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. 

As we said in the introduction, the crucial question of stability of a dynamical system is related to the structure and density of invariant tori which foliate the phase space. This is in fact the geometrical representation of the Nekhoroshev theorem (1977). We recall that in a quasi-integrable system with Hamiltonian ... 

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 ... 

It should be noted that the Nekhoroshev s theory is not limited to two-dimensional systems, but rather it holds in general dimensions. Therefore, in the system with many degrees of freedom, the Nekhoroshev s theorem may explain sticky motions. This is also the case with the KAM theory. As given before, the statement of the KAM theorem is not limited to the Hamiltonian with few degrees of freedom. In Section IV, we will discuss to what extend these perturbation theories have capability to predict the slow motions in manydimensional systems. 

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