Lagrangian minimization

Arbitrary-Lagrangian-Eulerian (ALE) codes dynamically position the mesh to optimize some feature of the solution. An ALE code has tremendous flexibility. It can treat part of the mesh in a Lagrangian fashion (mesh velocity equation to particle velocity), part of the mesh in an Eulerian fashion (mesh velocity equal to zero), and part in an intermediate fashion (arbitrary mesh velocity). All these techniques can be applied to different parts of the mesh at the same time as shown in Fig. 9.18. In particular, an element can be Lagrangian until the element distortion exceeds some criteria when the nodes are repositioned to minimize the distortion. 

Ehf from equation (1-20) is obviously a functional of the spin orbitals, EHF = E[ XJ]. Thus, the variational freedom in this expression is in the choice of the orbitals. In addition, the constraint that the % remain orthonormal must be satisfied throughout the minimization, which introduces the Lagrangian multipliers e in the resulting equations. These equations (1-24) represent the Hartree-Fock equations, which determine the best spin orbitals, i. e., those (xj for which EHF attains its lowest value (for a detailed derivation see Szabo and Ostlund, 1982)... 

To show that this constrained minimization is indeed equivalent to the steady-state formulation, let us adjoin the equality constraints to the objective function to form the Lagrangian function,... 

The PDF of an inert scalar is unchanged by the first two steps, but approaches the well mixed condition during step (3).108 The overall rate of mixing will be determined by the slowest step in the process. In general, this will be step (1). Note also that, except in the linear-eddy model (Kerstein 1988), interactions between Lagrangian fluid particles are not accounted for in step (1). This limits the applicability of most mechanistic models to cases where a small volume of fluid is mixed into a much larger volume (i.e., where interactions between fluid particles will be minimal). 

The Ay are Lagrangian multipliers arising from the side conditions of Eq. (7) which maintain orbital orthonormality during the minimization process. Solution of Eq. (17) in conjunction with Eq. (7) determines simul-... 

Step 1. For a given set of Lagrange multipliers and penalty parameter minimize the Lagrangian function L R) to obtain an improved estimate of the factorized 2-RDM at the energy minimum. 

The constraint is enforced by introducing a Lagrangian multiplier X in the minimization function given by... 

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 effect of the Higgs mechanism can be seen most clearly by minimizing the Lagrangian (251) with respect to A ... 

The calculation of the equilibrium composition of a system of chemical reactions with equcalcc is based on minimizing the Gibbs energy subject to the conservation condition An = nc. This is accomplished by using a Lagrangian L defined by... 

If we are interested simply in minimizing equation (5) with respect to T, subject to the constraint of equation (9), then from the classical theory of optimization we know that we can incorporate this constraint by constructing the lagrangian... 

The standard procedure for rendering F an extremal, subject to the requirement that the nA remain fixed, consists in introducing Lagrangian multipliers, Ai( one for each species, and minimizing the enlarged function Fg - (i) ini That is, we require... 

In most electronegativity equalization models, if the energy is quadratic in the charges (as in Eq. [36]), the minimization condition (Eq. [41]) leads to a coupled set of linear equations for the charges. As with the polarizable point dipole and shell models, solving for the charges can be done by matrix inversion, iteration, or extended Lagrangian methods. 

The minimization is carried out using the Lagrangian multiplier technique we minimize the enlarged function Fg — A, n,, where the 1, are the undetermined multipliers. Thus, we write... 

For a system with N degrees of freedom, q, i = 1 to N, this equation is obtained for each of the N coordinates qi. These are Lagrange s equations of motion, the equations of motion for a system obeying classical mechanics. Thus, the Lagrangian, which minimizes the value of the action integral along the true trajectory between the times tj and fj, is also the function which yields the equations of motion when inserted into the Euler equation (8.50). 

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