In 1999 Chenciner and Montgomery found a remarkably simple choreographic motion for the planar 3-body problem (see \cite{CM}). In this solution 3 equal masses travel on a eight shaped planar curve; this orbit is obtained minimizing the action integral on the set of simple planar choreographies with some special symmetry constraints. In this work our aim is to study the problem of $n$ masses moving in $\RR^d$ under an attractive force generated by a potential of the kind $1/r^α$, $α>0$, with the only constraint to be a simple choreography: if $q_1(t),...,q_n(t)$ are the $n$ orbits then we impose the existence of $x \in H^1_{2 π}(\RR,\RR^d)$ such that $q_i(t)=x(t+(i-1) τ), i=1,...,n, t \in \RR, $ where $τ= 2π/ n$. In this setting, we first prove that for every $d,n \in \NN$ and $α>0$, the lagrangian action attains its absolute minimum on the planar circle. Next we deal with the problem in a rotating frame and we show a reacher phenomenology: indeed while for some values of the angular velocity minimizers are still circles, for others the minima of the action are not anymore rigid motions.
The aim of this paper is to study the motion of $2+n$-body problem where two equal masses are assumed to be fixed. We assume that the value of each fixed mass is equal to $M>0$ and the remaining $n$ moving particles have equal masses $m>0$. According to Newton's second law and the universal gravitation law, the $n$ particles move under the interaction of each other and the affection of the two fixed particles. Also, this motion has a natural variational structure. Under the simple choreography constraint, we show that the Lagrangian action functional attains its absolute minimum on a uniform c…
We try to prove the existence of choreography solutions for the $n-$body problem on $S^2$. For the three-body problem, we show the existence of the 8-shape orbit on $S^2$.
In this work we introduce a planar restricted four-body problem where a massless particle moves under the gravitational influence due to three bodies following the eight figure choreography, and we explore some symmetric periodic orbits of this system which turns out to be non autonomous. We demonstrate that certain families of involutions are reversing symmetries of certain N-body problem. We consider a particular case of such families of reversing symmetries and we use it to study both theoretically and numerically certain type of symmetric periodic orbits of this system. The symmetric perio…
We use numerical continuation and bifurcation techniques in a boundary value setting to follow Lyapunov families of periodic orbits. These arise from the polygonal system of $n$ bodies in a rotating frame of reference. When the frequency of a Lyapunov orbit and the frequency of the rotating frame have a rational relationship then the orbit is also periodic in the inertial frame. We prove that a dense set of Lyapunov orbits, with frequencies satisfying a diophantine equation, correspond to choreographies. We present a sample of the many choreographies that we have determined numerically along t…
In this note, we consider the entropy of unit area translation surfaces in the $SL(2, \mathbb R)$ orbits of square tiled surfaces that are the union of squares, where the singularities occur at the vertices and the singularities have a common cone angle. We show that the entropy over such orbits is minimized at those surfaces tiled by equilateral triangles where the singularities occur precisely at the vertices.
For exact symplectic twist maps of the annulus, we etablish a choice of weak K.A.M. solutions $u_c=u(\cdot, c)$ that depend in a Lipschitz-continuous way on the cohomology class $c$. This allows us to make a bridge between weak K.A.M. theory of Fathi, Aubry-Mather theory for semi-orbits as developped by Bangert and existence of backward invariant pseudo-foliations as seen by Katnelson \& Ornstein. We deduce a very precise description of the pseudographs of the weak K.A.M. solutions and many interesting results as --the Aubry-Mather sets are contained in pseudographs that are vertically ordered…
In this paper, we apply the variational method with the Structural Prescribed Boundary Conditions (SPBC) to prove the existence of periodic and quasi-periodic solutions for planar four-body problem with $m_1=m_3$ and $m_2=m_4$. A path $q(t)$ in $[0,T]$ satisfies SPBC if the boundaries $q(0)\in \mathbf{A}$ and $q(T)\in \mathbf{B}$, where $\mathbf{A}$ and $\mathbf{B}$ are two structural configuration spaces in $(\mathbf{R}^2)^4$ and they depend on a rotation angle $θ\in (0,2π)$ and the mass ratio $μ=\frac{m_2}{m_1}\in \mathbf{R}^+$. We show that there is a region $Ω\subseteq (0,2π)\times R^+$ su…
We show the existence of some infinite families of periodic solutions of the planar Newtonian n-body problem --with positive masses-- which are symmetric with respect to suitable actions of finite groups (under a strong--force assumption, or just numerically). The method is by minimizing a discretization of the action functional under symmetry constraints.
This paper introduces a new difference scheme to the difference equations for N-body type problems. To find the non-collision periodic solutions and generalized periodic solutions in multi-radial symmetric constraint for the N-body type difference equations, the variational approach and the method of minimizing the Lagrangian action are adopted and the strong force condition is considered correspondingly, which is an efficient method in studying those with singular potentials. And the difference equation can also be taken into consideration of other periodic solutions with symmetric or choreog…
We continue the variational approach to parabolic trajectories introduced in our previous paper [5], which sees parabolic orbits as minimal phase transitions. We deepen and complete the analysis in the planar case for homogeneous singular potentials. We characterize all parabolic orbits connecting two minimal central configurations as free-time Morse minimizers (in a given homotopy class of paths). These may occur for at most one value of the homogeneity exponent. In addition, we link this threshold of existence of parabolic trajectories with the absence of collisions for all the minimizers of…
We consider periodic and quasi-periodic solutions of the three-body problem with homogeneous potential from the point of view of the equivariant calculus of variations. First, we show that symmetry groups of the Lagrangian action functional can be reduced to groups in a finite explicitly given list, after a suitable change of coordinates. Then, we show that local symmetric minimizers are always collisionless, without any assumption on the group other than the fact that collisions are not forced by the group itself. Moreover, we describe some properties of the resulting symmetric collisionless …
We prove several new results concerning action minimizing periodic orbits of Tonelli Lagrangian systems on an oriented closed surface $M$. More specifically, we show that for every energy larger than the maximal energy of a constant orbit and smaller than or equal to the Mañé critical value of the universal abelian cover, the Lagrangian system admits a minimal boundary, i.e. a global minimizer of the Lagrangian action on the space of smooth boundaries of open sets of $M$. We also extend the celebrated graph theorem of Mather in this context: in the tangent bundle $TM$, the union of the support…
Periodic and quasi-periodic solutions of the n-body problem can be found as minimizers of the Lagrangian action functional restricted to suitable spaces of symmetric paths. The main purpose of this paper is to develop a systematic approach to the equivariant minimization for the three-body problem in the three-dimensional space. First we give a finite complete list of symmetry groups fit to the minimization of the action, with the property that any other symmetry group can be reduced to be isomorphic to one of these representatives. A second step is to prove that the resulting (local and globa…
The concept of sphere of influence of a planet is useful in both the context of impact monitoring of asteroids with the Earth and of the design of interplanetary trajectories for spacecrafts. After reviewing the classical results, we propose a new definition for this sphere that depends on the position and velocity of the small body for given values of the Jacobi constant $C$. Here we compare the orbit of the small body obtained in the framework of the circular restricted three-body problem, with orbits obtained by patching two-body solutions. Our definition is based on an optimisation process…
