109 research outputs found
Central configurations of three nested regular polyhedra for the spatial 3n–body problem
Three regular polyhedra are called nested if they have the same number of vertices
n, the same center and the positions of the vertices of the inner polyhedron ri, the
ones of the medium polyhedron Ri and the ones of the outer polyhedron Ri satisfy
the relation Ri = ri and Ri = Rri for some scale factors R > > 1 and for all
i = 1, . . . , n. We consider 3n masses located at the vertices of three nested regular
polyhedra. We assume that the masses of the inner polyhedron are equal to m1,
the masses of the medium one are equal to m2, and the masses of the outer one
are equal to m3. We prove that if the ratios of the masses m2/m1 and m3/m1 and
the scale factors and R satisfy two convenient relations, then this configuration is
central for the 3n–body problem. Moreover there is some numerical evidence that,
first, fixed two values of the ratios m2/m1 and m3/m1, the 3n–body problem has
a unique central configuration of this type; and second that the number of nested
regular polyhedra with the same number of vertices forming a central configuration
for convenient masses and sizes is arbitrary
Central configurations of nested rotated regular tetrahedra
In this paper we prove that there are only two different classes of central configura-
tions with convenient masses located at the vertices of two nested regular tetrahedra:
either when one of the tetrahedra is a homothecy of the other one, or when one of
the tetrahedra is a homothecy followed by a rotation of Euler angles =
= 0 and
= of the other one.
We also analyze the central configurations with convenient masses located at the
vertices of three nested regular tetrahedra when one them is a homothecy of the
other one, and the third one is a homothecy followed by a rotation of Euler angles
=
= 0 and = of the other two.
In all these cases we have assumed that the masses on each tetrahedron are equal
but masses on different tetrahedra could be different
On centered co-circular central configurations of the n-body problem
We study the co-circular central configurations of the n-body problem for which the center of mass and the center of the common circle coincide. In particular, we prove that there are no central configurations of this type with all the masses equal except one. This provides more evidences for the veracity of the conjecture that the regular n-gon with equal masses is the unique co-circular central configuration of the n-body problem whose center of mass is the center of the circle. Our result remains valid if we consider power-law potentials
Survey on central configurations related with regular polyhedra
This paper is a survey on the known results about central configurations related
with regular polyhedra
Global phase portraits of ℤ2-symmetric planar polynomial Hamiltonian systems of degree three with a nilpotent saddle at the origin
We characterize the phase portraits in the Poincaré disk of all planar polynomial Hamiltonian systems of degree three with a nilpotent saddle at the origin and ℤ2-symmetric with (x,y)→(-x,y)
Generation of symmetric periodic orbits by a heteroclinic loop formed by two singular points and their invariant manifolds of dimension 1 and 2 in R3
In this paper we will find a continuous of periodic orbits passing
near infinity for a class of polynomial vector fields in R3. We consider
polynomial vector fields that are invariant under a symmetry with
respect to a plane and that possess a “generalized heteroclinic loop”
formed by two singular points e+ and e− at infinity and their invariant
manifolds � and . � is an invariant manifold of dimension 1 formed
by an orbit going from e− to e+, � is contained in R3 and is transversal
to . is an invariant manifold of dimension 2 at infinity. In fact, is
the 2–dimensional sphere at infinity in the Poincar´e compactification
minus the singular points e+ and e−. The main tool for proving the
existence of such periodic orbits is the construction of a Poincar´e map
along the generalized heteroclinic loop together with the symmetry
with respect to
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