Lagrange’s method

The use of this theory in studies of nonlinear oscillations was suggested in 1929 (by Andronov). At a later date (1937) Krylov and Bogoliubov (K.B.) simplified somewhat the method of attack by a device resembling Lagrange s method of the variation of parameters, and in this form the method became useful for solving practical problems. Most of these early applications were to autonomous systems (mainly the self-excited oscillations), but later the method was extended to... 

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... 

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-... 

Rather than minimize the energy function o(P) = (P, H) by varying over the set of fe-matrices, there is a dual formulation where the bottom eigenvalue /lo(H + S) of the matrix H + S is maximized over the set of Pauli matrices S e The dual formulation can be derived using Lagrange s method, which requires converting the constrained energy problem to an unconstrained one. If... 

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... 

The problem may be seen as that of minimizing an objective function, subject to a set of nonlinear constraints. It may be solved by Lagrange s method... 

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. 

In order to integrate we use Simpson s formula. We obtain U and a as we would like to see it, with the help of a four-point interpolation following Lagrange s method. 

If we have a constrained maximum or minimum problem with more than two variables, the direct method of substituting the constraint relation into the function is usually not practical. Lagrange s method finds a constrained maximum or minimum without substituting the constraint relation into the function. If the constraint is written in the form g(x, y) = 0, the method for finding the constrained maximum or minimum in f x, y) is as follows ... 

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... 

The classical building blocks that we need are Newton s method (Newton 1687) for unconstrained optimization, Lagrange s method (Lagrange 1788) for optimization with equality constraints, and Fiacco and McCormick s barrier method (Fiacco and McCormick 1968) for optimization with inequality constraints. Let us review these. A good general reference is Bazarra and Shetty 1979. 

This is now a problem with equality constraints, so we can use Lagrange s method. The Lagrangian function is... 

This could have been obtained by applying Newton s second law, but in more complicated systems, Lagrange s method is convenient to apply. Lagrange was bom in Turin, Italy and his book on Analytical Mechanics was published in 1788 (Oliveira 2013). 

By using the calculus of variations, the integral form of Lagrange s method can be obtained. This is known as Hamilton s principle (Goldstein et al. 2002). It can be stated as follows ... 

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. 

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