## Periodic Reporting for period 1 - StableChaoticPlanetM (Stable and Chaotic Motions in the Planetary Problem)

**Reporting period:**2016-03-01

**to**2017-02-28

## Summary of the context and overall objectives of the project

The planetary problem consists in determining the motions of n planets, interacting among themselves and with a sun, via gravity only. Its deep comprehension has relevant consequences in Mathematics, Physics, Astronomy and Astrophysics. The problem is by its nature perturbative, being well approximated by the much easier (and in fact exactly solved since the XVII century) problem where each planet interacts only with the sun. However, when the mutual interactions among planets are taken into account, the dynamics of the system is much richer and, up to nowadays, essentially unsolved. Stable and unstable motions coexist as well. In general, perturbation theory allows to describe qualitative aspects of the motion, but it does not apply directly to the problem, because of its deep degeneracies. During my PhD, I obtained important results on the stability of the problem, based on a new symplectic description, that allowed me to write, for the first time, in the framework of close to be integrable systems, the Hamilton equations governing the dynamics of the problem, made free of its integral of motions, and degeneracies related. By such results, I was an invited speaker to the ICM of 2014, in Seoul. The goal of this research is to use such recent tools, develop techniques, ideas and wide collaborations, also by means of the creation of post-doc positions, assistant professorships (non-tenure track), workshops and advanced schools, in order to find results concerning the long-time stability of the problem, as well as unstable or diffusive motions.

## Work performed from the beginning of the project to the end of the period covered by the report and main results achieved so far

The Principal Investigator Gabriella Pinzari was in charge of the direction of the team and of research activity since 1/03/2016. Her research focused on the study of geometric part in n-body problem. Specifically, she focused on the study of canonical coordinates that allow for the complete reduction of the symmetries of its Hamiltonian, related to the conservation of the total angular momentum. The presence of non-commuting integrals causes degeneracies that make the application of standard perturbative techniques impossible to planetary systems. Moreover, the planetary problem possess a degeneracy (called “proper degeneracy”, consisting in the fact that it is a perturbation of a super—integrable system) that allows (by a Theorem of Nehorosev, Mitchenko and Fomenko in the 70s) for the possibility of reducing such symmetries in not a unique way. Pinzari produced three different kinds of reduction that give different results and hence allow to highlight different properties. These results are completely new in the literature, since the question had been never considered before. Pinzari started considering the problem since her PhD, where she obtained the first direct proof of a theorem of stability of the planetary motions, that goes under the name of “Arnold’s Theorem”, stated and proved in a particular case by (V. I. Arnold, 1963, Russ. Math.Surv.). In her PhD she found global set of coordinates reducing the symmetries (and hence the degeneracies) that allowed her to obtain a direct proof of Arnold’s theorem, published in 2011 in Inventiones Mathematicae. The coordinates that she found in her successive work allowed her to obtain a generalization of her previous result, that has been accepted (since 2016) for publication in the monography series “Memoir of the American Mathematical Society” and will appear at mid 2018. The PI was invited by M. Guardia at UCP, Barcelona, for a scientific collaboration (June 2016); she organized one special session to the AIMS Conference in Orlando for dissemination (July 2016); attended an advanced course on Double resonances in the problem of Arnold at the Institut Henri Poincaré, Paris (December 2016) and was invited to a winter school in “Conservative dynamics” organised in Engelberg, Swiss by the ETH Zurich (February 2017).

Dr Alexandre Pousse joined the team on 1/11/2016. He has been awarded of a yearly post-doc position. Alexandre Pousse’s research focused on the study of the co-orbital dynamics in the Three-Body Problem (for example two planets that gravitate around a Star with the same period). He completed a numerical study that revisits the restricted case in order to clarify the notion of « quasi-satellite » configuration . This work is now published in the scientific journal «Celestial Mechanics and Dynamical Astronomy » (Pousse&al.2017).

He began a work based on the development of an analytic formalism adapted to quasi-satellite motion. This new method allows to highlight a stable quasi-satellite fixed point family as well as to study their neighbourhood via Birkhkoff’s normal forms. All the previous results involve the averaged problem that is an approximation of the Three-Body Problem. Hence, in order to prove rigorous theorems on the behaviour of co-orbital motions over a finite but large timescale, he started a scientific collaboration with Philippe Robutel (Paris Observatory) and Laurent Niederman (Paris Observatory, Paris XI University).

Pousse is currently pursuing this collaboration in order to provide a rigorous proof (and up to our knowledge, the first one) of existence of the ``horseshoe » dynamics over infinite times thanks to KAM theory. He was invited at the Paris Observatory for a scientific collaboration linked to the project. Afterwards, he also attended an advanced course on Double resonances in the problem of Arnold diffusion (December 2016) at the Institut Henri Poincaré , Paris, and a winter school in “Conservati

Dr Alexandre Pousse joined the team on 1/11/2016. He has been awarded of a yearly post-doc position. Alexandre Pousse’s research focused on the study of the co-orbital dynamics in the Three-Body Problem (for example two planets that gravitate around a Star with the same period). He completed a numerical study that revisits the restricted case in order to clarify the notion of « quasi-satellite » configuration . This work is now published in the scientific journal «Celestial Mechanics and Dynamical Astronomy » (Pousse&al.2017).

He began a work based on the development of an analytic formalism adapted to quasi-satellite motion. This new method allows to highlight a stable quasi-satellite fixed point family as well as to study their neighbourhood via Birkhkoff’s normal forms. All the previous results involve the averaged problem that is an approximation of the Three-Body Problem. Hence, in order to prove rigorous theorems on the behaviour of co-orbital motions over a finite but large timescale, he started a scientific collaboration with Philippe Robutel (Paris Observatory) and Laurent Niederman (Paris Observatory, Paris XI University).

Pousse is currently pursuing this collaboration in order to provide a rigorous proof (and up to our knowledge, the first one) of existence of the ``horseshoe » dynamics over infinite times thanks to KAM theory. He was invited at the Paris Observatory for a scientific collaboration linked to the project. Afterwards, he also attended an advanced course on Double resonances in the problem of Arnold diffusion (December 2016) at the Institut Henri Poincaré , Paris, and a winter school in “Conservati

## Progress beyond the state of the art and expected potential impact (including the socio-economic impact and the wider societal implications of the project so far)

We expect to obtain concrete results concerning dynamics of gravitational systems especially for what concerns the existence of extremely unstable orbits, as well as extremely stable ones.