### Session S03 - Geometric and Variational Methods in Celestial Mechanics

## Talks

Monday, July 12, 12:00 ~ 12:25 UTC-3

## Hyperbolic Solutions and Scattering in Newtonian n-Body Problem

### Guowei Yu

#### Nankai University, China - This email address is being protected from spambots. You need JavaScript enabled to view it.

In the Newtonian n-body problem, the hyperbolic solutions are those all mutual distances between the masses tend to infinity with nonzero speed as time goes to negative or positive infinity. By using a variation of the McGehee coordinates, we show these solutions form the stable or unstable manifolds of some equilibria at infinity. This allows us to give a new proof of Chazy's classical asymptotic formulas for these solutions. In the second part of the talk we will consider bi-hyperbolic solutions, which means solutions that are hyperbolic in both forward and backward time. We will discuss the possible pairs of equilibria at infinity that are connected by a given bi-hyperbolic solution. This is a joint work with N. Duignan, R. Moeckel and R. Montgomery.

Monday, July 12, 12:30 ~ 13:10 UTC-3

## Celestial Mechanics tools for studying the hydrogen atom

### Amadeu Delshams

#### Universitat Politècnica de Catalunya, Spain - This email address is being protected from spambots. You need JavaScript enabled to view it.

We consider the Rydberg electron in a circularly polarized microwave field, whose dynamics is described by a 2 d.o.f. Hamiltonian depending on one parameter $K>0$, which is a perturbation of the standard Kepler problem. The associated Hamiltonian system has two equilibria: $L_1$ (center-saddle for all $K$) and $L_2$ (center-center for small $K$ and complex-saddle otherwise). Associated to $L_1$ there is a family of Lyapunov periodic orbits that form a normally hyperbolic invariant manifold (NHIM). In this talk, we compute the primary transversal homoclinic orbits to the NHIM (and therefore the associated scattering maps) combining Poincaré-Melnikov methods with numerical methods. It should be noted that the transversality of these homoclinic orbits is exponentially small in $K$ (in analogy with the libration point $L_3$ of the R3BP).

Joint work with Mercè Ollé and Juan R. Pacha (Universitat Politècnica de Catalunya).

Monday, July 12, 13:15 ~ 13:40 UTC-3

## Nonlinear stability of equilibria: The satellite problem

### Patricia Yanguas

#### Universidad Pública de Navarra, Spain - This email address is being protected from spambots. You need JavaScript enabled to view it.

We study the nonlinear stability of elliptic equilibria in Hamiltonian systems. Specifically, a kind of formal stability, the so-called Lie stability, is described. The concept of Lie stability appears as a strong alternative in cases where a sort of nonlinear stability is required but Liapunov stability cannot be ensured. Moreover, Lie stability is obtained even for some Hamiltonian systems that do not satisfy Nekhoroshev's theory necessary conditions [1].

The determination of Lie stability passes through the obtention of the required convexity of the Hamiltonian function restricted to a certain subspace that is contained in the orthogonal space related to the frequency vector.

In the Lie stable cases, the error estimates of the solutions over exponentially long times are obtained through a result based in the determination of error bounds for adiabatic invariants in Hamiltonian systems.

As an application we analyse the attitude nonlinear stability of the spatial satellite problem and enlarge previous results by Markeev and Sokol'skii [2]. Furthermore, KAM tori related to Lie stable, as well as unstable equilibria, are also calculated [3].

[1] Nonlinear stability of elliptic equilibria in Hamiltonian systems with exponential time estimates, D. Cárcamo-Díaz, J.F. Palacián, C. Vidal, P. Yanguas, accepted in Discrete & Continuous Dynamical Systems (2021).

[2] On the stability of relative equilibrium of a satellite in a circular orbit, A.P. Markeev, A.G. Sokol'skii, Kosmicheskie Issledovaniya, 13(2), 139-146 (1975); Cosm. Res., 13(2), 119-125 (1975).

[3] Nonlinear stability in the spatial attitude motion of a satellite in a circular orbit, D. Cárcamo-Díaz, J.F. Palacián, C. Vidal, P. Yanguas, preprint (2021).

Joint work with Daniela Cárcamo-Díaz (Universidad del Bío-Bío, Chile), Jesús F. Palacián (Universidad Pública de Navarra, Spain) and Claudio Vidal (Universidad del Bío-Bío, Chile).

Monday, July 12, 14:15 ~ 14:55 UTC-3

## Symbolic dynamics for the anisotropic $N$-centre problem at negative energies

### Susanna Terracini

#### University of Turin, Italy - This email address is being protected from spambots. You need JavaScript enabled to view it.

The planar $N$-centre problem describes the motion of a particle moving in the plane under the action of the force fields of $N$ fixed attractive centres: \[ \ddot{x}(t)=\sum_{j=1}^N\nabla V_j(x-c_j) \]

In this paper we prove symbolic dynamics at slightly negative energy for an $N$-centre problem where the potentials $V_j$ are positive, anisotropic and homogeneous of degree $-\alpha_j$: \[ V_j(x)=|x|^{-\al_j}V_j\left(\frac{x}{|x|}\right). \] The proof is based on a broken geodesics argument and trajectories are extremals of the Maupertuis' functional.

Compared with the classical $N$-centre problem with Kepler potentials, a major difficulty arises from the lack of a regularization of the singularities. We will consider both the collisional dynamics and the non collision one. Symbols describe geometric and topological features of the associated trajectory.

Joint work with Vivina Barutello (University of Turin) and Gian Marco Canneori (University of Turin).

Monday, July 12, 15:00 ~ 15:25 UTC-3

## The anisotropic $N$-centre problem

### Gian Marco Canneori

#### Università di Torino, Italy - This email address is being protected from spambots. You need JavaScript enabled to view it.

We consider an anisotropic version of the planar $N$-centre problem of Celestial Mechanics and we prove the existence of symbolic dynamics on constant negative energy shells. The proof relies on a broken geodesics technique, where trajectories are extremals for the Maupertuis' functional. A deep analysis of collision trajectories reveals also to be essential in this context. Close to every centre, minimal collision arcs span the local stable manifold of an equilibrium point lying in the McGehee collision manifold.

Joint work with Vivina Barutello (Università di Torino) and Susanna Terracini (Università di Torino).

Monday, July 12, 15:30 ~ 15:55 UTC-3

## Projective Dynamics and an integrable Billiard System of Boltzmann-Gallavotti-Jauslin

### Lei Zhao

#### Univeritat Augsburg, Germany - This email address is being protected from spambots. You need JavaScript enabled to view it.

A biliard model in a half plane, with the boundary line of the half plane as wall of reflection, defined via the planar Kepler problem, is shown to be integrable by Gallavotti and Jauslin who explicitely constructed an independent integral additional to the energy. The bounded dynamics of the system has been shown by Felder to carry periodic and quasi-periodic dynamics. The model is proposed as a limiting case of a set of more general models proposed by Boltzmann in order to illustrate his "ergodic hypothesis". In this talk, I shall explain that the integral of Gallavotti-Jauslin is ultimately related to the energy of an associated Kepler problem on the sphere.

Wednesday, July 14, 16:00 ~ 16:25 UTC-3

## Symmetric bicircular central configurations of the $3n$--body problem

### Montserrat Corbera

#### Universitat de Vic-Universitat central de Catalunya, Spain - This email address is being protected from spambots. You need JavaScript enabled to view it.

We consider symmetric central configurations of the planar $3n$--body problem consisting of $n$ equal masses at the vertices of a regular $n$-gon inscribed on a circle of radius $r$ and $2n$ masses at the vertices of a second concentric $2n$--gon inscribed in a circle of radius $ar$. We call this kind of central configurations bicircular central configurations of the $3n$--body problem. We analyze two different types of such configurations. In the first type, called regular bicircular central configurations of the $3n$--body problem, the second $2n$--gon is regular and the masses at the vertices of this $2n$--gon alternate values. In the second type, called semiregular bicircular central configurations of the $3n$--body problem, the second $2n$--gon is semiregular and the masses at its vertices are all of them equal. A semiregular $2n$--gon is a $2n$--gon having $n$ pairs of vertices symmetric by a reflection of an angle $\beta$ with respect to the axis of symmetry of the regular $n$--gon. Our aim is to analyze the set of values of the parameter $a$ providing regular bicircular central configurations of the $3n$--body problem and the set of values of the parameters $(a,\beta)$ providing semiregular bicircular central configurations of the $3n$--body problem. In particular, we prove analytically the existence of regular bicircular central configurations with $a$ sufficiently large for all $n\geq 2$ and with $a$ sufficiently close to the origin for all $n\geq 3$, and we compute numerically the entire set of values of $a$ providing regular bicircular central configurations for fixed values of $n$. Furthermore we prove analytically the existence of two families when $n=2$ and four families when $n\geq 2$ of semiregular bicircular central configurations with $\beta$ sufficiently close to $\pi/2n$ and the existence of one family when $n=2$ and two families when $n\geq 3$ of semiregular bicircular central configurations with $\beta$ sufficiently close to $\pi/n$. We also study numerically the entire families of semiregular bicircular central configurations for fixed values of $n$.

Joint work with Claudia Valls (Instituto Superior Técnico, Universidade de Lisboa).

Wednesday, July 14, 16:30 ~ 17:10 UTC-3

## On the global dynamics of the planar hyperbolic two-body problem

### Mario Dias Carneiro

#### Federal University of Minas Gerais, Brazil - This email address is being protected from spambots. You need JavaScript enabled to view it.

We discuss some results about the global dynamics of the hyperbolic two-body problem in the context of symplectic reduction and reconstruction of Hamiltonian with zero curvature principal bundle. We also show how the bifurcation of relative equilibria is an example of recent results by N. de Paula and P. Salomão [SP} about the global dynamics of Hamiltonians with a saddle-center singularity.

[SP] Pedro A. S. Salomão; DE PAULO, N. V. . Systems of transversal sections near-critical energy levels of Hamiltonian systems in R^4. Memoirs of the American Mathematical Society, v. 252, p. 1-105, 2018.

Joint work with Justino Muniz Jr ( Federal University of Viçosa) and Matthew Perlmutter (federal University of Minas Gerais-Brazil).

Wednesday, July 14, 17:15 ~ 17:40 UTC-3

## An Existence Proof of a Symmetric Periodic Orbit in the Octahedral Six-Body Problem

### Anete Soares Cavalcanti

#### Universidade Federal Rural de Pernambuco, Brazil - This email address is being protected from spambots. You need JavaScript enabled to view it.

The $\textit{Variational Methods}$ applied to the n-body Newtonian problem allows to prove the existence of periodic orbits, in most cases with some symmetry. It was exploited by the Italian school in the 1990's (Coti-Zelati, Degiovanni-Gianonni-Marino, Serre-Terracini). They gave new periodic solutions for a mechanical systems with potentials satisfying a hypothesis called $\textit{strong force}$, which excludes the Newtonian potential. The $\textit{strong force hypothesis}$ was introduced by Poincaré. In this talk we will give a variational existence proof of a periodic orbit in the $\textit{octahedral}$ $\textit{six-body problem}$ with equal masses. Next we explain the main ideas of the proof, more datails could be find at [1].

Let us consider six bodies with equal masses in $\mathbb{R}^3$. We assume that every coordinate axis contains a couple of bodies, and they are symmetric with respect to the origin, which is the center of mass of the system. This is the octahedral six-body problem. Our orbit starts with a collision in the $x$-axis and the other $4$ bodies form a square on the orthogonal plane. Let us denote by $T$ the period of the solution. During the $\textit{first sixth}$ of the period, the bodies on the $x$-axis move away, the two bodies on the $z$-axis also move away, while the bodies on the $y$-axis approach. At the time $t=T/6$ the bodies on the $z$-axis are on a turning of the period and they approach each other as $t\in [T/6,T/3]$. At time $t=T/3$ the bodies on the $y$-axis are at a double collision, and the other bodies form a square on the orthogonal plane. In the second third of the period, the motion is the same as above, after having exchanged $x$ by $y$, $y$ by $z$ and $z$ by $x$. That is to say, the solution satisfies the symmetry condition $x(t-T/3)=y(t)$, $y(t-T/3)=z(t)$ and $z(t-T/3)=x(t)$.

At the work, we introduce the formal aspects of the problem, we will write the equations of motion and explain the variational setting. In particular, we will prove that the lagrangian action functional is $\textit{coercive}$. Aftewords, we show that a minimizer has no other collisions besides those imposed by the choice of set of loops where we minimize the action functional, and that all collisions are double ($\textit{i.e.}$ there are no quadruple collisions). Finally, we regularize all possible collisions.

Bibliography

[1] Cavalcanti, A. S., $\textit{An Existence Proof of a Symmetric Periodic Orbit in the Octahedral Six-Body Problem}$, Discrete and Continuous Dynamical Systems 37, 2017, p. 1903-1922.

[2] Venturelli, A., $\textit{ A Variational proof of the existence of Von Schubart's Orbits}$, Discrete and Continuous Dynamical Systems B 10, 2008, p. 699-717.

Wednesday, July 14, 18:15 ~ 18:40 UTC-3

## Geodesic rays of the N-body problem

### Juan Manuel Burgos

#### Conacyt - Cinvestav, México - This email address is being protected from spambots. You need JavaScript enabled to view it.

In the context of the Newtonian N-body problem, we will show that geodesic rays of the nonnegative energy Jacobi-Mapertuis metric are expansive in the sense that all mutual distances between the bodies are divergent functions. We will also comment on the state of the art concerning the existence of these rays, recent achievements and some open problems.

J. M. Burgos, E. Maderna, Geodesic rays of the N-body problem, arXiv:2002.06153.

J. M. Burgos, Existence of partially hyperbolic motions in the N-body problem, arXiv:2008.08762.

Joint work with Ezequiel Maderna (Universidad de la República, Uruguay).

Wednesday, July 14, 18:45 ~ 19:10 UTC-3

## Weak KAM theorem on the N-center problem

### Eddaly Guerra Velasco

#### CONACyT-Universidad Autónoma de Chiapas., México - This email address is being protected from spambots. You need JavaScript enabled to view it.

In this talk we consider the Hamiltonian system defined by a particle, subordinated to a potential of attractive $N$-center type. We prove the existence of a variety of minimizing motions by studying the viscosity solutions of the Hamiton-Jacobi equation asociated to the system.

Wednesday, July 14, 19:15 ~ 19:40 UTC-3

## Perspectives on the N-center problem and viscosity solutions

### Boris Asdrubal Percino Figueroa

#### Universidad Autónoma de Chiapas, México - This email address is being protected from spambots. You need JavaScript enabled to view it.

In this talk we study the differentiability of the viscosity solutions of the Hamilton-Jacobi equiation asociated to the $N$-center problem, such solutions are not differentiable on the centers of the system, we prove the existence of a broader set of non differentiabilty. We also present some open questions regarding the Hamilton-Jacobi equation of the system.

Wednesday, July 21, 16:00 ~ 16:25 UTC-3

## Parabolic orbits in Celestial Mechanics: a functional-analytic approach

### Alberto Boscaggin

#### University of Turin, Italy - This email address is being protected from spambots. You need JavaScript enabled to view it.

We show a new functional analytic formulation for half-entire parabolic solutions of a class of systems of ODEs driven by singular potentials. A key point for this is represented by the choice of a suitable functional space supporting an Hardy-type inequality. Applications to various equations of Celestial Mechanics are presented, yielding, in a perturbative setting, the existence of parabolic solutions asymptotic to a (possibly non-minimal) prescribed central configuration.

Joint work with Walter Dambrosio (University of Turin), Guglielmo Feltrin (University of Udine) and Susanna Terracini (University of Turin).

Wednesday, July 21, 16:30 ~ 17:10 UTC-3

## Resonances and their collapse in moving binaries

### Mark Levi

#### Penn State, USA - This email address is being protected from spambots. You need JavaScript enabled to view it.

When two coupled point masses move in a periodic potential the energy of their collective motion may get transferred, via resonance, to the energy of their relative motion. Two pendula with torsional coupling is a prime example. I will describe the role of the symplectic group played in this effect, a connection with the finite-zone Lame potential and a certain universality in the resonant behavior.

Joint work with Jing Zhou.

Wednesday, July 21, 17:15 ~ 17:40 UTC-3

## Projective geometry of the planar Kepler problem

### Connor Jackman

#### CIMAT, Mexico - This email address is being protected from spambots. You need JavaScript enabled to view it.

The Kepler orbits are a 3-parameter family of plane curves: conics with a fixed focus. The group of (local) transformations of the plane preserving this family is 7-dimensional: the maximum possible for a 3-parameter family of plane curves. We will explain the construction of this action, and how these point symmetries can be useful to study the geometric properties of Kepler orbits.

Joint work with Gil Bor.

Wednesday, July 21, 18:15 ~ 18:40 UTC-3

## A variational principle for stable orbits.

### Daniel Offin

#### Queen's University, Canada - This email address is being protected from spambots. You need JavaScript enabled to view it.

We discuss a general theorem concerning stable orbits, and the critical values which correspond to them. We also discuss applications of this to several interesting examples from classical and celestial mechanics.

Joint work with Henry Kavle (Queen's University, Canada).

Wednesday, July 21, 18:45 ~ 19:10 UTC-3

## Satellite-pair orbits in a Four-Body Problem

### Lennard Bakker

#### Brigham Young University, USA - This email address is being protected from spambots. You need JavaScript enabled to view it.

We consider a four-body planar problem in which two primaries move in uniform circular motion around their center of mass and, in the gravitational field generated by the two primaries, two bodies with positive but small masses (in comparison with the masses of the primaries) move and interact gravitationally with each other. We investigate through both numerical and analytical approaches the existence of satellite-pair orbits which is when one of the smaller mass bodies moves in a bounded motion about one of the primaries and the other smaller mass body moves in a bounded motion about the other primary.

Joint work with Nick Freeman (Brigham Young University).