1,237 research outputs found
On the Equivalence Between Type I Liouville Dynamical Systems in the Plane and the Sphere
Producción CientÃficaSeparable Hamiltonian systems either in sphero-conical coordinates on an S2 sphere or in elliptic coordinates on a R2 plane are described in a unified way. A back and forth route connecting these Liouville Type I separable systems is unveiled. It is shown how the gnomonic projection and its inverse map allow us to pass from a Liouville Type I separable system with a spherical configuration space
to its Liouville Type I partners where the configuration space is a plane and back. Several selected spherical separable systems and their planar cousins are discussed in a classical context
Some Problems on the Classical N-Body Problem
Our idea is to imitate Smale's list of problems, in a restricted domain of
mathematical aspects of Celestial Mechanics. All the problems are on the n-body
problem, some with different homogeneity of the potential, addressing many
aspects such as central configurations, stability of relative equilibrium,
singularities, integral manifolds, etc. Following Steve Smale in his list, the
criteria for our selection are: (1) Simple statement. Also preferably
mathematically precise, and best even with a yes or no answer. (2) Personal
acquaintance with the problem, having found it not easy. (3) A belief that the
question, its solution, partial results or even attempts at its solution are
likely to have great importance for the development of the mathematical aspects
of Celestial Mechanics.Comment: 10 pages, list of mathematical problem
Relative Equilibria in the Four-Vortex Problem with Two Pairs of Equal Vorticities
We examine in detail the relative equilibria in the four-vortex problem where
two pairs of vortices have equal strength, that is, \Gamma_1 = \Gamma_2 = 1 and
\Gamma_3 = \Gamma_4 = m where m is a nonzero real parameter. One main result is
that for m > 0, the convex configurations all contain a line of symmetry,
forming a rhombus or an isosceles trapezoid. The rhombus solutions exist for
all m but the isosceles trapezoid case exists only when m is positive. In fact,
there exist asymmetric convex configurations when m < 0. In contrast to the
Newtonian four-body problem with two equal pairs of masses, where the symmetry
of all convex central configurations is unproven, the equations in the vortex
case are easier to handle, allowing for a complete classification of all
solutions. Precise counts on the number and type of solutions (equivalence
classes) for different values of m, as well as a description of some of the
bifurcations that occur, are provided. Our techniques involve a combination of
analysis and modern and computational algebraic geometry
Projective dynamics and classical gravitation
Given a real vector space V of finite dimension, together with a particular
homogeneous field of bivectors that we call a "field of projective forces", we
define a law of dynamics such that the position of the particle is a "ray" i.e.
a half-line drawn from the origin of V. The impulsion is a bivector whose
support is a 2-plane containing the ray. Throwing the particle with a given
initial impulsion defines a projective trajectory. It is a curve in the space
of rays S(V), together with an impulsion attached to each ray. In the simplest
example where the force is identically zero, the curve is a straight line and
the impulsion a constant bivector. A striking feature of projective dynamics
appears: the trajectories are not parameterized.
Among the projective force fields corresponding to a central force, the one
defining the Kepler problem is simpler than those corresponding to other
homogeneities. Here the thrown ray describes a quadratic cone whose section by
a hyperplane corresponds to a Keplerian conic. An original point of view on the
hidden symmetries of the Kepler problem emerges, and clarifies some remarks due
to Halphen and Appell. We also get the unexpected conclusion that there exists
a notion of divergence-free field of projective forces if and only if dim V=4.
No metric is involved in the axioms of projective dynamics.Comment: 20 pages, 4 figure
Rosette Central Configurations, Degenerate central configurations and bifurcations
In this paper we find a class of new degenerate central configurations and
bifurcations in the Newtonian -body problem. In particular we analyze the
Rosette central configurations, namely a coplanar configuration where
particles of mass lie at the vertices of a regular -gon, particles
of mass lie at the vertices of another -gon concentric with the first,
but rotated of an angle , and an additional particle of mass lies
at the center of mass of the system. This system admits two mass parameters
and \ep=m_2/m_1. We show that, as varies, if ,
there is a degenerate central configuration and a bifurcation for every
\ep>0, while if there is a bifurcations only for some values of
.Comment: 16 pages, 6 figure
Symmetry, bifurcation and stacking of the central configurations of the planar 1+4 body problem
In this work we are interested in the central configurations of the planar
1+4 body problem where the satellites have different infinitesimal masses and
two of them are diametrically opposite in a circle. We can think this problem
as a stacked central configuration too. We show that the configuration are
necessarily symmetric and the other sattelites has the same mass. Moreover we
proved that the number of central configuration in this case is in general one,
two or three and in the special case where the satellites diametrically
opposite have the same mass we proved that the number of central configuration
is one or two saying the exact value of the ratio of the masses that provides
this bifurcation.Comment: 9 pages, 2 figures. arXiv admin note: text overlap with
arXiv:1103.627
Seven-body central configurations: a family of central configurations in the spatial seven-body problem
The main result of this paper is the existence of a new family of central
configurations in the Newtonian spatial seven-body problem. This family is
unusual in that it is a simplex stacked central configuration, i.e the bodies
are arranged as concentric three and two dimensional simplexes.Comment: 15 pages 5 figure
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