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Eight-shaped Lissajous orbits in the Earth-Moon system

Grégory Archambeau, Philippe Augros, Emmanuel Trélat (2011)

MathematicS In Action

Euler and Lagrange proved the existence of five equilibrium points in the circular restricted three-body problem. These equilibrium points are known as the Lagrange points (Euler points or libration points) L 1 , ... , L 5 . The existence of families of periodic and quasi-periodic orbits around these points is well known (see [20, 21, 22, 23, 37]). Among them, halo orbits are 3-dimensional periodic orbits diffeomorphic to circles. They are the first kind of the so-called Lissajous orbits. To be selfcontained,...

Nekhoroshev stability for the D’Alembert problem of Celestial Mechanics

Luca Biasco, Luigi Chierchia (2002)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

The classical D’Alembert Hamiltonian model for a rotational oblate planet revolving near a «day-year» resonance around a fixed star on a Keplerian ellipse is considered. Notwithstanding the strong degeneracies of the model, stability results a là Nekhoroshev (i.e. for times which are exponentially long in the perturbative parameters) for the angular momentum of the planet hold.

Optimal stability and instability results for a class of nearly integrable Hamiltonian systems

Massimiliano Berti, Luca Biasco, Philippe Bolle (2002)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

We consider nearly integrable, non-isochronous, a-priori unstable Hamiltonian systems with a (trigonometric polynomial) O µ -perturbation which does not preserve the unperturbed tori. We prove the existence of Arnold diffusion with diffusion time T d = O 1 / μ log 1 / μ by a variational method which does not require the existence of «transition chains of tori» provided by KAM theory. We also prove that our estimate of the diffusion time T d is optimal as a consequence of a general stability result proved via classical perturbation...

The n -centre problem of celestial mechanics for large energies

Andreas Knauf (2002)

Journal of the European Mathematical Society

We consider the classical three-dimensional motion in a potential which is the sum of n attracting or repelling Coulombic potentials. Assuming a non-collinear configuration of the n centres, we find a universal behaviour for all energies E above a positive threshold. Whereas for n = 1 there are no bounded orbits, and for n = 2 there is just one closed orbit, for n 3 the bounded orbits form a Cantor set. We analyze the symbolic dynamics and estimate Hausdorff dimension and topological entropy of this hyperbolic set....

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