Euler integral and perihelion librations

Abstract

We discuss dynamical aspects of an analysis of the two-centre problem started in [15]. The perturbative nature of our approach allows us to foresee applications to the three-body problem.

Euler integral as a source of chaos in the three-body problem

“…As, in our experiments, we are going to consider a global region of phase space, we need to establish the singularities of U. Below, we shall briefly show that Proposition 2.2 ( [22,21,3]) The function U is singular if and only if 0 < r < 2 and (G, g) ∈ S 0 (r).…”

“…The full three-body (in general, many-body) problem inherits much of its reputation -especially in Hamiltonian mechanicsafter the breakthrough paper by V. I. Arnold [1] which will be recalled below. In fact, this paper is motivated by previous research [20,9,22,21], which here we briefly recall, in order to keep the paper self-contained.…”

“…are also level sets of U. On the other hand, the phase portrait of E can be studied exactly, and this has been done in [21]. We report the main results here.…”

“…

In this paper we address, from a purely numerical point of view, the question, raised in [20,21], and partly considered in [22,9,3], whether a certain function, referred to as "Euler Integral", is a quasi-integral along the trajectories of the three-body problem. Differently from our previous investigations, here we focus on the region of the "unperturbed separatrix", which turns to be complicated by a collision singularity.

…”

Euler integral as a source of chaos in the three–body problem

“…The full three-body (in general, many-body) problem inherits much of its reputation -especially in Hamiltonian mechanics -after the breakthrough paper by V. I. Arnold [1] which will be recalled below. In fact, this paper is motivated by previous research [2][3][4][5], which here we briefly recall, in order to keep the paper self-contained.…”