• shooting
• periodic orbit
• N body problem
• stability
• platonic polyhedra

In the last few years, many interesting periodic motions of the classical Newtonian
N-body problem have been discovered as minimizers of the Lagrangian action functional
on a particular subset of T-periodic loops. The
interest in this classical problem was revived by the numerical discovery of the now famous
figure eight solution of the three-body problem, by C. Moore in 1993.
In 2000 A. Chenciner and R. Montgomery rediscover this particular orbit, giving a formal
proof of its existence that uses the direct method of Calculus of Variations. The figure eight is a first
example of a N-body choreography, that is a solution of the classical N-body problem
in which N equal masses chase each other around a fixed closed curve, equally spaced in
phase along the curve: since 2000 many other choreographies that present a strong
symmetrical structure have been found. Moreover, in 2007 T. Kapela and C. Simò proved the linear stability of the figure eight: this fact is quite surprising, since that no other stable
choreographies are known.

In this thesis we prove the existence of a number of
periodic motions of the classical N-body problem which, up to relabeling of the
N particles, are invariant under the rotation group R of one of the five
Platonic polyhedra. The number N coincides with the order of the rotation group
R and the particles have all the same mass. We use again variational
techniques to minimize the Lagrangian action A on a suitable subset
K of the H^1 T-periodic maps. The set K is a cone and is
determined by imposing on $u$ both topological and symmetry constraints which are defined
in terms of the rotation group R. For a certain number of cones
K, using level estimates and local perturbations, we show that minimizers are
free of collisions and therefore they are classical solutions on the N-body problem.

A natural question that comes out in presence of a periodic orbit is whether is it stable
or not. To perform a study of the linear stability we use numerical methods, since our
problem is not integrable. In fact we know only that periodic orbits with the previous
symmetries exist, but we do not have their analytic
expression. These particular solutions were found numerically,
using a method described by C. Moore and called relaxation dynamics.
The numerical implementation of this method boils down to a gradient search of the minima in some
finite-dimensional approximation of the path space: in short, it is a numerical
implementation of a direct method of Calculus of Variations. Starting from these numerical solutions,
we can propagate numerically the variational equation in order to produce an
approximation of the monodromy matrix, from which we can determine the
linear stability studying its spectral properties: this is a first method that we
develop. However, because of the convergence of the gradient search is slow, especially when the orbit presents some close approaches,
this method could result inefficient. An alternative approach is to find an initial
condition of the periodic orbit and then propagate numerically the equation of motion and the
variational equation coupled together. A classical method to find initial conditions is
the well known multiple shooting method. This method has been successfully used by T.
Kapela and C. Simò to find
initial conditions for the figure eight and some other non-symmetric choreographies.
However, since this is an iterative method too, it could fail to converge and
this typically happens when the orbit passes close to a collisions. Therefore, it is clear that
the problem of close approaches must be treated with more care.

The thesis is structured as follows:
Chapter 1. It contains results on the existence of periodic orbits of the
classical N-body problem with the symmetries of Platonic polyhedra.
Chapter 2. In this chapter we try to develop an automatic procedure in order to
find all the periodic orbits described in Chapter 1.
Chapter 3. We present the
classical theory of linear stability for periodic solutions of autonomous systems.
In particular, we introduce here the monodromy matrix, the Floquet multipliers and
the Poincaré map.
Chapter 4. It is the heart of the work, in which we develop the two different
numerical methods to study the linear stability of periodic orbits found in Chapters 1 and
2. Tests of the software written are reported at the end of the chapter.
Chapter 5. In this chapter we list all the results obtained with our software,
from which we can get some conclusions. At the end we suggest some improvements of
the methods of Chapter 4 and of the software, which could represent a continuation
of the present work, in order to produce a true computer assisted proof of stability
or instability of these orbits.