Lagrange equilibrium equation

Since all other terms of the Euler-Lagrange equations remain unaffected, it is sufficient to examine the resulting generalized equilibrium equations obtained from the above Lagrangian ... 

There are several methods for imposing the loop closure conditions. The Lagrange multipliers method uses a vector of unknown reaction forces that act at the cut joints. The dynamic equilibrium equations together with the constraint equations form a set of DAE s that must be integrated. In order to reduce the size of the final system of equations, Avello et al.(1993) presented an approach based on the penalty formulation developed by Bayo et al. (1988). In this paper, the reduction of the original system of DAE s to a set of ODE s is achieved through a second velocity transformation. The numerical efficiency of the last two approaches in terms of CPU time per function evaluation is very similar, as it is shown by the practical examples solved in section 4. 

A solution will now be sought which will correspond to a periodic array of discli-nation lines separated by the distance L as indicated in Fig. 3.24. The disclinations are perpendicular to the page and are located at the peaks and troughs appearing in the Figure. The Euler-Lagrange equation for Fgurf provides the equilibrium equation... 

The components of 11 can be obtained explicitly from the general vector formula (6.92) if desired, with, of course, a and c given by (6.120). It is a simple exercise to use (6.125), (6.130) and (6.131) to verify that V x = 0 where y is defined by (6.132), indicating that the a-equations (6.123) are fulfilled. It now follows that solving the equilibrium equation (6.129) for 0(z) will result in a full solution to the equilibrium equations for a and c given by (6.120). Although we have calculated all the Lagrange multipliers explicitly in this example, it is not often required in simple static problems to know IS explicitly since it is frequently sufficient to know that 7 and fS can be found if desired once a solution to the c-equations has been determined, as mentioned previously. 

STANJAN The Element Potential Method for Chemical Equilibrium Analysis Implementation in the Interactive Program STANJAN, W.C. Reynolds, Thermosciences Division, Department of Mechanical Engineering, Stanford University, Stanford, CA, 1986. A computer program for IBM PC and compatibles for making chemical equilibrium calculations in an interactive environment. The equilibrium calculations use a version of the method of element potentials in which exact equations for the gas-phase mole fractions are derived in terms of Lagrange multipliers associated with the atomic constraints. The Lagrange multipliers (the element potentials ) and the total number of moles are adjusted to meet the constraints and to render the sum of mole fractions unity. If condensed phases are present, their populations also are adjusted to achieve phase equilibrium. However, the condensed-phase species need not be present in the gas-phase, and this enables the method to deal with problems in which the gas-phase mole fraction of a condensed-phase species is extremely low, as with the formation of carbon particulates. 

The equilibrium state of fullerite is determined from the conditions for minimum of free energy, that are easily found by the method of the indeterminate Lagrange factor X. Factor X is correlated with the equation of the relation between numbers n )(s = O1,O2,0)Q1,Q2,D1,D2,D3). After substitution of free energy F into the conditions for free energy minimum, we find equations whereby the numbers Ns of hydrogen atoms in the interstitial sites s = Ob 02, 0, Qi, Q2, D,. D2, D3 with /th configuration of O, 02 fiillerenes are evaluated... 

This is equivalent to the equilibrium condition pv = 0, and so the objective is to calculate the Lagrange multipliers. The number of Lagrange multipliers is equal to the number of components. Once the Lagrange multipliers have been obtained, the chemical potentials can be calculated using equation 5.3-2 and the equilibrium mole fractions can be calculated using... 

Now, by using the Lagrange multiplicators method, the equation governing the maximalization of the entropy required in order to obtain the statistical equilibrium, in view of the four constraints ... 

Solve the nine equations from steps 4 and 5 simultaneously, to find yco, yco. ycH4> yn2> yn/>, c, X0, XB, and nT. This step should, of course, be done on a computer. The Lagrange multipliers have no physical significance and should be eliminated from the solution scheme. The equilibrium composition is thus found to be as follows yco2 = 0.122 ycH4 = 0.166 yco = 0.378 vh, = 0.290 and yn2o = 0.044. 

Rational thermodynamics is formulated based on the following hypotheses (i) absolute temperature and entropy are not limited to near-equilibrium situations, (ii) it is assumed that systems have memories, their behavior at a given instant of time is determined by the history of the variables, and (iii) the second law of thermodynamics is expressed in mathematical terms by means of the Clausius-Duhem inequality. The balance equations were combined with the Clausius-Duhem inequality by means of arbitrary source terms, or by an approach based on Lagrange multipliers. 

The chemical potential p, = 6J f6p enters the respective Euler-Lagrange equation obtained by minimizing the grand ensemble thermodynamic potential — p J pd a , which defines the equilibrium particle density distribution... 

But we can make no further progress until we choose standard states for all species i the options are discussed in 10.4.1. Nevertheless, the results (10.3.37) and (10.3.38) represent (C + nig) coupled algebraic equations that can be solved for C unknown equilibrium mole numbers and nig unknown Lagrange multipliers. The final form will be developed in 10.4.5. 

Recall the fugadties depend on the unknown mole fractions, but the standard state fugacities f° are constants whose values are obtained via judicious choices for standard states ( 10.4.1). Note that the standard states must be applied consistently to both fugacities and chemical potentials. Once those choices have been made, the six equations (10.3.43)-(10.3.45) and (10.3.49)-(10.3.51) can be solved for the three equilibrium mole fractions, the total number of moles (relative to the selected basis), and the two Lagrange multipliers. The calculation will be finished in 10.4.6. 

In this nonstoichiometric method, part of the solution is the set of values for the Lagrange multipliers In most situations these multipliers have little physical significance they merely serve to ensure conservation of atoms, so their values are a necessary but nonphysical by-product of the calculation. When the number of elements Mg is less than the number of species C, the C equations (10.4.35) could be combined to eliminate the nig multipliers X, so their values would not obtained explicitly. However, if such a combination is done, the result is equivalent to the stoichiometric expression for the equilibrium constant, and the computational advantages of the nonstoichiometric method are lost. 

Finally note that we could combine the three equations (10.4.43)-(10.4.45) to eliminate the two Lagrange multipliers. But doing so produces the stoichiometric equation (10.4.31) that relates the equilibrium constant to the mole fractions. In other words, the stoichiometric and nonstoicdiiometric developments are merely two different formulations of the same equations, though in particular applications one approacdi or the other may be easier to use. 

The equilibrium density profile p z) is the one that minimises the surface tension functional thus by the standard methods of calculus of variations we can write down an Euler-Lagrange equation that the density profile must satisfy ... 

Substitution of Eqs. (3.7a), (3.7b), (3.8a), (3.8b) and (3.9) into Eq. (3.5) allows to obtain free energies Gy and Gs as the functions of polarization P3 only. It is obvious that they have to be different for the particles of cylindrical and spherical shapes as well as for the films. Variation of these free energies over F3 allows deriving Euler-Lagrange equations, which define the equilibrium values of order parameter. 

F7 /8 y = Oij in the form of Euler-Lagrange equations (ojk is the stress tensor) along with the boundary conditions (see e.g. Refs. [47, 58, 59]). This system of differential equations should be solved along with the equations of mechanical equilibrium daij x) /9x, = 0 and compatibility equations equivalent to the mechanical displacement vector , continuity [100]. 

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