Vivina Barutello

Action minimizing orbits in the n-body problem with simple choreography constraint

In 1999 Chenciner and Montgomery found a remarkably simple choreographic motion for the planar three-body problem (see [11]). In this solution, three equal masses travel on an figure-of-eight shaped planar curve; this orbit is obtained by 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 under an attractive force generated by a potential of the kind 1/rα, α > 0, with the only constraint to be a simple choreography: if q1(t),...,qn(t) are the n orbits then we impose the existence of such that where τ = 2π/n. In this setting, we first prove that for every and α > 0, the Lagrangian action attains its absolute minimum on the planar regular n-gon relative equilibrium. Next, we deal with the problem in a rotating frame and show a richer phenomenology: indeed, while for some values of the angular velocity, the minimizers are still relative equilibria, for others, the minima of the action are no longer rigid motions.

A note on the radial solutions for the supercritical Hénon equation

We prove the existence of a positive radial solution for the Henon 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.

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

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

\alpha \in (0, 2)

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

α∈(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.