In the planar N-body problem, N point masses move in the plane under their mutual gravitational attraction. It is classical that the dynamics of this motion conserves the intgrals of motion: center of mass, linear momentum, angular momentum c, and energy h. Further, the motion has a rotational symmetry. The dynamics thus takes place on a (4N?7)-dimensional open manifold, known as the reduced in...

Let M be a compact k-dimensional riemmanian manifold minimally immersed in the unit n-dimensional sphere S. It is easy to show that for any p ∈ S the boundary of the geodesic ball in S with radius π2 and center at p (in this case this boundary is an equator) must intercept the manifold M . When the codimension is 1, i.e. k = n − 1, it is known that the ricci curvature, is not greater than 1. We...

By means of a center manifold analysis we investigate the averaged mean curvature flow near spheres. In particular, we show that there exist global solutions to this flow starting from non-convex initial hypersurfaces.

The present work studies the robustness of certain basic homoclinic motions in an equilateral restricted four-body problem. problem can be viewed as a two-parameter family conservative autonomous vector fields. main tools are numerical continuation techniques for and periodic orbits, well formal series methods computing normal forms center stable/unstable manifold parameterizations. After caref...