Generalized momenta

V X is the curl operator). Only the kinetic energy of the electrons are considered within the Bom-Oppenheimer approximation, and the generalized momentum becomes... 

Here /(R) and pflX) denote the shift and generalized momentum for the molecular vibration of the low frequency a>9 and reduced mass m, at the Rth site of the adsorbate lattice bi+(K) and K) are creation and annihilation operators for the collectivized mode of the adsorbate that is characterized by the squared frequency /2(K) = ml + d>, a,(K)/m , with O / iat(K) representing the Fourier component of the force constant function /jat(R). Shifts i//(R) for all molecules are assumed to be oriented in the same arbitrary direction specified by the unit vector e they are related to the corresponding normal coordinates, ue (K), and secondary quantization operators ... 

We consider a balance equation for a generalized momentum density na q,t), defined by... 

This means that the generalized momentum operator is... 

Solution A general momentum equation for the trailing particle takes the form... 

In Eq. (8.54), Upt is the particle terminal velocity, and n is the modified Richardson-Zaki index for gas-solid systems. Thus, on the basis of Eqs. (8.52) and (8.53), a general momentum equation for the solid phase becomes... 

Equation (8.62) is the general momentum balance equation. For moving bed solids transport in the standpipe, a is constant with a value of amb. Thus, Eq. (8.62) can be simplified to... 

By equating the pressure gradient and the gas-particle drag force, the general momentum balance for the gas phase can be expressed as... 

In the case of a hydraulic jump on a sloping channel, it is simply necessary to add the sine component of the weight of the water to the general momentum equation. Although there exist more refined methods for doing this [41], a good approximation may be obtained by assuming the jump section to be a trapezoid with bases jq and y2 and altitude of about 6Y2. 

In the third formulation, the so-called Hamiltonian formulation, the velocities Lagrangian form are replaced by the so-called generalized momenta pi via a Legendre transformation. The generalized momentum pi, conjugate to the coordinate qi, is defined as... 

From L one can get the generalized momentum p conjugate to the generalized coordinate q ... 

Generalized momentum operator, 248 Genetic Algorithms (GA) for global—... 

In order to use these general momentum conservation equations to calculate the velocity field, it is necessary to express viscous stress terms in terms of the velocity field. The equations which relate the stress tensor to the motion of the continuous fluid are called constitutive equations or rheological equations of state. Although the governing momentum conservation equations are valid for all fluids, the constitutive equations, in general, vary from one fluid material to another and possibly also from one type of flow to another. Fluids, which follow Newton s law of viscosity (although it is referred to as a law, it is just an empirical proposition) are called Newtonian fluids. For such fluids, the viscous stress at a point is linearly dependent on the rates of strain (deformation) of the fluid. With this assumption, a general deformation law which relates stress tensor and velocity components can be written ... 

Generalized momentum operators as defined by Eq. (2.77) can be used in wave mechanical as well as in matrix mechanical formulations. It ensures that the operators are Hermitian, and that momenta, 7r, conjugated to generalized coordinates, qh fulfil commutation relations similar to the canonical relations of Cartesian coordinates and momenta,... 

For one component fluids the Bernoulli equation for inviscid flow along a streamline can either be formulated by direct application of Newton s second law to a fluid particle moving along a streamline [114] [10] or derived projecting the generalized momentum equation (1.78) onto a streamline. Applying the latter approach, the Navier-Stokes equation for non-viscous fluids becomes ... 

Prom the introductory courses in fluid flow one recalls that the simple parabolic profile for laminar flow in a pipe, the Hagen-Poiseuille law, is derived by integration of a sufficiently simplified form of the generalized momentum equation (see e.g., [13], Example 3.6-1) ... 

Unfortunately, it is not possible to derive an analogue velocity profile for turbulent flow in an anal dical manner based on the generalized momentum equations. However, a number of entirely empirical relations of similar simplicity exist for the velocity profile in turbulent pipe flow. One such relation often found in introductory textbooks on engineering fluid flow is the power law velocity profile. ... 

Expressions for the calculation of magnetic properties at the relativistic level of theory can be derived by replacing the momentum operator p in the Dirac equation (1) with the generalized momentum operator n... 

In order to derive expressions which can be used for calculating magnetic properties at the quasi-relativistic level of theory, we analogously replace p in equations (13)-(18) by the corresponding generalized momentum %. The expressions are inserted into equation (20) and one proceeds as described in Section 3. The total quasi-relativistic energy as a function of the perturbation strength parameter then becomes... 

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