Matrices Euler

In Section V.B, we discussed to some extent the 3x3 adiabatic-to-diabatic transformation matrix A(= for a tri-state system. This matrix was expressed in terms of three (Euler-type) angles Y,y,r = 1,2,3 [see Eq. (81)], which fulfill a set of three coupled, first-order, differential equations [see Eq. (82)]. 

The Euler angles are often used to describe the orientations of a molecule. There are thre Euler cmgles d and ip. is a rotation about the Cartesian z axis this has the effec of moving the x and y axes, d is a rotation about the new x axis. Finally, ip is rotation about the new z axis (Figure 8.4). If the Euler angles are randomly changed b small amounts S[Pg.437]

Solution of Batch-Mill Equations In general, the grinding equation can be solved by numerical methods—for example, the Luler technique (Austin and Gardner, l.st Furopean Symposium on Size Reduction, 1962) or the Runge-Kutta technique. The matrix method is a particiilarly convenient fornmlation of the Euler technique. 

An alternative procedure is the dynamic programming method of Bellman (1957) which is based on the principle of optimality and the imbedding approach. The principle of optimality yields the Hamilton-Jacobi partial differential equation, whose solution results in an optimal control policy. Euler-Lagrange and Pontrya-gin s equations are applicable to systems with non-linear, time-varying state equations and non-quadratic, time varying performance criteria. The Hamilton-Jacobi equation is usually solved for the important and special case of the linear time-invariant plant with quadratic performance criterion (called the performance index), which takes the form of the matrix Riccati (1724) equation. This produces an optimal control law as a linear function of the state vector components which is always stable, providing the system is controllable. 

While the matrix R does not explicitly involve the axis of rotation along which e extends, nor the angle of rotation 6, it is known (from a theorem of Euler s) that every R is equivalent to a single rotation definable geometrically by an e and a 0. 

Figure 10.1 Schematic diagram of the sequential solution of model and sensitivity equations. The order is shown for a three parameter problem. Steps l, 5 and 9 involve iterative solution that requires a matrix inversion at each iteration of the fully implicit Euler s method. All other steps (i.e., the integration of the sensitivity equations) involve only one matrix multiplication each.

Note that the first column in the transformation matrix is just the last column of Table 1, while the second column is the same as the second column of R(x) [Eq. (16)]. Of course, as x is along-the z axis, its coefficient is equal to one. Substitution of the angular velocity components given by Eq. (18) allows the rotational energy JEq. (10)] to be expressed in terms of the velocities with respect to Euler s angles (see problem 7). 

Figure 4.12 [E] Computer-simulated ESR spectra for a hypothetical low-spin Mn(n) radical with g = (2.100, 2.050, 2.000), AMn = (150, 25, 25) x 10-4 cm-1, for various values of / , the Euler angle between the g-matrix and hyper-...

The Euler-Lagrange equations can he formed for the dynamical variables q—Rji, Pji, Zph, Zph and collected into a matrix equation which, when solved, yields the wave function for the compound system at each time step. 

The reader can easily check that, as claimed above, Z XgZ = Y j. Note that the lirsl rotation performed is the right-hand factor in the matrix multiplication. The proof that any element of 5(9(3) can be written as Z, X()Z for some real (, Q and iff is harder and is left as Exercise 4.24. The angles , d and i/r are known in the physics literature as Euler angles. For their geometric interpretation, see Figure 4,2,... 

Other than linear molecules. If molecules of symmetry other than axial are considered, it is not possible to describe their orientation by an azimuthal and polar angle, Euler angles, Q = a pi, y and Wigner rotation matrices are then needed as Eq. 4.8 suggests. In that case, besides the set of parameters X, X2, A, L that has been used for linear molecules, two new parameters, u, with i = 1,2, occur that enter through the rotation matrices. These must be chosen so that the dipole moment is invariant under any rotation belonging to the molecular symmetry group. The rotation matrix is expressed as a linear combination of such... 

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