Methods of algebraic manipulation in perturbation theory
We give a short introduction to the methods of representing polynomial and trigonometric series that are often used in Celestial Mechanics. A few applications are also illustrated.
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arXiv subjects
Marco Sansottera
Publications and source records attributed to Marco Sansottera.
Kolmogorov and Nekhoroshev theory for the problem of three bodies
We investigate the long time stability in Nekhoroshev's sense for the Sun-Jupiter-Saturn problem in the framework of the problem of three bodies. Using computer algebra in order to perform huge perturbation expansions we show that the stability for a time comparable with the age of the universe is actually reached, but with some strong truncations on the perturbation expansion of the Hamiltonian at some stage. An improvement of such results is currently under investigation.
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A Semi-Analytic Algorithm for Constructing Lower Dimensional Elliptic Tori in Planetary Systems
We adapt the Kolmogorov's normalization algorithm (which is the key element of the original proof scheme of the KAM theorem) to the construction of a suitable normal form related to an invariant elliptic torus. As a byproduct, our procedure can also provide some analytic expansions of the motions on elliptic tori. By extensively using algebraic manipulations on a computer, we explicitly apply our method to a planar four-body model not too different with respect to the real Sun--Jupiter--Saturn--Uranus system. The frequency analysis method allows us to check that our location of the initial conditions on an invariant elliptic torus is really accurate.
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On the stability of the secular evolution of the planar Sun-Jupiter-Saturn-Uranus system
We investigate the long time stability of the Sun-Jupiter-Saturn-Uranus system by considering the planar, secular model. Our method may be considered as an extension of Lagrange's theory for the secular motions. Indeed, concerning the planetary orbital revolutions, we improve the classical circular approximation by replacing it with a torus which is invariant up to order two in the masses; therefore, we investigate the stability of the elliptic equilibrium point of the secular system for small values of the eccentricities. For the initial data corresponding to a real set of astronomical observations, we find an estimated stability time of $10^7$ years, which is not extremely smaller than the lifetime of the Solar System ($\sim 5$ Gyr).
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