Lagrange’s multipliers

It is also possible to derive the procedure as a constrained maxiinization problem, by using the method of Lagrange s multipliers. This procedure is more elaborate, and can be applied also to second and higher order response surface models. This is in essence the method of Ridge anafysis[2]. We do not go into this here. 

By means of Lagrange s multipliers (see subsection 1.2.1), the equilibrium conditions can be obtained ... 

Here, Wp, Wg are the collective fields experienced by the monomers and solvent, respectively, and Qp,Qg represent their respective collective densities. All charged species (excluding the ion-pairs formed due to adsorption of counterions) experience a field t ) (which is equivalent to the electrostatic potential), t) and u are Lagrange s multipliers corresponding to, respectively, the incompressibility and net charge constraints in the partition function. 

Iris type of constrained minimisation problem can be tackled using the method of Lagrange nultipliers. In this approach (see Section 1.10.5 for a brief introduction to Lagrange nultipliers) the derivative of the function to be minimised is added to the derivatives of he constraint(s) multiplied by a constant called a Lagrange multiplier. The sum is then et equal to zero. If the Lagrange multiplier for each of the orthonormality conditions is... 

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). 

Lagrange multipliers 255-256 Lagrange s moan-value theorem 30-32 Lagperre polynomials 140, 360 Lambert s law 11 Langevin function 61n Laplace transforms 279—286 convolution 283-284 delta function 285 derivative of a function 281-282 differential equation solutions 282-283... 

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 ... 

Use Lagrange s method of multipliers to derive the law of refraction of light from Fermat s principle of least time between two fixed points. 

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-... 

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... 

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

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 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... 

The step 8 towards the minimum is found by minimizing E(x + 8) with respect to 8 imposing the constraint from Eq. (42). Introducing a Lagrange s undetermined multiplier X we obtain the functional... 

The problem is to find the set u which minimizes G for specified T and P, subject to the constraints of the material balances. The standard solution to tliis problem is based on the method of Lagrange s undetermined multipliers. The procedure for gas-phase reactions is described as follows. 

Specific expressions for molecular properties can be developed for both variational and non-variational methods, the latter through the use of Lagrange s method of undetermined multipliers. 

Find the maximum in the function of the previous problem subject to the constraint jc -f y = 2. Do this by substitution and by Lagrange s method of undetermined multipliers. 

This is a problem in constrained extremals that can be solved using Lagrange s method. The constraining equations (6.6) and (6.7) are multiplied by two arbitrary constraints a and / , added to the logarithm of (6.4) and the desired maximum is given by... 

The direct method of obtaining the change in the quotient eqn ( 3.6) has not even been considered as it is far too involved to be manageable. The traditional method is to use Lagrange s method of undetermined multipliers to form a linear combination of the two expressions which are required to vanish, and require this linear combination to vanish for each degree of variational freedom. In Our case this is to combine eqns ( 3.20) and ( 3.22) using a linear combination... 

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