ENG 
ESP Session S03 - Geometric and Variational Methods in Celestial Mechanics
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.