Chaotic quasi-collision trajectories in the 3-centre problem

Abstract

We study a particular kind of chaotic dynamics for the planar 3-centre problem on small negative energy level sets. We know that chaotic motions exist, if we make the assumption that one of the centres is far away from the other two (see Bolotin and Negrini, J. Diff. Eq. 190 (2003), 539--558): this result has been obtained by the use of the Poincar\'e-Melnikov theory. Here we change the assumption on the third centre: we do not make any hypothesis on its position, and we obtain a perturbation of the 2-centre problem by assuming its intensity to be very small. Then, for a dense subset of possible positions of the perturbing centre on the real plane, we prove the existence of uniformly hyperbolic invariant sets of periodic and chaotic almost collision orbits by the use of a general result of Bolotin and MacKay (see Cel. Mech. & Dyn. Astr. 77 (2000), 49--75). To apply it, we must preliminarily construct chains of collision arcs in a proper way. We succeed in doing that by the classical regularisation of the 2-centre problem and the use of the periodic orbits of the regularised problem passing through the third centre.Comment: 22 pages, 6 figure

Scattering Parabolic Solutions for the Spatial N-Centre Problem

“…the conclusion follows from (11). Now we are in position to prove the main result of this section, giving asymptotic estimates for large parabolic solutions defined on the half-line [t 1 , +∞).…”

“…[33]). The planar case of N -centre with N ≥ 3 is known to be non integrable on non-negative energy levels and has positive entropy; some partial extensions are available also for the spatial case (see [5,6,7,11,18,19]). Recently, in [24] Soave and Terracini have shown the presence of a chaotic subsystem for the planar N -centre problem also at negative energies.…”

Topologically Distinct Collision-Free Periodic Solutions for the $${N}$$ N -Center Problem

“…The paper [29] concerns the non existence of an analytic independent integral for the N ≥ 3 centre problem in the space, when the energy is larger than a threshold E th . In the case of negative energy it has been proved that there exists chaotic dynamics for the 3-center problem on small negative energy level sets when one centre is far away from the other two [10] or when the mass of one of the centres is much smaller then the others [19]. Both of these results rely on perturbative methods.…”

Symbolic dynamics: from the N-centre to the (N + 1)-body problem, a preliminary study

“…where x = x(t) ∈ R 2 denotes the position of the particle at time t ∈ R; basic references for such a problem are [3,4,[6][7][8][9]11] and the references therein. In this paper we consider α-gravitational potentials of type…”

“…In Sect. 2 we will perform a suitable rescaling in order to pass from problem (6) to an equivalent problem where the parameter Jacobi constant will be replaced by the parameter given by the maximal distance of the centres from the origin. This leads to the study of a rotating problem with a rescaled potential…”