A first integral to the partially averaged Newtonian potential of the three-body problem

Abstract

We consider the partial average i.e., the Lagrange average with respect to just one of the two mean anomalies, of the Newtonian part of the perturbing function in the three-body problem Hamiltonian. We prove that such a partial average exhibits a non-trivial first integral. We show that this integral is fully responsible of certain cancellations in the averaged Newtonian potential, including a property noticed by Harrington in the 60s. We also highlight its joint rôle (together with certain symmetries) in the appearance of the so called "Herman resonance". Finally, we discuss an application and an open problem.

Perihelion Librations in the Secular Three-Body Problem

“…Here we have used that F ε does not depend explicitly on Λ, since both U and E depend on Λ only via G Λ . We claim that 5 Beware not to confuse the coordinate γ in (26) with its homonymous in (15). The latter is a cyclic coordinate for the Hamiltonians H i in (2) and (5), and hence has no rôle in the paper.…”

“…In the planar case, the relation (23) becomes very special. Instead of U and E, it is convenient to switch to the functions if a = a(Λ) is as in (15). We rewrite relation (23) as…”

“…To prove Proposition 3.1, we need to recall the following result from [15]. Then f renormalizably integrable by g through f , and f can be chosen to be…”

“…The formula in (31) (and its consequences below) is pretty specific for the planar case. In [15,Equation (49)] we proposed a general formula (holding for the planar and the spatial case), which, unfortunately, does not seem equally exploitable.…”

Euler integral and perihelion librations

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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.

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