Nearly integrable system freedom, Hamiltonian systems

In this way, the above statement may be acceptable not only in a qualitative level, but the most important implication of the mathematical statement is that the character of freezing has been shown to be quantitatively the same as adiabaticity predicted by the Nekhoroshev estimate in the nearly integrable system. This solves, at least partially, question (iii) posed in the previous section, because the model Hamiltonian [Eq. (1)] just describes the situation where not all the degrees of freedom do not necessarily show adiabaticity but only a limited number of variables, just the energies of two subsystems in this case, are almost frozen. It is true that there may be, in principle, many other possibilities and the proposed one is not a unique way as for the division of phase space into lower-dimensional subspaces, but the separation induced by the internal structure of molecules is the most natural and plausible candidate. 

The hierarchy of tori is theoretically predicted by the Poincare-Birkhoff theorem [10] in nearly integrable systems with two degrees of freedom. For instance, the hierarchy in the Henon-Heiles system represented by the Hamiltonian is shown in Fig. 1. 

In the previous section, we have shown that switching the picture from the nearly integrable Hamiltonian to the Hamiltonian with internal structures may make it possible to solve several controversial issues listed in Section IV. In this section we shall examine the validity of an alternative scenario by reconsidering the analyses done in MD simulations of liquid water. As mentioned in Section III, since a water molecule is modeled by a rigid rotor, and has both translational and rotational degrees of freedom. So, the equation of motion involves the Euler equation for the rigid body, coupled with ordinary Hamiltonian equations describing the translational motions. The precise Hamiltonian is therefore different from that of the Hamiltonian in Eq. (1), but they are common in that the systems have internal structures, and the separation of the time scale between subsystems appears if system parameters are appropriately set. 

Abstract The periodic orbits play an important role in the study of the stability of a dynamical system. The methods of study of the stability of a periodic orbit are presented both in the general case and for Hamiltonian systems. The Poincare map on a surface of section is presented as a powerful tool in the study of a dynamical system, especially for two or three degrees of freedom. Special attention is given to nearly integrable dynamical systems, because our solar system and the extra solar planetary systems are considered as perturbed Keplerian systems. The continuation of the families of periodic orbits from the unperturbed, integrable, system to the perturbed, nearly integrable system, is studied. 

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