Coordinate Transformations: Definitions of Vectors and Tensors

2020 ◽  
pp. 51-71
Author(s):  
Otto D. L. Strack
Keyword(s):  
Coordinate Transformations

Poisson Brackets & Angular Momentum

2018 ◽  
Author(s):  
Peter Mann
Keyword(s):  
Generating Functions ◽  
First Integrals ◽  
Lagrangian Mechanics ◽  
Poisson Brackets ◽  
Coordinate Transformations ◽  
Canonical Transformations ◽  
Gauge Transformations ◽  
The Third ◽  
Point Transformations ◽  
Canonical Coordinate

This chapter discusses canonical transformations and gauge transformations and is divided into three sections. In the first section, canonical coordinate transformations are introduced to the reader through generating functions as the extension of point transformations used in Lagrangian mechanics, with the harmonic oscillator being used as an example of a canonical transformation. In the second section, gauge theory is discussed in the canonical framework and compared to the Lagrangian case. Action-angle variables, direct conditions, symplectomorphisms, holomorphic variables, integrable systems and first integrals are examined. The third section looks at infinitesimal canonical transformations resulting from functions on phase space. Ostrogradsky equations in the canonical setting are also detailed.

Fitting Weibull Strength Data and Applying It to Stochastic Mechanical Design

1992 ◽  
Vol 114 (1) ◽  
pp. 35-41 ◽  
Author(s):  
C. R. Mischke
Keyword(s):  
Mechanical Design ◽  
Least Square ◽  
Stochastic Methods ◽  
Coordinate Transformations ◽  
Strength Data ◽  
The Third ◽  
Tension Tests ◽  
Property Data ◽  
Simple Tension ◽  
Best Fit

This is the second paper in a series relating to stochastic methods in mechanical design. The first is entitled, “Some Property Data and Corresponding Weibull Parameters for Stochastic Mechanical Design,” and the third, “Some Stochastic Mechanical Design Applications.” When data are sparse, many investigators prefer employing coordinate transformations to rectify the data string, and a least-square regression to seek the best fit. Such an approach introduces some bias, which the method presented here is intended to reduce. With mass-produced products, extensive testing can be carried out and prototypes built and evaluated. When production is small, material testing may be limited to simple tension tests or perhaps none at all. How should a designer proceed in order to achieve a reliability goal or to assess a design to see if the goal has been realized? The purpose of this paper is to show how sparse strength data can be reduced to distributional parameters with less bias and how such information can be used when designing to a reliability goal.

On the factorization of rotations with examples in diffractometry

1990 ◽  
Vol 428 (1875) ◽  
pp. 451-472 ◽  
Keyword(s):  
Rotation Angle ◽  
Rotation Axis ◽  
Axial Direction ◽  
The Other ◽  
Rotation Vector ◽  
Coordinate Transformations ◽  
Rotation Axes ◽  
Analytic Expressions ◽  
Direction Cosines

An analysis of compound rotations, such as occur in eulerian cradles, is presented in terms of a calculus of rotation axes, without reference to the associated coordinate transformations. The general case of three rotation shafts mounted on one another, with any relation between them at datum zero, is presented. The problem and its solution may be represented entirely in terms of a plane octagon in which four sides have directions that are instrumental constants and the other four sides have lengths that are instrumental constants. When the first four sides are given lengths that express both the rotation angle and the axial direction of the required rotation, then the remaining four sides have directions that directly express the rotations in the drive shafts, that will generate the required rotation. Analytic expressions are given for the shaft setting angles in the general case. If the first and third axes are parallel and the intermediate one perpendicular to these at datum zero (as in the four-circle diffractometer) then these reduce to θ 1 = arctan ( μ, σ ) + [arctan ( λ , v ) - ψ -½8π], θ 2 = 2 s arcsin ( λ 2 + v 2 )½, θ 3 = ( μ, σ ) - [arctan ( λ , v ) - ψ - ½8π], s = ± 1, 0 ≤ arcsin ( λ 2 + v 2)½ ≤ ½π, in which λ, μ, v and σ are the four components of a rotation vector constructed such that λ, μ and v are the direction cosines of the rotation axis multiplied by sin½ θ for a rotation angle θ and σ is cos½ θ . ψ is a constant determined by the choice of directions to which λ and v are measured. The results for the general case are also expressed in terms of more conventional variables.

scholarly journals GRAVITATIONAL FIELD OF SPHERICAL BRANES

2008 ◽  
Vol 23 (35) ◽  
pp. 2979-2986
Author(s):  
MERAB GOGBERASHVILI
Keyword(s):  
Black Holes ◽  
Gravitational Field ◽  
Analytic Form ◽  
Gravitational Mass ◽  
Moving Frame ◽  
Coordinate Transformations ◽  
Einstein's Equations ◽  
Five Dimensions ◽  
Physical Fields ◽  
Active Gravitational Mass

The warped solution of Einstein's equations corresponding to the spherical brane in five-dimensional AdS is considered. This metric represents interiors of black holes on both sides of the brane and can provide gravitational trapping of physical fields on the shell. It is found that the analytic form of the coordinate transformations from the Schwarzschild to co-moving frame that exists only in five dimensions. It is shown that in the static coordinates active gravitational mass of the spherical brane, in agreement with Tolman's formula, is negative, i.e. such objects are gravitationally repulsive.

scholarly journals On the classification of isotropic tensors

1987 ◽  
Vol 29 (2) ◽  
pp. 185-196 ◽  
Author(s):  
P. G. Appleby ◽  
B. R. Duffy ◽  
R. W. Ogden
Keyword(s):  
Linear Group ◽  
General Linear Group ◽  
Final Section ◽  
Complete Classification ◽  
Coordinate Transformations ◽  
Tensor Fields ◽  
Isotropic Tensor ◽  
Unimodular Group ◽  
Derivatives Of

A tensor is said to be isotropic relative to a group of transformations if its components are invariant under the associated group of coordinate transformations. In this paper we review the classification of tensors which are isotropic under the general linear group, the special linear (unimodular) group and the rotational group. These correspond respectively to isotropic absolute tensors [4, 8] isotropic relative tensors [4] and isotropic Cartesian tensors [3]. New proofs are given for the representation of isotropic tensors in terms of Kronecker deltas and alternating tensors. And, for isotropic Cartesian tensors, we provide a complete classification, clarifying results described in [3].In the final section of the paper certain derivatives of isotropic tensor fields are examined.

scholarly journals The relativity of color

2015 ◽  
Vol 30 (35) ◽  
pp. 1550211 ◽  
Author(s):  
Paul D. Stack ◽  
Robert Delbourgo
Keyword(s):  
Space Time ◽  
Coordinate Transformations ◽  
Gauge Transformations ◽  
Time Property ◽  
Property Space ◽  
Lorentz Scalar ◽  
Color Property

By attaching three anticommuting Lorentz scalar (color) property coordinates to space–time, with an appropriate extended metric, we unify gravity with chromodynamics: gauge transformations then just correspond to coordinate transformations in the enlarged space–time-property space.

Power-based control: Canonical coordinate transformations, integral and adaptive control

Automatica ◽  
2012 ◽  
Vol 48 (6) ◽  
pp. 1045-1056 ◽  
Author(s):  
Daniel A. Dirksz ◽  
Jacquelien M.A. Scherpen
Keyword(s):  
Adaptive Control ◽  
Coordinate Transformations ◽  
Canonical Coordinate
Sign in / Sign up

Export Citation Format

Share Document