Spectral stability, spectral flow and circular relative equilibria for the Newtonian n-body problem

Reduction, Relative Equilibria and Stability for a Gyrostat in the N-body Problem

We consider the non-canonical Hamiltonian dynamics of a gyrostat in the nbody problem. Using the symmetries of the system we carry out a reduction process in two steps, giving explicitly at each step the Poisson structure of the reduced system. Next, we obtain general properties of the relative equilibria of the problem and if we restrict to different approximations of the gravitational potenti...

Central configurations and relative equilibria in the n–body problem

Stability of Relative Equilibria in the Planar n-Vortex Problem

We study the linear and nonlinear stability of relative equilibria in the planar n-vortex problem, adapting the approach of Moeckel from the corresponding problem in celestial mechanics. After establishing some general theory, a topological approach is taken to show that for the case of positive circulations, a relative equilibrium is linearly stable if and only if it is a nondegenerate minimum...

Singularities in the Newtonian n-body Problem

In this note, we study the structure of singularities of the Newtonian n-body problem where we establish certain relations between collisions and noncollision singularities. Based on our results, we conjecture that the set of all accumulation points of collisions in the extended phase space is exactly the set of initial conditions that lead to a noncollision singularity.

Bifurcations and Stability of Lagrangian Relative Equilibria for a Gyrostat in the Three Body Problem

In this paper we consider the non–canonical Hamiltonian dynamics of a gyrostat in the frame of the three body problem. Using geometric/mechanics methods we study the approximate dynamics of the truncated Legendre series representation of the potential of an arbitrary order. Working in the reduced problem, we study the existence of relative equilibria that we call of Lagrange type following the ...