By B.A. Steves, A.J. Maciejewski, M. Hendry
According to the hot NATO complicated research Institute "Chaotic Worlds: From Order to sickness in Gravitational N-Body Dynamical Systems", this state-of-the-art textbook, written by means of the world over popular specialists, offers a useful reference quantity for all scholars and researchers in gravitational n-body structures. The contributions are specifically designed to offer a scientific improvement from the basic arithmetic which underpin glossy reports of ordered and chaotic behaviour in n-body dynamics to their software to actual movement in planetary structures. This quantity provides an up to date synoptic view of the topic.
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Additional resources for Chaotic Worlds: from Order to Disorder in Gravitational N-Body Dynamical Systems (NATO Science Series II: Mathematics, Physics and Chemistry)
The standard scheme of proof of the theorem is composed by a so called analytic part, based on the construction of a local Birkhoﬀ normal form in suitably chosen domains, and of a geometric part which takes care of covering the phase space with good domains. A remarkable exception to this scheme is the paper by Lochak (1992), where the geometric part is replaced by a clever use of convexity and of the simultaneous approximations of real numbers with rationals. We shall sketch the traditional scheme.
This is true in the coordinates of the normal form. If we look at the original coordinates then we must take into account the deformation due to the canonical transformations —as we already remarked while discussing the case of an elliptic equilibrium. Moreover, we must consider also the noise due to the remainder, but in this case too we have Φ˙ = O(εr ), so that the noise causes only a slow drift that becomes comparable with the deformation only after a time T (ε) ∼ 1/εr . The conclusion is illustrated in Figure 4: until the orbit remains in the non-resonance domain V it lies in a small neighborhood of the plane of fast drift ΠM (p0 ).
The approximate construction of the torus is a rather lengthy calculation, but may be performed with the computers available nowadays. The reliability of the construction may be appreciated by plotting the distance between the orbit calculated via numerical integration of the original system and the orbit calculated via the normal form of Kolmogorov. The result is reported in Figure 1 for diﬀerent orders of approximation of the torus. One sees that the successive applications of the algorithm of Kolmogorov give a better and better agreement, until the small diﬀerence between the frequencies of the integrated orbit and the ﬁxed frequencies of the motion on the torus shows up, causing a very slow drift.