Transitive decomposition of symmetry groups for the nn-body problem

Periodic and quasi-periodic orbits of the $n$-body problem are critical points of the action functional constrained to the Sobolev space of symmetric loops. Variational methods yield collisionless orbits provided the group of symmetries fulfills certain conditions (such as the \emph{rotating circle property}). Here we generalize such conditions to more general group types and show how to constructively classify all groups satisfying such hypothesis, by a decomposition into irreducible transitive components. As examples we show approximate trajectories of some of the resulting symmetric minimizers

On the dihedral n-body problem

Consider n=2l>=4 point particles with equal masses in space, subject to the following symmetry constraint: at each instant they form an orbit of the dihedral group D_l, where D_l is the group of order 2l generated by two rotations of angle pi around two secant lines in space meeting at an angle of pi/l. By adding a homogeneous gravitational (Newtonian) potential one finds a special $n$-body problem with three degrees of freedom, which is a kind of generalisation of Devaney isosceles problem, in which all orbits have zero angular momentum. In the paper we find all the central configurations and we compute the dimension of the stable/unstable manifolds.Comment: Second version. In the first there was a mistake in a proof: some section had been omitte

Stratified fibre bundles

A stratified bundle is a fibered space in which strata are classical bundles and in which attachment of strata is controlled by a structure category of fibers. Well known results on fibre bundles are shown to be true for stratified bundles; namely the pull back theorem, the bundle theorem and the principal bundle theorem.Comment: LaTeX file. Revised version. Accepted for publication on "Forum Mathematicum

K-theory of stratified vector bundles

We show that the Atiyah-Hirzebruch K-theory of spaces admits a canonical generalization for stratified spaces. For this we study algebraic constructions on stratified vector bundles. In particular the tangent bundle of a stratified manifold is such a stratified vector bundle.Comment: LaTeX file, 22 page