## Periodic Reporting for period 5 - StableChaoticPlanetM (Stable and Chaotic Motions in the Planetary Problem)

Reporting period: 2021-09-01 to 2022-08-31

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 perturbative, but perturbation does not apply directly to the problem, because of its degeneracies. The goal of this research is to develop techniques in order to prove long-time stability or diffusive motions.

The PI of the project, Gabriella Pinzari, has been in charge in the direction of the team and advance the project’s scientific purposes.

In the publication [2], extending results in [Arnold, 1963], [Laskar & Robutel, 1996], [Fejoz 2004], [Chierchia & Pinzari, 2011], she proved the stability of a planetary system, without the constraint of small eccentricities and inclinations. In [3] she discussed the existence of a zone, in the three-body problem, where chaos and stability are possible simultaneously. In [7] G. Pinzari proved the existence of a first integral to a suitable average of the Newtonian potential; in [10, 11] she discussed that the first integral characteristic of the two centre problem affords small variations over exponentially long times also in the three-body problem, and shew that in the three-body problem, if the total angular momentum vanishes and the three bodies are constrained on a plane, the the perihelia of the smaller masses maklibrations about their center of mass for very long times.

In [arXiv: 2209.07097] G. Pinzari proves a conjecture on the periods of the two-centre problem. She hopes to apply the result to the study of the stability of the three-body problem.

Finally, in [arXiv: 2209.07457] the PI underlines the central role of the choice oof coordinates when dealing with the n-body problem.

Alexandre Pousse has been a post-doctoral fellow. His research attained the motions of two small moons of Saturn, Janus and Epimetheus that evolve on close circular orbits with a small radius difference and periodically get closer and exchange their orbits without collision [1, 12, 16].

Santiago Barbieri has been a PhD Student. His activity has been focused on Nekhoroshev theory applied to the three or four body problem. His research activity has been focused on refined Nekhoroshev estimates for quasi-integrable systems. His results appeared in [6]. He also studied the so-called “many-jet” conditions, a main ingredient to prove the stability of “degenerate variables” in many body systems, which is one of the primary objectives of this project [22].

Sara Di Ruzza has been a senior researcher and a post-doctoral fellow. Her research has been motivated by the PI's papers [7, 10, 11].

Indeed, in collaboration with the PI, she studied the existence of unstable orbits in the three body problem. The point of view of this research in completely new: the three-body problem is regarded as a “perturbation” of the problem where the two most massive bodies are fixed. This is a integrable problem, called two-centre problem. The purpose of the study of Pinzari and Di Ruzza was to understand whether it is possible to detect chaos in such framework. The answer is affermative [17, 26, arXiv 2209.03114]. In [15, 27] she discusses parallel properties of the classical and relativistic planetary problem.

Rocio Isabel Paez has been a post-doctoral fellow. Her research has been focused on orbital dynamics. She studied fast transitions in phase space associated to close encounters between gravitationally interacting bodies, including both natural and artificial hyperbolic dynamics [5, 8, 13].

Jerome Daquin is a post-doctoral fellow. His research attained long time dynamics of a Earth-Satellite-Moon-Sun system, including oblateness of one of the major bodies. He found a systematic lack of correlation between local hyperbolicty and global instabilty in the phase-space. However, using classical techniques of phase-space visualisation he has shown that local hyperbolicity is generally connected to large drift. Some of their numerical experiments have revealed that, in the regime of strong chaos, the time evolution of the second moment (variance) of some actions variables may depart significantly from normal diffusion (at least on the timescale of interest) [4, 9, 28].

Qinbo Chen has been a post-doctoral fellow. He has collaborated with the PI to the proof of the slow variations of Euler integral (see [7, 10, 11]) in the averaged three-body problem in a configuration where one of the two smaller bodies is close to have a collision with the primary [18].

Dongchen Li has been a post-doctoral fellow. He has been working in the framework of chaos in dynamical systems. His research attained to show that any C^r dynamical system having a heterodimensional cycle can be approximated in the C^r topology by those having robust heterodimensional dynamics [arXiv: 2105.03739]. In [arXiv: 2203.14075] it is proved that robust heterodimensional cycles emerge near homoclinic tangencies to a periodic orbit with complex central multipliers.

Alessio Troiani has been a post-doctoral fellow. He has collaborated with the PI and B. Scoppola (a collaborator to the project) to prove stability of a planetary system using tools of statistical mechanics [19]. In [23, 25, 29].

The list of dissemination activities is here: https://ercprojectpinzari.wordpress.com/activities/

In the publication [2], extending results in [Arnold, 1963], [Laskar & Robutel, 1996], [Fejoz 2004], [Chierchia & Pinzari, 2011], she proved the stability of a planetary system, without the constraint of small eccentricities and inclinations. In [3] she discussed the existence of a zone, in the three-body problem, where chaos and stability are possible simultaneously. In [7] G. Pinzari proved the existence of a first integral to a suitable average of the Newtonian potential; in [10, 11] she discussed that the first integral characteristic of the two centre problem affords small variations over exponentially long times also in the three-body problem, and shew that in the three-body problem, if the total angular momentum vanishes and the three bodies are constrained on a plane, the the perihelia of the smaller masses maklibrations about their center of mass for very long times.

In [arXiv: 2209.07097] G. Pinzari proves a conjecture on the periods of the two-centre problem. She hopes to apply the result to the study of the stability of the three-body problem.

Finally, in [arXiv: 2209.07457] the PI underlines the central role of the choice oof coordinates when dealing with the n-body problem.

Alexandre Pousse has been a post-doctoral fellow. His research attained the motions of two small moons of Saturn, Janus and Epimetheus that evolve on close circular orbits with a small radius difference and periodically get closer and exchange their orbits without collision [1, 12, 16].

Santiago Barbieri has been a PhD Student. His activity has been focused on Nekhoroshev theory applied to the three or four body problem. His research activity has been focused on refined Nekhoroshev estimates for quasi-integrable systems. His results appeared in [6]. He also studied the so-called “many-jet” conditions, a main ingredient to prove the stability of “degenerate variables” in many body systems, which is one of the primary objectives of this project [22].

Sara Di Ruzza has been a senior researcher and a post-doctoral fellow. Her research has been motivated by the PI's papers [7, 10, 11].

Indeed, in collaboration with the PI, she studied the existence of unstable orbits in the three body problem. The point of view of this research in completely new: the three-body problem is regarded as a “perturbation” of the problem where the two most massive bodies are fixed. This is a integrable problem, called two-centre problem. The purpose of the study of Pinzari and Di Ruzza was to understand whether it is possible to detect chaos in such framework. The answer is affermative [17, 26, arXiv 2209.03114]. In [15, 27] she discusses parallel properties of the classical and relativistic planetary problem.

Rocio Isabel Paez has been a post-doctoral fellow. Her research has been focused on orbital dynamics. She studied fast transitions in phase space associated to close encounters between gravitationally interacting bodies, including both natural and artificial hyperbolic dynamics [5, 8, 13].

Jerome Daquin is a post-doctoral fellow. His research attained long time dynamics of a Earth-Satellite-Moon-Sun system, including oblateness of one of the major bodies. He found a systematic lack of correlation between local hyperbolicty and global instabilty in the phase-space. However, using classical techniques of phase-space visualisation he has shown that local hyperbolicity is generally connected to large drift. Some of their numerical experiments have revealed that, in the regime of strong chaos, the time evolution of the second moment (variance) of some actions variables may depart significantly from normal diffusion (at least on the timescale of interest) [4, 9, 28].

Qinbo Chen has been a post-doctoral fellow. He has collaborated with the PI to the proof of the slow variations of Euler integral (see [7, 10, 11]) in the averaged three-body problem in a configuration where one of the two smaller bodies is close to have a collision with the primary [18].

Dongchen Li has been a post-doctoral fellow. He has been working in the framework of chaos in dynamical systems. His research attained to show that any C^r dynamical system having a heterodimensional cycle can be approximated in the C^r topology by those having robust heterodimensional dynamics [arXiv: 2105.03739]. In [arXiv: 2203.14075] it is proved that robust heterodimensional cycles emerge near homoclinic tangencies to a periodic orbit with complex central multipliers.

Alessio Troiani has been a post-doctoral fellow. He has collaborated with the PI and B. Scoppola (a collaborator to the project) to prove stability of a planetary system using tools of statistical mechanics [19]. In [23, 25, 29].

The list of dissemination activities is here: https://ercprojectpinzari.wordpress.com/activities/

The most significant results of the project are the following publications:

[2] =[Gabriella Pinzari. Memoirs of the American Mathematical Society, 2018]

This paper proves the existence of Lagrangian full-dimensional quasi-periodic motions in a regime far from small eccentricities and inclinations (previously considered by Arnold, Laskar, Robutel, Herman, Féjoz, Chierchia and the PI).

[7] =[Gabriella Pinzari. Celestial Mechanics and Dynamical Astronomy, 2019]

Here it is proved that the averaged perturbing function is a function with one degree of freedom (hence, integrable), and is a function of another function, called Euler Integral, whose dynamics can be simply studied. It opens the way the possibility of continuing the motions of the Euler Integral in the three-body problem.

[17] =[Sara Di Ruzza, Jérôme Daquin, Gabriella Pinzari. Communications in Nonlinear Science and Numerical Simulation, 2020]

The averaged Hamiltonian of the three-body problem is considered and we ask whether, in the situation when the two secondaries are far one from the other, librational motions which are proper of the Euler Integral do survive in the averaged three-body system.

[18]=[ Qinbo Chen, Gabriella Pinzari. Nonlinear Analysis, 2021].

This paper studies the same question as in [17], but from with rigorous tools. The averaged Hamiltonian of the three-body problem is considered, in the so called "collisional region", where the two secondaries are very close one to the other. The interest in such region comes from the fact that in the phase space of the Euler Integral it corresponds to a separatrix through a saddle point. We prove that closely to such separatrix, hence, closely to collisions, the variations of the Euler integral are slow.

[19] =[Gabriella Pinzari, Benedetto Scoppola, Alessio Troiani. Annales Henri Poincaré, 2022]

The novelty of this paper relies in the application of statistical mechanics methods to celestial mechanics, a task never considered before by the unboundedness of the Newtonian potential.

[2] =[Gabriella Pinzari. Memoirs of the American Mathematical Society, 2018]

This paper proves the existence of Lagrangian full-dimensional quasi-periodic motions in a regime far from small eccentricities and inclinations (previously considered by Arnold, Laskar, Robutel, Herman, Féjoz, Chierchia and the PI).

[7] =[Gabriella Pinzari. Celestial Mechanics and Dynamical Astronomy, 2019]

Here it is proved that the averaged perturbing function is a function with one degree of freedom (hence, integrable), and is a function of another function, called Euler Integral, whose dynamics can be simply studied. It opens the way the possibility of continuing the motions of the Euler Integral in the three-body problem.

[17] =[Sara Di Ruzza, Jérôme Daquin, Gabriella Pinzari. Communications in Nonlinear Science and Numerical Simulation, 2020]

The averaged Hamiltonian of the three-body problem is considered and we ask whether, in the situation when the two secondaries are far one from the other, librational motions which are proper of the Euler Integral do survive in the averaged three-body system.

[18]=[ Qinbo Chen, Gabriella Pinzari. Nonlinear Analysis, 2021].

This paper studies the same question as in [17], but from with rigorous tools. The averaged Hamiltonian of the three-body problem is considered, in the so called "collisional region", where the two secondaries are very close one to the other. The interest in such region comes from the fact that in the phase space of the Euler Integral it corresponds to a separatrix through a saddle point. We prove that closely to such separatrix, hence, closely to collisions, the variations of the Euler integral are slow.

[19] =[Gabriella Pinzari, Benedetto Scoppola, Alessio Troiani. Annales Henri Poincaré, 2022]

The novelty of this paper relies in the application of statistical mechanics methods to celestial mechanics, a task never considered before by the unboundedness of the Newtonian potential.