Lagrange’s method of undetermined

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... 

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... 

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... 

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. 

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... 

P must be maximized subject to the constraints of equations 2.106 and 2.108. This is accomplished using Lagrange s method of undetermined multipliers. Equation 2.106 is multiplied by the arbitrary constant (a - 1) and equation 2.108 is multiplied by the arbitrary constant p ot the sam/8 as defined in equatioiiL90]. Then these are added to equation 2.104 to give... 

Lagrange s method of undetermined multipliers is now applied to find the maximum in P. Two new variables (a for equation 2A.10 and p for equation 2A.11), the undetermined multipliers, are introduced as restrictions on equation 2A.9. Thus... 

Using Lagrange s method of undetermined multipliers described in Chapter l,we therefore minimize, with respect to the coefficients c,-, the following functional... 

To find the spinodal condition, the first step requires the determination of the second differential (5 t, of the segment-molar Helmholtz energy with respect to 5Vs, S J/b and 5[il/B s,Bir)]- The second step is to find that variation function B s sir)] minimizing the second differential S A. In this the condition BWs B(r)]dr = SipB has to be taken into account by using Lagrange s method of undetermined multipliers as minimization procedure. Setting b sM )] the quantity S As SVs,SiJ/Bi [ B s,Bi )T found. 

Such constrained optimizations are conveniently carried out by Lagrange s method of undetermined multipliers. Introducing one Lagrange multiplier for each constraint in equation (33), we arrive at the SCF Lagrangian ... 

Recalling our discussion of SCF molecular gradients in Section 5, we set up a variational Cl energy functional by using Lagrange s method of undetermined multipliers. For each... 

There are a number of problems in physical chemistry for which it is necessary to maximize (or minimize) a function under specific restrictive conditions. For example, suppose we wished to maximize some function/ x, y) subject to the restriction that another function of x and y, 0(jc, y), always equals zero. We can do this by a method known as Lagrange s method of undetermined multipliers. In order to maximize f(x, y) by this method, consider the total differentials... 

The variational function L a, X, X) is a function of the auxiliary set of parameters X as well as the original set X. To arrive at this function, we use Lagrange s method of undetermined multipliers and regard the original electronic energy E(a, X) as variationally optimized subject to the constraints that the variational parameters X satisfy (4.2.76) at each value of a. Formally, therefore, the electronic energy may be viewed as obtained by an unconstrained optimization of the Lagrangian... 

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