Rotational coordinate transformations

Recently, an Eulerian derivation of the Coriolis force has been reported by Kageyama and Hyodo [45]. They present a general procedure to derive the transformed equations in the rotating frame of reference based on the local Galilean transformation and rotational coordinate transformation of field quantities. 

As discussed in section 8.2.1, the lineshape of a deuteron spectrum of an oriented material is determined by the distribution of C-D bonds relative to the magnetic field, and can be calculated from equation (8.1) with knowledge of the distribution of coordinates jc, y, z or equivalently of and Q> for equation (8.2). This distribution is obtained through a series of rotational coordinate transformations ... 

The main difference between the adiabatic-to-diabatic transformation and the Wigner matrices is that whereas the Wigner matiix is defined for an ordinary spatial coordinate the adiabatic-to-diabatic transformation matrix is defined for a rotation coordinate in a different space. 

When the operation is a rotation by an angle 4> about a C axis, in this case by an angle of 2n/3 radians about the C3 axis, the resulting coordinate transformation, a result which will not be derived here, is given by... 

Phasor rotator to transform the field frame coordinates to the stator frame coordinates. 

This is the hypoelastic constitutive equation considered by Truesdell (see Truesdell and Noll [20]). In large deformations, this equation should be independent of the motion of the observer, a property termed objectivity, i.e., it should be invariant under rigid rotation and translation of the coordinate frame. In order to investigate this property, a coordinate transformation (A.50) is applied. If the elastic stress rate relation is to be unchanged in the new coordinate system denoted x, then... 

In Section 5.2 the set of internal state variables k was introduced. In the referential theory, a similar set of referential internal state variables K will be introduced in the same way without further physical identification at this stage. It will merely be assumed that each member of the set K is invariant under the coordinate transformation (A.50) representing a rigid rotation and translation of the coordinate frame. 

The objectivity of the spatial stress rate relation (5.154) may be investigated by applying the coordinate transformation (A.50) representing a rotation and translation of the coordinate frame. The spatial strain and its convected rate are indifferent by (A.58) and (A.62). So are the stress and its Truesdell rate. It is readily verified from (5.151), (5.152), and the fact that K has been assumed to be invariant, that k and its Truesdell rate are also indifferent. Using these facts together with (A.53) in (5.154) with c and b given by (5.155)... 

It is expected that constitutive equations should be invariant to relative rigid rotation and translation between the material and the coordinate frame with respect to which the motion is measured, a property termed objectivity. In order to investigate this invariance, the coordinate transformation... 

A linear coordinate transformation may be illustrated by a simple two-dimensional example. The new coordinate system is defined in term of the old by means of a rotation matrix, U. In the general case the U matrix is unitary (complex elements), although for most applications it may be chosen to be orthogonal (real elements). This means that the matrix inverse is given by transposing the complex conjugate, or in the... 

Since the individual coordinate transformations T depend continuously and differentially on some rotation angles specifying these transformations, the same must hold for the combined transformations, Xk > as well, since transposition and matrix... 

The model of a reacting molecular crystal proposed by Luty and Eckhardt [315] is centered on the description of the collective response of the crystal to a local strain expressed by means of an elastic stress tensor. The local strain of mechanical origin is, for our purposes, produced by the pressure or by the chemical transformation of a molecule at site n. The mechanical perturbation field couples to the internal and external (translational and rotational) coordinates Q n) generating a non local response. The dynamical variable Q can include any set of coordinates of interest for the process under consideration. In the model the system Hamiltonian includes a single molecule term, the coupling between the molecular variables at different sites through a force constants matrix W, and a third term that takes into account the coupling to the dynamical variables of the operator of the local stress. In the linear approximation, the response of the system is expressed by a response function X to a local field that can be approximated by a mean field V ... 

That is, the transformation is represented as successive rotations of y, P, a about the e3, e2, and axes. A positive rotation is a counterclockwise rotation.1 Since R is unitary, it follows that the Cartesian coordinates transform as... 

These same rotation matrices arise when the transformation properties of spherical harmonics are examined for transformations that rotate coordinate systems. For example,... 

HD-X induced dipoles. The HD molecule differs from H2 by its greater mass (which is of little concern here), the weak permanent dipole moment which arises from a non-adiabatic mechanism, and a center of mass which does not coincide with the center of electronic charge. Dipole moments are defined with respect to an origin that coincides with the center of mass. The presence of the permanent dipole leads to well-known rotational and rotvibrational spectra of RV(J) lines which show an interesting dispersion shape, arising from the interference of the induced and the allowed dipole spectra. For a theoretical analysis, one needs the induced dipole components of pairs like HD-X, with X = He, Ar, H2 or HD. These have been obtained previously [59] from the ab initio data for the familiar isotopes summarized above, using a simple coordinate transform familiar from potential studies of the isotopes. 

If the principal stresses had had shear components, which by definition they don t, then, in general, those shear components would have contributed to the stress vector on the rotated z plane. The a vector completely defines the stress state on the rotated z face. However, our objective is to determine the stress-state vector on the z face that aligns with the rotated coordinate system (z,r,G) x--, x-r, and x-e. The a vector itself has no particular value in its own right. Therefore one more transformation from cs to r is required ... 

In addition to representing a vector in a rotated coordinate system, the matrix of direction cosines can also be used to transform a tensor (e.g., the stress tensor) into a rotated coordinate system as... 

Dividing by n yields the desired transformation of a tensor into a rotated coordinate system, Eq. A.126. 

For the special case of S2 s i, the mirror image is produced by the inversion operation, but must be rotated by 180° to bring it into an exact reflective relationship to the original. This can be seen in Figure 3.4 and is conveniently expressed by using the matrices for the coordinate transformations. (Readers unfamiliar with matrix algebra may consult Appendix I.) Thus, we represent the operation S2 35 i hy the first matrix shown below and a rotation by n... 

In principle, the algorithms for color constancy, which are described in the following chapters, can also be applied to the rotated coordinates (Figure 2.20). Land notes that, in practice, the transformation may not be that simple if the algorithm is nonlinear, i.e. contains a thresholding operation. We see later, e.g. in Section 6.6, Section 7.5, or Section 11.2, that some of the algorithms for color constancy are also based on a rotated coordinate system where the gray vector, i.e. the achromatic axis, plays a central role. 

Figure 2.20 The transformation performed by the color opponency mechanism may be a simple rotation of the coordinate system. In this case, the mechanism for color constancy may operate on the rotated coordinates. (Reprinted from Vision Res., Vol. 26, No. 1, Edwin H. Land, Recent advances in retinex theory, pp. 7-21, Copyright 1986, with permission from Elsevier.)...

We are not going to review here the transformation properties of spatial wave functions under the symmetry operations of molecular point groups. To prepare the discussion of the transformation properties of spinors, we shall put some effort, however, in discussing the symmetry operations of 0(3)+, the group of proper rotations in 3D coordinate space (i.e., orthogonal transformations with determinant + 1). Reflections and improper rotations (orthogonal transformations with determinant -1) will be dealt with later. 

To obtain the coordinate transformations for rotation operators placed at the origin (without loss of generality, assume that the rotation axis is parallel to the z axis, Fig. 7.3) we can define the arbitrary rotation though an angle 6, as given by... 

We have derived the total Hamiltonian expressed in a space-fixed (i.e. non-rotating) coordinate system in (2.36), (2.37) and (2.75). We can now simplify the electronic Hamiltonian 3Q,i by transforming the electronic coordinates to the molecule-fixed axis system defined by (2.40) because the Coulombic potential term, when expressed as a function of these new coordinates, is independent of 0, ip and x From a physical standpoint it is obviously sensible to transform the electronic coordinates in this way because under the influence of the electrostatic interactions, the electrons rotate in space with the nuclei. We shall take the opportunity to refer the electron spins to the molecule-fixed axis system in this section also, and leave discussion of the alternative scheme of space quantisation to a later section. Since we assume the electron spin wave function to be completely separable from the spatial (i.e. orbital) wave function,... 

In chapter 2 we showed how the wave equation of a vibrating rotator was derived through a series of coordinate transformations. We discussed the solutions of this wave equation in section 2.8, and the particular problem of representing the potential in which the nuclei move. We outlined the relatively simple solutions obtained for a harmonic oscillator, the corrections which are introduced to take account of anharmonicity, and derived an expression for the rovibrational energies. Our treatment was relatively brief, so we now return to this subject in rather more detail. 

Note that although this discussion will be limited to the translational case, the major results can be applied immediately to rotational dynamics. Transforming translation into angular velocities and axis coordinates into angular coordinates leaves unchanged the form of the equations involved. The only caveat which has to be used concerns the boundary conditions that have to be imposed so as to respect the periodic character of the rotational problems (rotation by 27t leaves these systems unchanged). 

The local frame for rigid molecules, once chosen, will always be clearly defined. The necessary transformation of the separation vector from the laboratory to the local frame is usually accomplished by multiplication by the rotation matrix of the central molecule. The construction of this rotation matrix is usually a straightforward task. In fact, it will already be available in any program that describes molecular orientations in terms of quaternions (coordinate-transformed eulerian angles) [3,24]. 

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