Morse Index and Linear Stability of the Lagrangian Circular Orbit in a Three-Body-Type Problem Via Index Theory

It is well known that the linear stability of the Lagrangian elliptic solutions in the classical planar three-body problem depends on a mass parameter β and on the eccentricity e of the orbit. We consider only the circular case (e = 0) but under the action of a broader family of singular potentials: α-homogeneous potentials, for \(\alpha \in (0, 2)\), and the logarithmic one. It turns out indeed that the Lagrangian circular orbit persists also in this more general setting. We discover a region of linear stability expressed in terms of the homogeneity parameter α and the mass parameter β, then we compute the Morse index of this orbit and of its iterates and we find that the boundary of the stability region is the envelope of a family of curves on which the Morse indices of the iterates jump. In order to conduct our analysis we rely on a Maslov-type index theory devised and developed by Y. Long, X. Hu and S. Sun; a key role is played by an appropriate index theorem and by some precise computations of suitable Maslov-type indices.     

Linear instability of relative equilibria for n-body problems in the plane

Following Smale, we study simple symmetric mechanical systems of n point particles in the plane. In particular, we address the question of the linear and spectral stability properties of relative equilibria, which are special solutions of the equations of motion.     

A New Branch of Mountain Pass Solutions for the Choreographical 3-Body Problem

We prove the existence of a new branch of solutions of Mountain Pass type for the periodic 3-body problem with choreographical constraint. At first we describe the variational structure of the action functional associated to the choreographical three body problem in . In the second part, using a bisection algorithm, we provide a numerical non-rigorous solution of Mountain Pass type for this problem in a rotating frame with angular velocity 1.5. The last step consists in the rigorous computer-assisted proof of the existence of a full branch of solutions for the problem starting from the Mountain Pass solution detected numerically. Electronic Supplementary Material Supplementary material is available for this article at and is accessible for authorized users. This work was supported by the MIUR project “Metodi Variazionali ed Equazioni Differenziali non Lineari.”     

Symmetry Groups of the Planar Three-Body Problem and Action-Minimizing Trajectories

We consider periodic and quasi-periodic solutions of the three-body problem with homogeneous potential from the point of view of 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 minimizers (Lagrange, Euler, Hill-type orbits and Chenciner–Montgomery figure-eights).     

Morse index properties of colliding solutions to the N-body problem

We study a singular Hamiltonian system with an α-homogeneous potential that contains, as a particular case, the classical N-body problem. We introduce a variational Morse-like index for a class of collision solutions and, using the asymptotic estimates near collisions, we prove the non-minimality of some special classes of colliding trajectories under suitable spectral conditions provided α is sufficiently away from zero. We then prove some minimality results for small values of the parameter α.     

Entire parabolic trajectories as minimal phase transitions

For the class of anisotropic Kepler problems in $\mathbb{R }^d\setminus \{0\}$ with homogeneous potentials, we seek parabolic trajectories having prescribed asymptotic directions at infinity and which, in addition, are Morse minimizing geodesics for the Jacobi metric. Such trajectories correspond to saddle heteroclinics on the collision manifold, are structurally unstable and appear only for a codimension-one submanifold of such potentials. We give them a variational characterization in terms of the behavior of the parameter-free minimizers of an associated obstacle problem. We then give a full characterization of such a codimension-one manifold of potentials and we show how to parameterize it with respect to the degree of homogeneity.     

Entire Minimal Parabolic Trajectories: The Planar Anisotropic Kepler Problem

We continue the variational approach to parabolic trajectories introduced in our previous paper (Barutello et al., Entire parabolic trajectories as minimal phase transitions. arXiv:1105.3358v1, 2011), 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 fixed-end problems, and also with the existence of action minimizing periodic trajectories with nontrivial homotopy type.     

A bisection algorithm for the numerical Mountain Pass

We propose a constructive proof for the Ambrosetti-Rabinowitz Mountain Pass Theorem providing an algorithm, based on a bisection method, for its implementation. The efficiency of our algorithm, particularly suitable for problems in high dimensions, consists in the low number of flow lines to be computed for its convergence; for this reason it improves the one currently used and proposed by Y.S. Choi and P.J. McKenna in [3]. Susanna Terracini: This work is partially supported by M.I.U.R. project “Metodi Variazionali ed Equazioni Differenziali Nonlineari”.     

A note on the radial solutions for the supercritical Hénon equation

We prove the existence of a positive radial solution for the Hénon equation with arbitrary growth. The solution is found by means of a shooting method and turns out to be an increasing function of the radial variable. Some numerical experiments suggest the existence of many positive oscillating solutions.