Coordinate transformations rotations

The group of Poincare transformations consists of coordinate transformations (rotations, translations, proper Lorentz transformations...) linking the different inertial frames that are supposed to be equivalent for the description of nature. The free Dirac equation is invariant under these Poincare transformations. More precisely, the free Dirac equation is invariant under (the covering group of) the proper orthochronous Poincare group, which excludes the time reversal and the space-time inversion, but does include the parity transformation (space reflection). 

As is well known, the 3x3 matrix Oy can be diagonalized by an appropriate orthogonal coordinate transformation (rotational transformation), provided it is a symmetric matrix generally it is considered to be symmetric because of its physical meaning. If the principal-axes frame of o, where o is expressed by a diagonal matrix, is transformed to the laboratory frame by a rotational transformation R(o, /3, y) which is defined by three Eulerian angles a, /3 and y, then the representations of o in both frames are related to each other by the equation = (5)... 

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... 

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 special class of transformation, known as symmetry (or unitary) transformation, preserves the shape of geometrical objects, and in particular the norm (length) of individual vectors. For this class of transformation the symmetry operation becomes equivalent to a transformation of the coordinate system. Rotation, translation, reflection and inversion are obvious examples of such transformations. If the discussion is restricted to real vector space the transformations are called orthogonal. 

The most expensive parts of a conventional NSE instrument are the main solenoids providing the precession field. A closer look at Bloch s equation of motion for the spins (Eq. 2.11) shows that in a coordinate system that rotates with the precession frequency around Bg the spin is stationary, the coordinate system rotation is equivalent to the addition of - to all magnetic fields. By this means the large precession field inside the main coils may be transformed to zero - zero field spin-echo). The flippers are viewed as elements rotating... 

If individual atomic coordinate systems are used, as is common when chemical constraints are applied in the least-squares refinement, they must first be rotated to have a common orientation. The transformation of the population parameters under coordinate-system rotation is described in section D.5 of appendix D (Cromer et al. 1976, Su 1993, Su and Coppens 1994). 

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... 

This representation is in block form, and is obviously reducible. Consider another coordinate system, rotated in the a — y plane by 45°. Verify that in this new coordinate system the formulas giving the effect of cr are a —y and y —s- —x. Find the matrix relating the two coordinate systems and verify that a similarity transformation applied to the matrices of this new representation produces the old representation. How does this demonstrate the reducibility of the new representation ... 

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. 

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... 

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... 

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. 

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]. 

The critical stress intensity factor, Kc, obtained at different loading rates and environmental conditions was then reported as a function of fibre orientation. It was expected that the most evident relationship with the considered mechanical property would have been shown by the orientation factor in the applied stress direction (direction 1 in Fig. 2b). Therefore the orientation factor aj, previously measured on the plane 2 -3 was transformed into the orientation factor a, defined with respect to the applied stress direction 1 (Fig. 2 a and b) by a coordinate axis rotation of an angle a. Further details on such data handling can be found in ref [7, 13]. It is worth reminding here that a, and a go opposite when a increases from 0 to 90 degrees a, decreases from 1 to 2ero. 

The examination of coordinate transformations as local contractions and expansions of decorations about the poles of the principal rotational axes on the unit sphere for objects of Oh symmetry leads to intermediate geometries corresponding to particular Archimedean polyhedra related to the cube. In a similar manner, partial contractions and expansions of the decorations of the regular orbit of Ih point symmetry, i.e. the vertices of the great rhombicosidodecahedron, leads to the remaining polyhedra within the icosahedral family of Archimedean structures and orbits of Ih. 

Table 13.1 show how the cartesian coordinates x, y,z, combinations of cartesian coordinates and rotations Rx,Ry, Rz transform under the operations of the group. 

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... 

The Wigner rotations describe the coordinate transformations from the principal axis frame (P ) in which the tensor describing the interaction X is diagonal, via a molecule-fixed frame (C) and the rotor-fixed frame (R) to the laboratory frame (L) as illustrated in an ORTEP representation in Fig. 1. 

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. 

Fig. 5.4.5 Definition of q space in terms of position-change NMR. (a) Initial and final positions are encoded by fci and fea in the narrow gradient-pulse approximation. The transformation to a coordinate system where the difference wave number q defines one of the axes corresponds to a right-handed 45° rotation of the coordinate system (cf. eqn (5.4.21)). The perpendicular variable is proportional to the wave number k which encodes position, (b) 2D Fourier transformation of such a 2D position-change data set produces the displacement coordinate in a coordinate frame rotated by 45° on one axis and the space coordinate r on the other axis.

In velocity-change NMR the variables q and q 2 replace the axes ki and kz in Fig. 5.4.5, and ui and V2 replace r and r2 (Fig. 5.4.12). In a coordinate frame rotated by 45° the difference coordinate corresponds to acceleration a and the other to velocity v, so that this exchange experiment can be read as a velocity-acceleration correlation experiment. Following the coordinate transformations (5.4.21) for position-change spectroscopy, the following coordinate transformations apply for velocity-change spectroscopy. 

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