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 {\it 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.

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