The method of Lagrangian multipliers

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 unknown parameters Xp are chosen such that the velocity V determined by eqn (3.143) satisfies. 

The Smoluchowski equation is obtained if eqn (3.163) is substituted into the continuity equation  

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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. 

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. 

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

Assuming uniform prior probabilities, we maximise S subject to these constraints. This is a standard variation problem solved by the use of Lagrangian multipliers. A numerical solution using standard variation methods gives i.p6j=. 05435, 0.07877, 0.11416, 0.16545, 0.23977, 0.34749 with an entropy of 1.61358 natural units. 

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). 

The charges are not independent variables since there is a charge conservation constraint. In the following we constrain each molecule to be neutrtd, Qia = 0. We treat the charges undetermined multipliers to enforce the constraint. The Lagrangian is then... 

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

A stationary point for a general Lagrangian function may or may not be a loctil extremum. If, as described in Section 2, suitable convexity conditions hold, then the method of Lagrange multipliers will yield a global minimum. 

The time consumer in calculating chemical equilibria by one of the linear or nonlinear programming methods is usually considerably longer than the time needed for methods based on the theory of Lagrangian multipliers For this reason, linear and non-linear programming are rather seldom employed in calculating chemical equilibria. 

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... 

Since the x1 and x averages coincide by definition, to solve the posed problem it is necessary to find a minimum of dispersion (5.25) with conditions (5.22) satisfied. Let us use the method of uncertain Lagrangian multipliers and form an auxiliary expression ... 

As mentioned above, Ryckaert et al. introduced the method of undetermined parameters, which is essentially the analytical method modified to ensure that the constraints are satisfied exactly at each time step. The method of undetermined parameters is discussed in detail, in its most general form, in a later section. Basically, the aforementioned modification requires that the highest derivatives with respect to time of the Lagrangian multipliers (i.e., of order be replaced by a set of undetermined parameters with values to be determined such that the constraints are satisfied exactly at each time step. As will be seen, the algorithm does not introduce into the trajectories additional numerical errors any worse than the error already present in the integration scheme itself. 

The numerical procedure used to solve the final equations The analytical method leads to a system of equations linear in the unknowns (i.e., the Lagrangian multipliers and their time derivatives up to order Therefore standard numerical techniques for solving such systems can be employed. The method of undetermined parameters leads to an additional system of equations generally nonlinear in the unknowns (i.e., the derivatives of the Lagrange multipliers of order s ,3x)- The order of nonlinearity depends on the particular... 

The most general form of holonomic constraint is nonlinear in the particle positions. Even the simple bond-stretch constraint is nonlinear. Consequently, Eq. [39] is in general a system of / coupled nonlinear equations, to be solved for the / unknowns (7). This nonlinear system of equations must be contrasted with the linear system of equations Eqs. [10] and [11] (which is also in general part of the method of undetermined parameters) used in the analytical method to solve for the Lagrangian multipliers and their derivatives. A solution of Eq. [39] can be achieved in two steps ... 

For the Verlet scheme with = 0, the are replaced by the (k = 1,. ..,/), and the method of undetermined parameters incorporating the basic Verlet scheme could equally well be termed, and is often referred to in the literature as, the method of undetermined (Lagrangian) multipliers. However, in general where > 0, the Lagrangian multipliers and their derivatives [X °>(tQ),. . ., computed, in addition to the undeter-... 

The methods may be classified in two groups. The first group of methods is based on the classical mathematical theory of Lagrangian multipliers. The second includes methods in which use is made of the theory of linear or convex programming. 

This method makes use of the Clascal mathematical theory of Lagrangian multipliers. Differing from the White-Johnson-Dantzig method, the function F, defined by relation (5.60), is directly substituted into the conditions (5.61) instead of a quadratic approximation of the function Q from relation (5.60). In this way, we obtain the equations... 

The equations (5.77) now form a set of M non-linear equations for M variables Ai, 2, Ajvf. The equilibrium composition is now easily determined from the calculated values of Lagrangian multipliers and the value of t by substitution into the equations (5.74). The set of non-linear equations (5.77) is best solved by means of Newton s method with reduction parameter. The choice of the first approximation will be discussed in the course of a more detailed description of the method in section (5.5). 

This value will not lead in one step to the minimum of (2.6.3.1-2). The value Abs+i is then added to bs and so on, until convergence is achieved. The method converges rapidly close to the minimum of 5(P) but with poor initial values of bo it may not. To avoid divergence in this region, Marquardt [1963] worked out a compromise between the method of steepest descent and that of Newton-Gauss. The steepest descent method is most efficient far from the minimum, whereas it is the other way around close to the minimum. The compromise has the ability to change the size and the direction of the optimization step by means of a scalar parameter X, which is a Lagrangian multiplier, so that (2.6.3.2-4) becomes... 

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