On the Equivalence Between Type I Liouville Dynamical Systems in the Plane and the Sphere

Producción CientíficaSeparable 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

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

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

The Lie-Poisson structure of the reduced n-body problem

The classical n-body problem in d-dimensional space is invariant under the Galilean symmetry group. We reduce by this symmetry group using the method of polynomial invariants. As a result we obtain a reduced system with a Lie-Poisson structure which is isomorphic to sp(2n-2), independently of d. The reduction preserves the natural form of the Hamiltonian as a sum of kinetic energy that depends on velocities only and a potential that depends on positions only. Hence we proceed to construct a Poisson integrator for the reduced n-body problem using a splitting method.Comment: 26 pages, 2 figure

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

HIGH SPIN ISOMERIC STATES IN 197Pb

High spin isomeric states in 197Pb have been investigated by in-beam gamma ray technics. These states are produced in (heavy ion, xn) reactions. various experiments performed are summarized in table I

Glitz

The crystal structure of the orthorhombic and tetragonal phases of La(Ba 2-xLax)Cu3-yO 6+x/2-y+ z are determined on twinned crystals. The orthorhombic structure, obtained for low x, is close to the regular Y-Ba-Cu-O type (twin a * b * c-b * a * c), but is highly copper deficient on the Cu(1) site (~ 30 %). The local correlations (ξ ~ 20 Å) between copper atoms and vacancies, as deduced from X-ray diffuse scattering, correspond to a short-range segregation of vacancies in chains. As a consequence of the large amount of defects, these crystals are non-typical semiconductors. The tetragonal structure, x ≃ 0.50, leads to tri-twinned crystals with 90° faulting, a * a * 3 a-a * 3 a * a -3 a * a * a (a, the perovskite lattice constant). In these materials the copper sites are found to be strongly anharmonic. This is due to the disorder introduced by the La-Ba substitution. These crystals are also semiconductors with a T-1/4 activation law for the conductivity which indicates that variable range hopping is expected to set in, a consequence of localization by the disorder

Carbon Nanotubes Synthesized in Channels of Alpo4-5 Single Crystals : First X-Ray Scattering Investigations

Following the synthesis of aligned single-wall carbon nanotubes in the channels of AlPO4-5 zeolite single crystals, we present the first X-ray diffraction and diffuse scattering results. They can be analysed in terms of a partial filling of the zeolite channels by nanotubes with diameter around 4A. The possible selection of only one type of nanotube during the synthesis, due to the constraints imposed by the zeolite host, is discussed.Comment: to appear in Solid State Com

Projective dynamics and first integrals

We present the theory of tensors with Young tableau symmetry as an efficient computational tool in dealing with the polynomial first integrals of a natural system in classical mechanics. We relate a special kind of such first integrals, already studied by Lundmark, to Beltrami's theorem about projectively flat Riemannian manifolds. We set the ground for a new and simple theory of the integrable systems having only quadratic first integrals. This theory begins with two centered quadrics related by central projection, each quadric being a model of a space of constant curvature. Finally, we present an extension of these models to the case of degenerate quadratic forms.Comment: 39 pages, 2 figure