Coordinates, curvilinear generalized

Another curvilinear coordinate system of importance in two-centre problems, such as the diatomic molecule, derives from the more general system of confo-cal elliptical coordinates. The general discussion as represented, for instance by Margenau and Murphy [5], will not be repeated here. Of special interest is the case of prolate spheroidal coordinates. In this system each point lies at the intersection of an ellipsoid, a hyperboloid and and a cylinder, such that... 

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

For the example in Figure 2.14 it would be possible to perform the coordinate transformation analytically by introducing cylindrical coordinates. However, in general, geometries are too complex to be described by a simple analytical transformation. There are a variety of methods related to numerical curvilinear coordinate transformations relying on ideas of tensor calculus and differential geometry [94]. The fimdamental idea is to establish a numerical relationship between the physical space coordinates and the computational space curvilinear coordinates The local basis vectors of the curvilinear system are then given as... 

It is nice to have a distinctive notation for the curvilinear co-ordinates, which emphasizes their difference from and yet their one-to-one correlation with the Rt co-ordinates. Most authors reporting anharmonic calculations do not in fact make any distinction they denote the curvilinear co-ordinates by the same symbols customarily used to denote the corresponding rectilinear coordinates in harmonic calculations. For many purposes this is satisfactory, particularly since the harmonic force constants are not altered by the change from rectilinear to curvilinear co-ordinates. However, in a general discussion it is important to distinguish the two sets, and so for the remainder of this section we shall follow Hoy et al.12 and write the curvilinear co-ordinates with the symbol Hi. 

Symmetry, and the Number of Independent Force Constants.—As in harmonic calculations, the rather general discussion of the preceding section can be simplified in particular cases by making use of symmetry, as discussed by Hoy et a/.12 Thus we may choose the curvilinear co-ordinates Jfin linear combinations that span the irreducible representations of the point group we denote such symmetrized curvilinear co-ordinates by the symbol S, and we define them by means of a U matrix exactly analogous to that used for rectilinear coordinates ... 

There is finally the possibility of decay-like leakages between the particle-antiparticle spaces, and further that there could be an overall escape out of the presently defined spaces. If so associated Jordan blocks naturally appearing would decelerate this decay via the polynomial delay mechanism described earlier [7-10] with implications to subject matters like problems related to size of the cosmological constant. Also, the account given here should consider a more general decomposition of pz into curvilinear coordinates (cf. Eq. (30)) in order to yield a more appropriate analysis (see e.g. [22] and references therein). 

Adding the analogous contributions from the q2- and 3-directions and dividing by the volume dr, we obtain the general result for the divergence in curvilinear coordinates ... 

In this section we shall see how the principles outlined above are applied to evaluate the Wilson-Howard Hamiltonian1,2 However, most of the derivation may be worked out without explicitly assuming that rectilinear internal coordinates are used. We shall take advantage of this in that we will also examine the general consequences of the Eckart conditions as opposed to the special properties connected with the introduction of linearized coordinates. As an intermediate result we will therefore obtain a Hamiltonian which is exactly equivalent to the one which Quade derived for the case of geometrically defined curvilinear coordinates7 ... 

Let us consider surfaces in a Cartesian frame, whence these results can be generalized to any set of three coordinates x in an arbitrary coordinate system fixed in space. A surface in 3D space can generally be defined in several different ways. Explicitly, z = F x,y), implicitly, f x,y,z) = 0 or parametrically by defining a set of parametric equations of the form x = x C, rf), y = y C, v), z = z (, p) which contain two independent parameters Q, p called surface coordinates or curvilinear coordinates of a point on the surface. In this coordinate system a curve on the surface is defined by a relation f Q, p) = Q between the curvilinear coordinates. By eliminating the parameters Q, p one can derive... 

The transport equations can be written in many different forms, depending on the coordinate system used. Generally, we may select the orthogonal curvilinear Cartesian-, cylindrical-, and spherical coordinate systems, or the non-orthogonal curvilinear coordinate systems, which may be fixed or moving. In reactor engineering we frequently apply the simple curvilinear... 

Considering a generalized orthogonal coordinate system, the orthogonal curvilinear coordinates are defined as qa- In this O-system the base vectors Gq, are defined as unit vectors along the coordinates. The position of the point P is given by the coordinates, or by the position vector r = r qa,t). 

In this section the relevant differential operators are defined for generalized orthogonal curvilinear coordinate systems. 

Let qi,q2,q3) be curvilinear orthogonal coordinates connected with the Cartesian coordinates x,y,z) by the vector relation r = r qi, q2, qa), where r is the radius vector of the point P considered. The Cartesian coordinates are then related to the generalized curvilinear coordinates by ... 

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