Search results for "RIEMANN"

Geodesic flow of the averaged controlled Kepler equation

2008

A normal form of the Riemannian metric arising when averaging the coplanar controlled Kepler equation is given. This metric is parameterized by two scalar invariants which encode its main properties. The restriction of the metric to $\SS^2$ is shown to be conformal to the flat metric on an oblate ellipsoid of revolution, and the associated conjugate locus is observed to be a deformation of the standard astroid. Though not complete because of a singularity in the space of ellipses, the metric has convexity properties that are expressed in terms of the aforementioned invariants, and related to surjectivity of the exponential mapping. Optimality properties of geodesics of the averaged controll…

Non subanalyticity of sub-Riemannian Martinet spheres

2001

Abstract Consider the sub-Riemannian Martinet structure (M,Δ,g) where M= R 3 , Δ= Ker ( d z− y 2 2 d x) and g is the general gradated metric of order 0 : g=(1+αy) 2 d x 2 +(1+βx+γy) 2 d y 2 . We prove that if α≠0 then the sub-Riemannian spheres S(0,r) with small radii are not subanalytic.

Time Versus Energy in the Averaged Optimal Coplanar Kepler Transfer towards Circular Orbits

2015

International audience; The aim of this note is to compare the averaged optimal coplanar transfer towards circular orbits when the costs are the transfer time transfer and the energy consumption. While the energy case leads to analyze a 2D Riemannian metric using the standard tools of Riemannian geometry (curvature computations, geodesic convexity), the time minimal case is associated to a Finsler metric which is not smooth. Nevertheless a qualitative analysis of the geodesic flow is given in this article to describe the optimal transfers. In particular we prove geodesic convexity of the elliptic domain.

Sub-Riemannian geometry: one-parameter deformation of the Martinet flat case

1998

Integrable Systems and Factorization Problems

2002

The present lectures were prepared for the Faro International Summer School on Factorization and Integrable Systems in September 2000. They were intended for participants with the background in Analysis and Operator Theory but without special knowledge of Geometry and Lie Groups. In order to make the main ideas reasonably clear, I tried to use only matrix algebras such as $\frak{gl}(n)$ and its natural subalgebras; Lie groups used are either GL(n) and its subgroups, or loop groups consisting of matrix-valued functions on the circle (possibly admitting an extension to parts of the Riemann sphere). I hope this makes the environment sufficiently easy to live in for an analyst. The main goal is…

From finite-gap solutions of KdV in terms of theta functions to solitons and positons

2010

We degenerate the finite gap solutions of the KdV equation from the general formulation in terms of abelian functions when the gaps tends to points, to recover solutions of KdV equations in terms of wronskians called solitons or positons. For this we establish a link between Fredholm determinants and Wronskians.

The Role of Differential Parameters in Beltrami's Work

1997

Abstract Differential parameters play a relevant role in Beltrami's mathematical work. They are employed in different contexts, in order to express some well-known results in a new way and to extend potential theory and the theory of elasticity to a Riemannian manifold. The author aims to show that differential parameters enabled Beltrami to solve many mathematical questions and that they constitute the first step toward the conception of tensor calculus. Les parametres differentiels jouent un role important dans l'oeuvre mathematique de Beltrami. Ils sont employes en contextes differents, pour exprimer dans une maniere nouvelle quelques resultats bien-connus et pour generaliser la theorie …

Optimal control and applications to orbital transfer and almost-riemannian geometry

2010

In this thesis we focus on optimal control techniques as well as geometric control techniques applied to the orbital transfer problem and to almost-Riemannian geometry. In these cases, Pontryagin’s Maximum Principle allows to analyse the extremal flow of affine control systems.In the case of a satellite with low-thrust propulsion, averaging techniques give an approximated system. Averaging is explicit in the energy minimization case and is directly related to almost-riemannian problems. The geometric analysis of such problems is generalized by the study of metrics on the two-sphere of revolution. In this way it is possible to classify the situations considering the transcendance of the solu…

Analyyttinen jatke ja Riemannin pinnat

2014

Tämän tutkielman tavoitteena on esittää, miten analyyttisen funktion määrittelyjoukko laajennetaan Riemannin pinnaksi, joka sisältää informaation kaikista funktion analyyttisistä jatkeista kompleksitasossa. Tätä Riemannin pintaa sanotaan kyseisen analyyttisen funktion Riemannin pinnaksi. Konstruktiota varten täytyy ensin tarkastella analyyttisen jatkeen käyttäytymistä kompleksitasossa, Riemannin pintojen ja kompleksitason välistä yhteyttä sekä perusryhmän ja peitteiden käyttäytymistä Riemannin pintojen kontekstissa. Erityisesti tarkastellaan miten useat kompleksianalyysin perustulokset yleistyvät suoraan Riemannin pinnoille ja Riemannin pintojen analyyttisille funktioille sekä sitä, milloin…

Second order optimality conditions with applications

2007

International audience; The aim of this article is to present the algorithm to compute the first conjugate point along a smooth extremal curve. Under generic assump- tions, the tra jectory ceases to be optimal at such a point. An implementation of this algorithm, called cotcot, is available online and based on recent devel- opments in geometric optimal control. It is applied to analyze the averaged optimal transfer of a satellite between elliptic orbits.