Coordinate system reduced

The starting point of the theoretical treatment is the Lagrangian for an ensemble of charged particles in an exterior magnetic field II. Under the neglect of all intramolecular magnetic interactions this Lagrangian, if referred to the space fixed laboratory coordinate system reduces to D... 

Explicit forms of the coefficients Tt and A depend on the coordinate system employed, the level of approximation applied, and so on. They can be chosen, for example, such that a part of the coupling with other degrees of freedom (typically stretching vibrations) is accounted for. In the space-fixed coordinate system at the infinitesimal bending vibrations, Tt + 7 reduces to the kinetic energy operator of a two-dimensional (2D) isotropic haiinonic oscillator. 

In an axisymmetric flow regime all of the field variables remain constant in the circumferential direction around an axis of symmetry. Therefore the governing flow equations in axisymmetric systems can be analytically integrated with respect to this direction to reduce the model to a two-dimensional form. In order to illustrate this procedure we consider the three-dimensional continuity equation for an incompressible fluid written in a cylindrical (r, 9, 2) coordinate system as... 

The Hamiltonian in this problem contains only the kinetic energy of rotation no potential energy is present because the molecule is undergoing unhindered "free rotation". The angles 0 and (j) describe the orientation of the diatomic molecule s axis relative to a laboratory-fixed coordinate system, and p is the reduced mass of the diatomic molecule p=mim2/(mi+m2). 

Proof. By utilizing the local coordinate systems (1.135), the assertion of Lemma 1.13 reduces to the case... 

As previously mentioned, R, versus jj. is an invariant coordinate system. This means that the reduced power, jj., can be expressed as a function of pressure ratio, R, across the expander (Figure 7-5). Knowing this. Equation 7-5 can be rewritten as ... 

Calculating Pj It is necessary to determine Pi ft to be able to determine the inlet and bypass valve steps. As seen in the invariant coordinate systems, the reduced power, j. is a function of the pressure ratio, R ... 

The reduced stiffnesses, Qy, are defined in terms of the engineering constants in Equation (2.66). In any other coordinate system in the plane of the lamina, the stresses are... 

Some coordinate transformations are non-linear, like transforming Cartesian to polar coordinates, where the polar coordinates are given in terms of square root and trigonometric functions of the Cartesian coordinates. This for example allows the Schrodinger equation for the hydrogen atom to be solved. Other transformations are linear, i.e. the new coordinate axes are linear combinations of the old coordinates. Such transfonnations can be used for reducing a matrix representation of an operator to a diagonal form. In the new coordinate system, the many-dimensional operator can be written as a sum of one-dimensional operators. 

Thus, solving a problem in particle statics reduces to finding the unknown force or forces such that the resultant force will be zero. To facilitate this process it is useful to draw a diagram showing the particle of interest and all the forces acting upon it. This is called a free-body diagram. Next a coordinate system (usually Cartesian) is superimposed on the free-body diagram, and tbe force.s are decomposed into their... 

Principal component analysis (PCA) is a statistical method having as its main purpose the representation in an economic way the location of the objects in a reduced coordinate system where only p axes instead of n axes corresponding to n variables (p[Pg.94]

When treating CF parameters in any of the two formalisms, non-specialists often overlook that the coefficients of the expansion of the CF potential (i.e. the values of CF parameters) depend on the choice of the coordinate system, so that conventions for assigning the correct reference framework are required. The conventional choice in which parameters are expressed requires the z-direction to be the principal symmetry axis, while the y-axis is chosen to coincide with a twofold symmetry axis (if present). Finally, the x-axis is perpendicular to both y- and z-axes, in such a way that the three axes form a right-handed coordinate system [31]. For symmetry in which no binary axis perpendicular to principal symmetry axis exists (e.g. C3h, Ctt), y is usually chosen so as to set one of the B kq (in Wybourne s approach) or Aq with q < 0 (in Stevens approach) to zero, thereby reducing the number of terms providing a non-zero imaginary contribution to the matrix elements of the ligand field Hamiltonian. Finally, for even lower symmetry (orthorhombic or monoclinic), the correct choice is such that the ratio of the Stevens parameter is restrained to X = /A (0, 1) and equivalently k =... 

Attaching the five-coordinate system to a rigid support (i.e., silica gel), thereby reducing its mobility and ability to add a sixth ligand. 

This is not as useful as Eq. (19.4) because products of different coordinates appear in the second term. However, the symmetry properties of this term ensure the existence of a coordinate system in which the cross-terms can be eliminated and the nuclear Hamiltonian reduces to a sum of harmonic oscillator terms ... 

This is the general equation for an ellipse in two independent variables, as shown in Fig. 21. If a new coordinate system is defined that has its center at the point S and axes directed along Xt and X2 of Fig. 21, then Eq. (Ill) reduces to... 

By the introduction of the (x, y) coordinate system, one has reduced the problem to the motion of a particle of mass (i in a two-dimensional rectilinear space (x, y). Thus, the problem of the collision between an atom and a diatomic molecule in a collinear geometry has been converted into a problem of a single particle on the potential energy surface expressed in terms of the coordinates x and y rather than the coordinates rAB and rBc The coordinates x and y which transform the kinetic energy to diagonal form in such way that the kinetic energy contains only one (effective) mass are referred to as mass scaled Jacobi coordinates. 

As mentioned in the introduction to this chapter this is a necessary condition when approximating the cylindrical screw in the Cartesian coordinate system. The screw rotation theory, New Theory line, predicts that the rate should constantly increase as the channel gets deeper. When a fixed positive pressure occurs for the screw rotation model, the New Theory with Pressure line, the predictions fits the data very well for all H/Ws. Thus for modern screw designs with deeper channels, reduced energy dissipation, and lower discharge temperatures, the screw rotation model would be expected to provide a good first estimation of the performance of the extruder regardless of the channel depth for Newtonian polymers. 

At Mo = 0, Eq. (12.5) reduces to the standard wave equation. In the 2D Cartesian coordinate system, the Laplace operator A is given by... 

The lower path is somewhat more complicated. The first step in the path involves either PCA (83) or principal-coordinate analysis (PCO) (83). This step can be followed by optimization of a function that minimizes the error between the proximity measure computed in the reduced-dimension and full coordinate systems if desired. Xie et al. (84) recently published an interesting paper along these lines. Kruscal stress (79) is a widely used function in this regard... 

The skew-symmetric part S 4 is equivalent to a vector (x t)/2 with components (/. t),/2 = (/.jtk — /.ktj)/2, involving correlations between a libration and a perpendicular translation. The components of S 4 can be reduced to zero, and S made symmetric, by a change of origin. It can be shown that the origin shift that symmetrizes S also minimizes the trace of T. In terms of the coordinate system based on the principal axes of L, the required origin shifts p, are... 

In the new coordinate system, the x axis coincides with one of the vertical mirror planes of the density, reducing the size of M 1 from 5 x 5 to 4 x 4. 

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