525 research outputs found
On the Equivalence Between Type I Liouville Dynamical Systems in the Plane and the Sphere
ProducciĂłn CientĂficaSeparable 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
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
Kustaanheimo-Stiefel Regularization and the Quadrupolar Conjugacy
In this note, we present the Kustaanheimo-Stiefel regularization in a
symplectic and quaternionic fashion. The bilinear relation is associated with
the moment map of the - action of the Kustaanheimo-Stiefel
transformation, which yields a concise proof of the symplecticity of the
Kustaanheimo-Stiefel transformation symplectically reduced by this circle
action. The relation between the Kustaanheimo-Stiefel regularization and the
Levi-Civita regularization is established via the investigation of the
Levi-Civita planes. A set of Darboux coordinates (which we call
Chenciner-F\'ejoz coordinates) is generalized from the planar case to the
spatial case. Finally, we obtain a conjugacy relation between the integrable
approximating dynamics of the lunar spatial three-body problem and its
regularized counterpart, similar to the conjugacy relation between the extended
averaged system and the averaged regularized system in the planar case.Comment: 19 pages, corrected versio
Pauli graphs, Riemann hypothesis, Goldbach pairs
Let consider the Pauli group with unitary quantum
generators (shift) and (clock) acting on the vectors of the
-dimensional Hilbert space via and , with
. It has been found that the number of maximal mutually
commuting sets within is controlled by the Dedekind psi
function (with a prime)
\cite{Planat2011} and that there exists a specific inequality , involving the Euler constant , that is only satisfied at specific low dimensions . The set is closely related to
the set of integers that are totally Goldbach, i.e.
that consist of all primes ) is equivalent to Riemann hypothesis.
Introducing the Hardy-Littlewood function (with the twin prime constant),
that is used for estimating the number of
Goldbach pairs, one shows that the new inequality is also equivalent to Riemann hypothesis. In this paper,
these number theoretical properties are discusssed in the context of the qudit
commutation structure.Comment: 11 page
Qudits of composite dimension, mutually unbiased bases and projective ring geometry
The Pauli operators attached to a composite qudit in dimension may
be mapped to the vectors of the symplectic module
( the modular ring). As a result, perpendicular vectors
correspond to commuting operators, a free cyclic submodule to a maximal
commuting set, and disjoint such sets to mutually unbiased bases. For
dimensions , and 18, the fine structure and the incidence
between maximal commuting sets is found to reproduce the projective line over
the rings , , ,
and ,
respectively.Comment: 10 pages (Fast Track communication). Journal of Physics A
Mathematical and Theoretical (2008) accepte
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
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