Coordinate systems generalized coordinates

Atomic coordinates A set of numbers that defines the position of an atom with respect to a specified coordinate system. Atomic coordinates are generally expressed as the dimensionless quantities x, y, and 2 which are fractions of unit cell edges. 

However, in proper scientific terms there is only one way to precisely describe the structure of an object, be it simple, or intricate and complex. That is by specifying, as in Figure 1.1, the coordinates in three-dimensional space of each point within the object, each with respect to some defined and agreed-upon system of axes in space, namely a coordinate system. Generally, the system is chosen to be an orthogonal, Cartesian coordinate system, but it need not be. It may be nonorthogonal, cylindrical, spherical, or any number of other systems. 

One of the many achievements of Einstein s general relativity was to geometrize gravitational theory. This geometrization consists in the first instance therein that one views the world of physical events as a space-time continuum in four dimensions. Such a continuum is, by definition, represented by a coordinate system. A coordinate system is simply a mapping of a class of world points on a class of four-fold numbers, or what may be called number points x, , x ). 

The coordinate system generally adopted to describe the stress field around the crack is shown in Figure A7-4. 

For the interaction between a nonlinear molecule and an atom, one can place the coordinate system at the centre of mass of the molecule so that the PES is a fiinction of tlie three spherical polar coordinates needed to specify the location of the atom. If the molecule is linear, V does not depend on <() and the PES is a fiinction of only two variables. In the general case of two nonlinear molecules, the interaction energy depends on the distance between the centres of mass, and five of the six Euler angles needed to specify the relative orientation of the molecular axes with respect to the global or space-fixed coordinate axes. 

Ciccotti G, Ferrario M and Ryckaert J-P 1982 Molecular dynamics of rigid systems in cartesian coordinates. A general formulation Mol. Phys. 47 1253-64... 

Neuhauser D 1992 Reactive scattering with absorbing potentials in general coordinate systems Chem. 

As noted above, the coordinate system is now recognized as being of fimdamental importance for efficient geometry optimization indeed, most of the major advances in this area in the last ten years or so have been due to a better choice of coordinates. This topic is seldom discussed in the mathematical literature, as it is in general not possible to choose simple and efficient new coordinates for an abstract optimization problem. A nonlmear molecule with N atoms and no... 

Until now we have implicitly assumed that our problem is formulated in a space-fixed coordinate system. However, electronic wave functions are naturally expressed in the system bound to the molecule otherwise they generally also depend on the rotational coordinate 4>. (This is not the case for E electronic states, for which the wave functions are invariant with respect to (j> ) The eigenfunctions of the electronic Hamiltonian, v / and v , computed in the framework of the BO approximation ( adiabatic electronic wave functions) for two electronic states into which a spatially degenerate state of linear molecule splits upon bending. 

Next, Euler s angles are employed for deriving the outcome of a general rotation of a system of coordinates [86]. It can be shown that R(k, 0) is accordingly presented as... 

In this chapter the general equations of laminar, non-Newtonian, non-isothermal, incompressible flow, commonly used to model polymer processing operations, are presented. Throughout this chapter, for the simplicity of presentation, vector notations are used and all of the equations are given in a fixed (stationary or Eulerian) coordinate system. 

The practical and computational complications encountered in obtaining solutions for the described differential or integral viscoelastic equations sometimes justifies using a heuristic approach based on an equation proposed by Criminale, Ericksen and Filbey (1958) to model polymer flows. Similar to the generalized Newtonian approach, under steady-state viscometric flow conditions components of the extra stress in the (CEF) model are given a.s explicit relationships in terms of the components of the rate of deformation tensor. However, in the (CEF) model stress components are corrected to take into account the influence of normal stresses in non-Newtonian flow behaviour. For example, in a two-dimensional planar coordinate system the components of extra stress in the (CEF) model are written as... 

Mathematical derivations presented in the following sections are, occasionally, given in the context of one- or two-dimensional Cartesian coordinate systems. These derivations can, however, be readily generalized and the adopted style is to make the explanations as simple as possible. 

In the figure operation (M) represents a one-to-one transformation between the local and global coordinate systems. This in general can be shown as... 

Isoparametric transformation functions between a global coordinate system and local coordinates are, in general, written as... 

Similarly in the absence of body forces the Stokes flow equations for a generalized Newtonian fluid in a two-dimensional (r, 8) coordinate system are written as... 

Figure 5.21 Thin layer between curved surfaces and the general curvilinear coordinate system...

Diagonalizing the K matrix converts arbitrary systems in generalized coordinate systems q... 

The form of the jump conditions also depends on the coordinate system. Substituting (2.40) into the general Eulerian form of the momentum jump condition (Table 2.1) yields the Lagrangian jump condition... 

In Section 2.2, the stress-strain relations (generalized Hooke s law) for anisotropic and orthotropic as well as isotropic materials are discussed. These relations have two commonly accepted manners of expression compliances and stiffnesses as coefficients (elastic constants) of the stress-strain relations. The most attractive form of the stress-strain relations for orthotropic materials involves the engineering constants described in Section 2.3. The engineering constants are particularly helpful in describing composite material behavior because they are defined by the use of very obvious and simple physical measurements. Restrictions in the form of bounds are derived for the elastic constants in Section 2.4. These restrictions are useful in understanding the unusual behavior of composite materials relative to conventional isotropic materials. Attention is focused in Section 2.5 on stress-strain relations for an orthotropic material under plane stress conditions, the most common use of a composite lamina. These stress-strain relations are transformed in Section 2.6 to coordinate systems that are not aligned with the principal material... 

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