Cosine Euler

A relationship, known as Euler s formula, exists between a complex number [x + jy] (x is the real part, y is the imaginary part of the complex number (j = P )) and a sine and cosine function. Many authors and textbooks prefer the complex number notation for its compactness and convenience. By substituting the Euler equations cos(r) = d + e -")/2 and sin(r) = (d - e t )l2j in eq. (40.1), a compact complex number notation for the Fourier transform is obtained as follows ... 

Each term on the right-hand side of this equation consists of the product of a direction cosine and an orbital angular momentum operator. Of these two factors, only the first depends on the Euler angles which define the instantaneous orientation of the molecule. In the corresponding equation (5.151) for./,, however, both factors depend on the Euler angles. 

In Eq. (4.14), (a =x, y, z) must be considered as functions of the normal coordinates Q and the inversion coordinate p are the direction cosines between the molecule and space-fixed system of axes which are functions of the Euler angles 0, 4>, X only. 

A contour plot is shown in Fig. 7.8. Note that this function is cylindrically-symmetrical about the z-axis with a node in the x, y-plane. The eigenfunctions 21 1 are complex and not as easy to represent graphically. Their angular dependence is that of the spherical harmonics 7i i, shown in Fig. 6.4. As deduced in Section 4.2, any linear combination of degenerate eigenfunctions is an equally-valid alternative eigenfunction. Making use of the Euler formulas for sine and cosine,... 

Rotational Degrees of Freedom. We can evaluate the gradients of the Euler angles from the directional cosines. For example, we can obtain Va0 from Eq. (141) as... 

Euler Angles, JMQ) Basis Functions, Direction Cosines,... 

For a linear molecule, the position of the symmetry axis (the molecule-fixed. z-axis) in space is specified by only two Euler angles, / and 7, which are respectively identical to the spherical polar coordinates 6 and (see Fig. 2.4). The third Euler angle, a, which specifies the orientation of the molecule-fixed x- and y-axes, is unaffected by molecular rotation but appears explicitly as an O- dependent phase factor in the rotational basis functions [Eq. (2.3.41)]. Cartesian coordinates in space- and molecule-fixed systems are related by the geometrical transformation represented by the 3x3 direction cosine matrix (Wilson et al., 1980, p. 286). The direction cosine matrix a given by Hougen (1970, p. 18) is obtained by setting a = 7t/2 (notation of Wilson et al, 1980 6 fi,4)=, x = oi 7t/2). The direction cosine matrix is expressed in terms of sines and cosines of 9 and 4>. Matrix elements (J M O la JMQ), evaluated in the JMQ) basis, of the direction cosines, are expressed in terms of the J, M, and quantum numbers. The direction cosine matrix elements of Hougen (1970, p. 31), Townes and Schawlow (1955, p. 96), and Table 2.1 assume the basis set definition derived from Eq. (2.3.40) and the phase choice a = 7t/2 ... 

A disadvantage of the Euler angle approach is that the rotation matrix contains a total of six trigonometric functions (sine and cosine for each of the three Euler angles). These trigonometric functions are computationally expensive to calculate. An alternative is to use quaternions. A quaternion is a four-dimensional vector such that its components sum to 1 0 + 1 + <72 + = 1- quaternion components are related to the Euler angles as follows ... 

Collecting the real and imaginary contributions to Eq. (4.53) and comparing with the corresponding terms in Euler s theorem (4.48) result in power-series expansions for the sine and cosine ... 

When ri and C2 are imaginary numbers, namely, ik and — ik, the solution (8.54) contains complex exponentials. Since, by Euler s theorem, these can be expressed as sums and differences of sine and cosine, we can write... 

The two kinds of Bessel functions, thus, have the asymptotic dependence of slowly damped cosines and sines. In analogy with Euler s formula = cos X i sinx, we define Hankel functions of the first and second kinds ... 

Converting the cosine/sine form to the complex exponential form allows many manipulations that would be very difficult otherwise (for an example, see Section 2 of Appendix A Convolution and DFT Properties). But, if you re totally uncomfortable with complex numbers and Euler s Identity (or with the identities of IS"" century mathematicians in general), then you can write the DFT in real number terms as a form of the Fourier Series ... 

If we take the cosine sinusoid definition and write it in terms of complex exponentials using the inverse Euler equation 10.11,... 

A more practical objection to the spherical tensor formulation has been that it is inefficient to use in computationally-intensive applications such as molecular dynamics, because of the supposed need to describe the molecular orientations in terms of Euler angles and to evaluate trigonometrical functions of these angles. This objection is completely unfounded. The Euler angles have traditionally been used to describe orientation, but they are by no means necessary, and the formulae to be presented below will not mention them at all. It is possible, within the spherical tensor formalism, to describe orientation in terms of the same direction cosines /. ... 

The are functions of relative orientation, and we should now consider how the relative orientation is to.be described. It is common to describe relative orientation by means of Euler angles, but this is cumbersome and would entail the evaluation of trigonometric functions. Instead, we can describe the relative orientations that we need in terms of direction cosines, just as in Cartesian tensor theory. If the local axes of region a are described, by unit vectors e" and those of region b by unit vectors e then the relative orientation ofthe two regions is described by the direction cosines... 

When an angular momentum is directed toward arbitrary directions that have direction cosines cos a, cos P, and cos y, where a, P, and y are Euler angles, angular momentum M(a, P, y) may be written as follows ... 

Treatment of angular momentum in a multi-quantum regime natiually benefits fi-om this operation. We applied this theorem to quantum operator Af(aPy), defined as cosa + M cosP +cosy, in which cos a, cos P, and cosy are direction cosines of Euler angles a, p, and y. The resultant eigenvalues of the angular momentum are independent of direction, but the projection matrices are dependent. The matrix for an angular momentum can therefore be resolved as a product of the transformation matrix with a direction cosine and an eigenfunetion matrix that corresponds to the transition. The projection matrix, after removal of information on direction, is an intrinsic density matrix. The proeedure to obtain... 

Euler s analysis provided us with the complex exponential to use in the place of trigonometric functions in problems giving periodic functions. After a little practice it is often much easier to manipulate than sines and cosines. A useful excercise is to obtain Equation (A9.98) from the boundary conditions in Equation (A9.96) and trial functions in Equation (A9.97) following the same route. 

By this means, the partial differential equation is transferred into an ordinary differential equation in the discrete cosine space. A semi-implicit method is used to trade-off the stability, computing time, and accuracy [39,40]. In order to remove the shortcomings with the small time-step size associated with the exphcit Euler scheme to achieve convergence, the linear fourth-order operators can be treated implicitly while the nonlinear terms can be treated explicitly. The resulting first-order semi-implicit Fourier scheme is ... 

First, the kinematic equation is constructed by obtaining the direction of cosines tables. Direction of cosines is obtained by using Euler Rotational Series which is a series of three rotations used to define uniquely the orientation of rigid body in 3-dimensional space. Table 1 summarize the direction of cosines for reference frame of scapula (N) acting on reference frame of humerus (A). 

Euler s Identities These identities allow sines and cosines to be expressed in terms of complex numbers, and vice versa ... 

From the best fit of the simulated spectrum to the observed, Euler angles and y for the PAS of the tensor were obtained, and these are readily converted to the direction cosines 6i, 02, and Table I lists the dir on cosine angles of the PAS relative to the hehcal axis determined from the 3 P-NMR spectra of the A form of DNA (Nall et al., 1981), together with the A form of poly(dAdT)-poly(dAdT) (Shindo et al.. 1981). The results obtained for different specimens are in good agreement within 5° of each other, suggesting that the A form of DNA exhibits no base-sequence dependence of the backbone structure, or little if any. If the detailed structure of DNA is known, the orientation of the tensor can easily be evaluated with respect to the helical axis, assuming the relative orientation of the PAS and... 

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