Symbolic Dynamics for the Anisotropic N-Centre Problem at Negative Energies

Abstract

The planar N-centre problem describes the motion of a particle moving in the plane under the action of the force fields of N fixed attractive centres: $$\begin{aligned} \ddot{x}(t)=\sum _{j=1}^N\nabla V_j(x(t)-c_j). \end{aligned}$$ x ¨ ( t ) = ∑ j = 1 N ∇ V j ( x ( t ) - c j ) . In this paper we prove symbolic dynamics at slightly negative energy for an N-centre problem where the potentials $$V_j$$ V j are positive, anisotropic and homogeneous of degree $$-\alpha _j$$ - α j : $$\begin{aligned} V_j(x)=|x|^{-\alpha _j}V_j\left( \frac{x}{|x|}\right) . \end{aligned}$$ V j ( x ) = | x | - α j V j x | x | . The proof is based on a broken geodesics argument and trajectories are extremals of the Maupertuis’ functional. Compared with the classical N-centre problem with Kepler potentials, a major difficulty arises from the lack of a regularization of the singularities. We will consider both the collisional dynamics and the non collision one. Symbols describe geometric and topological features of the associated trajectory.

Minimal collision arcs asymptotic to central configurations

“…Indeed, in addition to answering natural questions about the nature of these motions, the variational approach is a fruitful tool when building complex trajectories exploiting gluing techniques (cf. [3]). This application is the original motivation for this work, although we believe that the obtained result is interesting in itself.…”

“…It is well known that, as |q(t)| → 0, the normalized configuration q(t)/|q(t)| has infinitesimal distance from the set of central configurations of S. In particular, if this set is discrete, any collision trajectory admits a limiting central configuration ŝ ∈ S d−1 (see for instance [4,13,7,25,27]), that is (4) lim t→T q(t) |q(t)| = ŝ, for some T > 0. Given a central configuration ŝ ∈ S d−1 for S, we define the set of initial conditions for (3) in H h which evolve to collision with limiting configuration ŝ S h (ŝ) = {(q, p) ∈ H h : the solution of (3), with q(0) = q, p(0) = p, satisfies (4)}.…”

Chaotic dynamics in refraction galactic billiards

“…To prove theorem 1.4 we shall use a broken geodesic method, reminiscent of the one used in [2,34]; the key part of the proof is the surjectivity of the map π; in practice, this means that for every ℓ ∈ L we need to find an initial condition in X that generates a trajectory realizing the world ℓ. This result is achieved by means of a shadowing lemma which consists in searching critical points of a suitable length functional.…”

“…While studying dynamical systems coming from Celestial Mechanics, it is not uncommon to come across examples of chaotic models; nevertheless, rigorously proving the chaotic nature of a physical system is often problematic and it is, for example, the subject of the recent papers [1,2,5,6,15,20,21,26,34]. In this paper we propose the proof of the chaoticity of a model describing a class of mechanical refraction billiards, which can be thought as general models for the motion of a particle subject to a discontinuous potential.…”

Variational proof of the existence of periodic orbits in the anisotropic Kepler problem

“…A number of approximate periodic solutions of the AKP have been obtained via numerical calculation, but none have been mathematically proved to exist. Variational approach has been done by [1,2,3,4]. In this paper, we use the variational method to prove the existence of simple periodic orbits with certain properties in the AKP.…”