Axes, principal

Given a state at a point (e.g., stress or strain rate) that can be described by a symmetric tensor in some orthogonal coordinate system, it is always possible to represent that particular state in a rotated coordinate system for which the tensor has purely diagonal components. The axes for such a rotated coordinate system are called the principal axes, and the diagonal components are called the principal components. Finding the principal states and the principal directions is an eigenvalue problem. 

Assume that a stress tensor T is known at a point. Assume further that there are three orthogonal surfaces passing through the point for which the stress vectors r are parallel to the outward-normal unit vectors n that describe the orientation of the surfaces. In other words, on each of these surfaces the normal stress vector is a scalar multiple of the outward-normal unit vector, 

The stress vector in any direction (e.g., n) can be found from the stress tensor (Eq. A. 100). In particular, here we seek the vector in the direction of n, 

This expression can be written equivalently in matrix form as 

Thus the determinant must vanish, det (T — A.1) = 0, which expanded, for example, in cylindrical coordinates, is written as 

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The expansion is done around the principal axes so only tliree tenns occur in the simnnation. The nature of the critical pomt is detennined by the signs of the a. If > 0 for all n, then the critical point corresponds to a local minimum. If < 0 for all n, then the critical point corresponds to a local maximum. Otherwise, the critical points correspond to saddle points. 

The least recognized fonns of the Porod approximation are for the anisotropic system. If we consider the cylindrical scattering expression of equation (B 1.9.61). there are two principal axes (z and r directions) to be discussed... 

Let be a body-fixed frame IX, whose axes are the principal axes of... 

The coordinates p,Tx are called the principal axes of inertia symmetrized hyperspherical coordinates. The nuclear kinetic energy operator in these coordinates is given by... 

We now consider planar molecules. The electronic wave function is expressed with respect to molecule-fixed axes, which we can take to be the abc principal axes of ineitia, namely, by taking the coordinates (x,y,z) in Figure 1 coincided with the principal axes a,b,c). In order to detemiine the parity of the molecule through inversions in SF, we first rotate all the electrons and nuclei by 180° about the c axis (which is peipendicular to the molecular plane) and then reflect all the electrons in the molecular ab plane. The net effect is the inversion of all particles in SF. The first step has no effect on both the electronic and nuclear molecule-fixed coordinates, and has no effect on the electronic wave functions. The second step is a reflection of electronic spatial coordinates in the molecular plane. Note that such a plane is a symmetry plane and the eigenvalues of the corresponding operator then detemiine the parity of the electronic wave function. 

We now consider planar molecules. The electronic wave function is expressed with respect to molecule-fixed axes, which we can take to be the abc principal axes of inertia, namely, by taking the coordinates (x,y,z) in Figure 1 coincided with the principal axes (a, b, c). In order to determine the parity of the molecule through inversions in SF, we first rotate all the displacement vectors... 

The new coordinates are found by rotation of the old ones in the x-y plane such that they lie along the principal axes of the ellipse. 

We have found the principal axes from the equation of motion in an arbitrary coordinate system by means of a similarity transformation S KS (Chapter 2) on the coefficient matrix for the quadratic containing the mixed terms... 

The components of the quantum mechanical angular momentum operators along the three principal axes are ... 

For molecules that are non-linear and whose rotational wavefunctions are given in terms of the spherical or symmetric top functions D l,m,K, the dipole moment Pave can have components along any or all three of the molecule s internal coordinates (e.g., the three molecule-fixed coordinates that describe the orientation of the principal axes of the moment of inertia tensor). For a spherical top molecule, Pavel vanishes, so El transitions do not occur. 

A A2 — transition, 278, 282 interstellar, 120 molecular orbitals, 265ff principal axes, 103 structure determination, 132ff vibrations, 90ff H2CS (thioformaldehyde) interstellar, 120 HNCO (fukninic acid) interstellar, 120 HNCS (thiofulminic acid) interstellar, 120... 

CH3I (methyl iodide) principal axes, 103 If rotation, 113 CH2NH (methanimine) interstellar, 120 Cr203 (chromium trioxide) in alexandrite laser, 347ff in ruby laser, 346ff HC3N (cyanoacetylene) interstellar, 120 HCOOH (formic acid) interstellar, 120 NH2CN (cyanamide) interstellar, 120... 

Auger electron spectmm, 320 multiphoton dissociation, 376 principal axes, 105 symmetry elements, 76, 85 "SFg... 

Figure 8 A joint principal coordinate projection of the occupied regions in the conformational spaces of linear (Ala) (triangles) and its conformational constraint counterpart, cyclic-CAla) (squares), onto the optimal 3D principal axes. The symbols indicate the projected conformations, and the ellipsoids engulf the volume occupied by the projected points. This projection shows that the conformational volume accessible to the cyclic analog is only a small subset of the conformational volume accessible to the linear peptide, (Adapted from Ref. 41.)...

Fortunately, it was found that in polypeptide systems the effective dimensionality of conformational spaces is significantly smaller than the dimensionality of the full space, with only a few principal axes contributing to the projection [38-41]. In fact, in many cases a projection quality of 70-90% can be achieved in as few as tliree dimensions [42], opening the way for real 3D visualization of molecular conformational space. Figure 8... 

In terms of the dimensions, a, b and t for the section, several area properties can be found about the x-x and y-y axes, such as the second moment of area, 4, and the product moment of area, 4y. However, because the section has no axes of symmetry, unsymmetrical bending theory must be applied and it is required to find the principal axes, u-u and v-v, about which the second moments of area are a maximum and minimum respectively (Urry and Turner, 1986). The principal axes are again perpendicular and pass through the centre of gravity, but are a displaced angle, a, from x-x as shown in Figure 4.63. The objective is to find the plane in which the principal axes lie and calculate the second moments of area about these axes. The following formulae will be used in the development of the problem. 

H is a real symmetric matrix, and its eigenvalues are therefore real. Its eigenvectors are referred to as the principal axes of curvature . 

The r vectors are the principal axes of inertia determined by diagonalization of the matrix of inertia (eq. (12.14)). By forming the matrix product P FP, the translation and rotational directions are removed from the force constant matrix, and consequently the six (five) trivial vibrations become exactly zero (within the numerical accuracy of the machine). 

Note that e is defined in terms of the components of the velocities, not the vector velocities, whereas the momentum balance is defined in terms of the vector velocities. To solve Equations 2-32 and 2-33 when all the velocities are not colinear, one writes the momentum balances along the principal axes and solves the resulting equations simultaneously. 

A further result of Sadler s 2D-simulation was a relation between the step density and growth rate on the one hand and the inclination of the surface with respect to the principal axes on the other. From this relation crystal shapes were derived which show considerable curvature. This result of an exact treatment stands in contrast to Frank s mean-field curvature expression which gives essentially flat profiles. We will return to the discussion of curved edges in Sect. 3.6.3. 

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