Method of undetermined multipliers

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 Lagrange Method of Undetermined Multipliers. To prove important statistical mechanical results in Chapter 5, we need the method of undetermined multipliers, due to Lagrange.42 This method can be enunciated as follows Assume that a function f(xu x. .., xn) of n variables X, x2,..., xn is subject to two auxiliary conditions ... 

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 solution can be obtained by application of the Lagrange method of undetermined multiplier (Sec. VII.2). 

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... 

Constraint dynamics is just what it appears to be the equations of motion of the molecules are altered so that their motions are constrained to follow trajectories modified to mclude a constraint or constraints such as constant (total) kinetic energy or constant pressure, where the pressure in a dense adsorbed phase is given by the virial theorem. In statistical mechanics where large numbers of particles are involved, constraints are added by using the method of undetermined multipliers. (This approach to constrained dynamics was presented many years ago for mechanical systems by Gauss.) Suppose one has a constraint g(R, V)=0 that depends upon all the coordinates R=rj,r2...rN and velocities V=Vi,V2,...vn of all N particles in the system. By differentiation with respect to time, this constraint can be rewritten as l dV/dt -i- s = 0 where I and s are functions of R and V only. Gauss principle states that the constrained equations of motion can be written as ... 

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. 

Since the method of undetermined multipliers requires only (Further Information 16.1) din IV, only the terms din //,- survive, The constant term, j in 2jr, drops out, as do all terms d in /V. The difference, then, is in terms arising from In //, We need to compare //,- In //, to In /< , as both these terms survive the differentiation. The derivatives are... 

Find the minimum in the function of the previous problem subject to the constraint X + y = 2.T)o this by substitution and by the 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... 

On the other hand, we can use the method of undetermined multipliers to find out the extremum. First, we normalize the constraint equation into x+y—1 =0. In general the constraint equation would then have the form g(x, y) = 0. Then we add the constraint equation with a factor to the equation where to find out the extremum. Thus, we form... 

The method of undetermined multipliers has the advantage that we do not need the constraint equation itself, but rather the derivatives are needed, because the procedure needs Adg(x, y). 

In the argument set more convenient method of undetermined multipliers by Lagrange. In the differential form this equation reads as... 

Using the constraint equation (2.56) together with the method of undetermined multipliers,... 

Thus the total entropy becomes stationary if the temperatures of both systems are equal. We emphasize that in more complicated systems we can handle the constraint of constant energy more conveniently by the method of undetermined multipliers. In our derivation we have used the chain rule for derivation. 

Using the method of undetermined multipliers, from Eqs. (5.26) and (5.27) the condition for stationary energy turns out as... 

These two conditions are fulfilled if the method of undetermined multipliers is used to get the maximum by variational calculus. 

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