Method of Lagrangian multipliers

By the method of Lagrangian multipliers (a and / in the following) it is found in all cases for arbitrary SNi that... 

A further advantage of using Lagrangian dynamics is that we can easily impose boundary conditions and constraints by applying the method of Lagrangian multipliers. This is particularly important for the dynamics of the electronic degrees of freedom, as we will have to impose that the one-electron wavefunctions remain orthonormal during their time evolution. The Lex of our extended system can then be written as ... 

Constraints may be imposed on a set of simultaneous linear equations by the method of Lagrangian multipliers. Let the Lagrangian multipliers be — - Therefore, add to equation (A.28) the quantity... 

The problem of finding extrema of a function / (xi, X2,..., x, ) = / (x) subject to n constraints may be solved by using the method of Lagrangian multipliers. In the absence of such constraints the necessary condition for the existence of extrema may be stated as... 

In such a case the method of Lagrangian multipliers is feasible such a Lagrangian multiplier e is chosen so that the functional... 

The method universally used in constrained variational problems is the method of Lagrangian multipliers. The basic idea is so simple that, on meeting it for the first time, one is immediately suspicious that it works so well. 

I.-S. Liu (1972). Method of Lagrangian multipliers for exploitation of the entropy principle. Arch. Rat. Mech. Anal, 46, 131-148. 

The constraints can be treated by the alternative method of Lagrangian multipliers. Here it are regarded as independent variables, and the constraining forces are explicitly added to the right-hand side of eqn (3.135) ... 

The numerical algorithm of the method of Lagrangian multipliers is shown in the block diagram in Fig. 22. The algorithm is composed of the following blocks ... 

It is more advantageous to apply methods based on minimalisation of the Gibbs function. In this case we may proceed as follows We determine the equilibrium composition, e.g. by means of the method of Lagrangian multipliers for several temperature values in the vicinity of the expected Tg value. To do this, we must know the dependence of c,- = G]jRT + In P values on temperature. At every temperature, for which we have calculated the equilibrium composition, we determine the values of AH = HE D START- Let AH < 0 apply for the temperature and AH > 0 for the temperature The required temperature Tg will then lie in the interval (Te Furthermore we can apply e.g. the interval halving method or the regula falsi method (see Appendix 3). The c = Cf(T) i = 1, 2,. ..,iV relationship is determined as follows values of — (G — Hp)IT, are tabulated in the literature for various values of T. The standard temperature is usually OK or 298.15 K. The quantity is independent of temperature, and polynomial development to at most the third or fourth degree will usually suffice to elucidate the value of —(Gy — Hy)/T. 

These constraints may be combined with = 0 by using the method of Lagrangian multipliers we multiply the equations (6,1.16) by arbitrary constants jk and respectively and subtract the results from (6.1.13). On collecting terms with dipk on the left, we obtain... 

The desired solution can be obtained by the method of Lagrangian multipliers we multiply 15 11 by the parameter a, 15 12 by the parameter /3, and add the three equations, obtaining... 

In order to find the minimum of A, we must differentiate (3.10) and (3.11) with respect to Ae, and eliminate Ae from the system of equations thus obtained, with the help of the method of Lagrangian multipliers. 

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