Coordinate system

In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine the position of the points or other geometric elements on a manifold such as Euclidean space. The order of the coordinates is significant, and they are sometimes identified by their position in an ordered tuple and sometimes by a letter, as in "the x-coordinate". The coordinates are taken to be real numbers in elementary mathematics, but may be complex numbers or elements of a more abstract system such as a commutative ring. The use of a coordinate system allows problems in geometry to be translated into problems about numbers and vice versa; this is the basis of analytic geometry. Common coordinate systems Number line Number line The simplest example of a coordinate system is the identification of points on a line with real numbers using the number line. In this system, an arbitrary point O (the origin) is chosen on a given line. The coordinate of a

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PHYS-101(g): General physics : mechanics

Le but du cours de physique générale est de donner à l'étudiant les notions de base nécessaires à la compréhension des phénomènes physiques. L'objectif est atteint lorsque l'étudiant est capable de prévoir quantitativement les conséquences de ces phénomènes avec des outils théoriques appropriés.

PHYS-427: Relativity and cosmology I

Introduce the students to general relativity and its classical tests.

PHYS-100: Advanced physics I (mechanics)

La Physique Générale I (avancée) couvre la mécanique du point et du solide indéformable. Apprendre la mécanique, c'est apprendre à mettre sous forme mathématique un phénomène physique, en modélisant la situation et appliquant les lois de la physique.

Euclidean space

Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, it was the three-dimensional space of Euclidean geometry, but i

Cartesian coordinate system

In geometry, a Cartesian coordinate system (UKkɑːrˈtiːzjən, USkɑːrˈtiʒən) in a plane is a coordinate system that specifies each point uniquely by a pair of real numbers called coordinates, which ar

Geometry

Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest b

Shape and deformation measurements of large objects by fringe projection

Anne-Isabelle Desmangles

The height and/or deformation of a surface of an object (10) defined in terms of x, y and z coordinates is measured by projecting a divergent beam of light 21) onto the surface to produce a periodic pattern of fringes (30. A CCD camera (40) views the surface from another angle and images the reflected light. Phase information is extracted from at least one image to produce an optical print (36) of the object containing phase information and image-coordinates information both being related to camera pixel position, the image-coordinates being associated with phase information corresponding to the object coordinates. A processor (50) calculates the object-coordinates of the surface points through mathematical functions describing x, y and z. These mathematical functions take account of the spatial configuration and specifications of the system which are established by calibration procedures involving eg. measurements of a theodolite (60) .

2004

Stability of Equilibria for the so(4) Free Rigid Body

Tudor Ratiu, Murat Turhan

The stability for all generic equilibria of the Lie-Poisson dynamics of the so(4) rigid body dynamics is completely determined. It is shown that for the generalized rigid body certain Cartan subalgebras (called of coordinate type) of so(n) are equilibrium points for the rigid body dynamics. In the case of so(4) there are three coordinate type Cartan subalgebras whose intersection with a regular adjoint orbit gives three Weyl group orbits of equilibria. These coordinate type Cartan subalgebras are the analogues of the three axes of equilibria for the classical rigid body in so(3). In addition to these coordinate type Cartan equilibria there are others that come in curves.

2012

On A Class Of Mean Field Solutions Of The Monge Problem For Perfect And Self-Interacting Systems

Philippe Choquard

The Monge problem (Monge 1781; Taton 1951), as reformulated by Kantorovich (2006a, 2006b) is that of the transportation at a minimum "cost" of a given mass distribution from an initial to a final position during a given time interval. It is an optimal transport problem (Villani, 2003, sects. 1, 2). Following the fluid mechanical solution provided by Benamou and Brenier (2000) for quadratic cost functions (Villani, 2003, sects. 5.4, 8.1), Lagrangian formulations are needed to solve this boundary value problem in time and to determine the Actions as time integral of Lagrangians that are measures of the "cost" of the transportations (Benamou and Brenier, 2000, prop. 1.1). Given canonical Hamilltonians of perfect and self-interacting systems expressed in function of mass densities and velocity potentials, four versions of explicit constructions of Lagrangians, with their corresponding generalized coordinates, are proposed: elimination of the velocity potentials as a function of the densities and their time derivatives by inversion of the continuity equations; elimination of the gradient of the velocity potentials from the continuity equations thanks to the introduction of vector fields such that their divergences give the mass densities; generalization in nD of Gelfand mass coordinate (1963) by the introduction of n-dimensional vector fields such that the determinant of their Jacobian matrices give the mass densities; and, last, introduction of the Lagrangian coordinates that describe the characteristics of the different models and are parametrized by the former auxiliary vector fields. Using this version, weak solutions of several models of Coulombian and Newtonian systems known in Plasma Physics and in Cosmology, with spherically symmetric boundary densities, are given as illustrations.

2010