Lagrange, Lagrangian

A better approach is the method of Lagrange multipliers. This introduces the Lagrangian fiinction [59]... 

The constraint force can be introduced into Newton s equations as a Lagrange multipli (see Section 1.10.5). To achieve consistency with the usual Lagrangian notation, we wri F y as —A and so F Ar equals Am. Thus ... 

Penalty functions with augmented Lagrangian method (an enhancement of the classical Lagrange multiplier method)... 

H.E. Trease, Three-Dimensional Free Lagrangian Hydrodynamics, in The Free-Lagrange Method (edited by M.J. Fritts, W.P. Crowley and H.E. Trease), Lecture Notes in Physics, Number 238, Springer-Verlag, New York, 1985. 

Using (4.26) for d/l/d(, we now have an expression for the derivative which involves the Lagrange multiplier AF and the Lagrangian ... 

On the analogy of simulating the process of adding blobs of a miscible liquid, two-phase flow in stirred tanks in a RANS context may be treated in two ways Euler-Lagrange or Euler-Euler, with the second, dispersed phase treated according to a Lagrangian approach and from a Eulerian point of view, respectively. 

The KTC are closely related to the classical Lagrange multiplier results for equality constrained problems. Form the Lagrangian... 

Define Lagrange multipliers A, associated with the equalities and Uj for the inequalities, and form the Lagrangian function... 

To accommodate the constraint (b)9 a Lagrangian function L is formed by augmenting/with Equation (b), using a Lagrange multiplier (o... 

The solution is obtained by means of the Lagrange multipliers method. The Lagrangian for this problem is... 

We solve the nonlinear formulation of the semidefinite program by the augmented Lagrange multiplier method for constrained nonlinear optimization [28, 29]. Consider the augmented Lagrangian function... 

Step 1. For a given set of Lagrange multipliers and penalty parameter minimize the Lagrangian function L R) to obtain an improved estimate of the factorized 2-RDM at the energy minimum. 

This means that all moving nuclei (atoms) are treated as classical particles which is a serious approximation, but which was found to work very well (60,61). Applying the Euler-Lagrange equation (Eq. 2) to the Lagrangian C (Eq. 1) leads to the same equations as the well-known Newton s second law (Eq. 3). Or in other words, in classical mechanics the derivative of the Lagrangian is taken with respect to the nuclear positions. 

Therefore the fact that 9 is arbitrary in U(l) theory compels that theory to assert that photon mass is zero. This is an unphysical result based on the Lorentz group. When we come to consider the Poincare group, as in section XIII, we find that the Wigner little group for a particle with identically zero mass is E(2), and this is unphysical. Since 9 in the U(l) gauge transform is entirely arbitrary, it is also unphysical. On the U(l) level, the Euler-Lagrange equation (825) seems to contain four unknowns, the four components of , and the field tensor H v seems to contain six unknowns. This situation is simply the result of the term 7/MV in the initial Lagrangian (824) from which Eq. (826) is obtained. However, the fundamental field tensor is defined by the 4-curl ... 

The Lagrangian (824), which is the same as the Lagrangian (839), gives the inhomogeneous equation (826) using the same Euler-Lagrange equation (843). Therefore the photon mass can be identified with the vacuum charge-current density as follows (in SI units) ... 

In order to be consistent with other chapters, R(Ct) is defined as a positive number if the chemical is produced in the river and T(Ct) is positive if the net flux is directed from the river into the atmosphere or sediment. Note that (F(Ct) is a flux per unit volume its relation to the usual flux per area as defined, for instance, in Chapter 20, is given below (Eq. 24-15). Again we suppress the compound subscript i wherever the context is clear. The subscript Lagrange refers to what fluid dynamicists call the Lagrangian representation of the flow in which the observer travels with a selected water volume (the river slice ) and watches the concentration changes in the volume while moving downstream. Later the notion of an isolated water volume will be modified when mixing due to diffusion and dispersion across the boundaries of the volume is taken into account. 

Using the Euler-Lagrange equation (122) with the Lagrangian (126) pro-... 

The following Euler-Lagrange equation is used next with the Lagrangian (454) ... 

The Lehnert field equation is obtained from this Lagrangian using the Euler-Lagrange equation... 

As discussed in Section I.C we will say that two scalars are dual or that they form a dual pair if they verify the duality constraint (15) or, equivalently, (119) for any given time.] According to the method of the Lagrange multipliers, let us vary, as independent fields, the two scalars < ) and 0 in the modified Lagrangian density... 

The middle term is a Proca Lagrangian for a massive photon. Here the mass of this photon is assumed to be larger than the masses of the W and W° bosons. The current / 31( is determined by the charged fermions with masses given by the Yukawa interactions with the Higgs held. These are yet to be explored. Now consider the term in the Euler-Lagrange equation... 

The derivation of the Lagrange relaxation master problem employs Lagrangian duality and considers the dualization of the i(ac,y) < 0 constraints only. The dual takes the following form ... 

Alternatively, an auxiliary variable wk can be introduced, constrained to be dynamically equal to qk using a Lagrange multiplier that turns out to be the substituted variable pk. The constraint condition in this ingenious procedure is Xk = Pk wk = 0. The modified Lagrangian is... 

To take this restriction into account, the Lagrange multiplier formalism is employed. We devise a Lagrangian by adding the restriction multiplied by a Lagrange multiplier X. 

The Lagrange multiplier X is determined by requiring that the derivatives of the Lagrangian with respect to all optimised variables like the structure coefficients C are zero ... 

Faith and Morari (1979) further develop the ideas of using dual bounding through the use of Lagrangian techniques for this problem. They describe refinements which allow one to make a good first estimate to the Lagrange multipliers (needed for the bounding) and to develop rather easily a "lower" lower bound. 

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