13 research outputs found
The Kepler Problem with Anisotropic Perturbations
We study a 2-body problem given by the sum of the Newtonian potential and an
anisotropic perturbation that is a homogeneous function of degree ,
. For , the sets of initial conditions leading to
collisions/ejections and the one leading to escapes/captures have positive
measure. For and , the flow on the zero-energy manifold
is chaotic. For , a case we prove integrable, the infinity manifold of
the zero-energy level is a disconnected set, which has heteroclinic connections
with the collision manifold
On the convex central configurations of the symmetric (â + 2)-body problem
For the 4-body problem there is the following conjecture: Given arbitrary positive masses, the planar 4-body problem has a unique convex central configuration for each ordering of the masses on its convex hull. Until now this conjecture has remained open. Our aim is to prove that this conjecture cannot be extended to the (â + 2)-body problem with â â©Ÿ 3. In particular, we prove that the symmetric (2n + 1)-body problem with masses m1 = ⊠= m2nâ1 = 1 and m2n = m2n+1 = m sufficiently small has at least two classes of convex central configuration when n = 2, five when n = 3, and four when n = 4. We conjecture that the (2n + 1)-body problem has at least n classes of convex central configurations for n > 4 and we give some numerical evidence that the conjecture can be true. We also prove that the symmetric (2n + 2)-body problem with masses m1 = ⊠= m2n = 1 and m2n+1 = m2n+2 = m sufficiently small has at least three classes of convex central configuration when n = 3, two when n = 4, and three when n = 5. We also conjecture that the (2n + 2)-body problem has at least [(n +1)/2] classes of convex central configurations for n > 5 and we give some numerical evidences that the conjecture can be true
Linear stability of the Lagrangian triangle solutions for quasihomogeneous potentials
In this paper we study the linear stability of the relative equilibria for
homogeneous and quasihomogeneous potentials. Firstly, in the case the potential
is a homogeneous function of degree , we find that any relative equilibrium
of the -body problem with is spectrally unstable. We also find a
similar condition in the quasihomogeneous case. Then we consider the case of
three bodies and we study the stability of the equilateral triangle relative
equilibria. In the case of homogeneous potentials we recover the classical
result obtained by Routh in a simpler way. In the case of quasihomogeneous
potentials we find a generalization of Routh inequality and we show that, for
certain values of the masses, the stability of the relative equilibria depends
on the size of the configuration.Comment: 21 pages 4 figure