Multipliers. Undetermined

There are two main methods for enforcing such constraints. One is the Penalty Function approach, the other the metlrod of Lagrange Undetermined Multipliers. 

We can combine these three into one equation by using Lagrange s method of undetermined multipliers. To do so, we multiply equation (10.22) by ft and... 

The method of Lagrange s undetermined multipliers is a useful analytical technique for dealing with problems that have equality constraints (fixed design values). Examples of the use of this technique for simple design problems are given by Stoecker (1989), Peters and Timmerhaus (1991) and Boas (1963a). 

This conceptual hitch is addressed by adjusting the values of the superfluous variables, subject to the constraints of (9.40), to make G stationary - minimal. Accounting for the constraints (9.40) by the standard procedure of Lagrange s undetermined multipliers [49] yields ... 

By taking the minimum in Eq. (69) subject to the restrictions specified and using Lagrange undetermined multipliers (see, for example, Ref. 6), one finds a set of relationships satisfied by the defect chemical potentials. The results for the three basic types of intrinsic lattice disorder are as follows ... 

Derivation of the Boltzmann distribution function is based on statistical mechanical considerations and requires use of Stirling s approximation and Lagrange s method of undetermined multipliers to arrive at the basic equation, (N,/No) = (g/go)exp[-A Ae/]. The exponential term /3 defines the temperature scale of the Boltzmann function and can be shown to equal t/ksT. In classical mechanics, this distribution is defined by giving values for the coordinates and momenta for each particle in three-coordinate space and the lin-... 

The variational method is then used to minimize the expectation value of total energy E = (cj) H (j)) under small variation of the ip s in (19), and subject to the normalization condition of cj) ()) H (1)) = 1. (This may be done by employing the method of Lagrangian undetermined multipliers). 

There are N spin orbitals, so there are N(N+ 1)/2 independent constraints (note a b) = (6 a) ), so we need that many undetermined multipliers in our lagrangian function, for which we present... 

This constraint on the choice of Mr) was incorporated into the variation integral (265) by means of Lagrangian undetermined multipliers. The invariant Sf was minimised with respect to a variation its minimum value is when j/ — m. By comparison with eqn. (263)... 

To develop a lower bound on the steady state, Reck and Prager [507] again considered the variational integral of eqn. (265). In this case, however, let the approximate solution j/ satisfy the diffusion equation (263) rather than the equation defining the macroscopic density M as previously done. Multiply eqn. (263) by j5(r), a Lagrangian undetermined multiplier and add it to the variational integral to give... 

The terminal spectrum is furnished by cooperative motions which extend beyond slow points on chain in the equivalent system. The modulus associated with the terminal relaxations is vEkT, which is smaller by a factor of two than the value from a shifted Rouse spectrum. It is consistent with a front factor g = j given by some recent theories of rubber elasticity (Part 7). The terminal spectrum for E 1 has the Rouse spacings for all practical purposes, shifted along the time axis by an undetermined multiplying factor (essentially the slow point friction coefficient). Thus, the model does not predict the terminal spectrum narrowing which is observed experimentally. 

A simple way of achieving this end is by application of Lagrange s method of undetermined multipliers. Let us consider the function F, such that... 

In solving for the extremum of a general function / subject to the constraints g = constant and h = constant, we can use the Lagrange s method of undetermined multipliers. That is, we can solve for... 

A more elegant and useful method was suggested by Lagrange. The fundamental difficulty is that there are fewer variables than the number of derivative conditions. As suggested by Lagrange, we can therefore introduce new constants Ai, A2,..., Ac ( Lagrange undetermined multipliers, one for each constraint) to define a new constrained function / given by... 

SIDEBAR 5.2 ILLUSTRATION OF LAGRANGE S METHOD OF UNDETERMINED MULTIPLIERS... 

Lagrange Solution We introduce an undetermined multiplier A to write the constrained function / as... 

The important aspect of (13.70b) is that each pa=Pa(U, V, N) has maximal ( most probable ) character with respect to the natural control variables of S. The constrained maximization procedure to find this optimal distribution by the method of Lagrange undetermined multipliers [see Schrodinger (1949), Sidebar 13.4, for further details] is very similar to that described in Section 5.2. In particular, the pa must be maximal with respect to variations in each control variable, leading to the usual second-derivative curvature conditions such as... 

Constructing G as in Eqn. (2.41) but imposing the equilibrium condition 8CP 7-= 0 and using Lagrange s method of undetermined multipliers (2A,Aj) in order to meet the structural constraints, we obtain... 

The way Lagrange s method of undetermined multipliers is interpreted here is not conventional. The approach is described in Appendix A. To guarantee that L is independent of the set [xj], set ... 

The Lagrange Method of Undetermined Multipliers. To prove important statistical mechanical results in Chapter 5, we need the method of undetermined multipliers, due to Lagrange.42 This method can be enunciated as follows Assume that a function f(xu x. .., xn) of n variables X, x2,..., xn is subject to two auxiliary conditions ... 

In pedestrian terms, absolute temperature seems to creep out of a Lagrange multiplier The other undetermined multiplier oc can be evaluated, when the dependence of the energy m on quantum numbers i has been established (this calculation is different for every physical problem). 

The process in detail is as follows. We use what is known as Lagrange s method of undetermined multipliers, introducing constants such that the quantity W, defined by... 

Write equations for minimization of total Gibbs free energy. This step employs the method of Lagrange undetermined multipliers for minimization under constraint for a discussion of this method, refer to mathematics handbooks. As for its application to minimization of total Gibbs free energy, see Perry and Chilton [7] and Smith and Van Ness [11]. 

Hi) Constrained Maximization Method of Lagrange Undetermined Multipliers The problem of constrained maximization may be posed in its most general form as follows ... 

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