Lagrange method

H.E. Trease, Three-Dimensional Free Lagrangian Hydrodynamics, in The Free-Lagrange Method (edited by M.J. Fritts, W.P. Crowley and H.E. Trease), Lecture Notes in Physics, Number 238, Springer-Verlag, New York, 1985. 

Minimization of the ErrF subject to the normalization constraint is handled by the Lagrange method (Chapter 14), and leads to the following set of linear equations, where A is the multiplier associated with the normalization. 

A more elegant method of enforcing constraints is the Lagrange method. The function to be optimized depends on a number of variables,/(xi,X2,... xn), and the constraint condition can always be written as another function, g x, X2,...xat) = 0. Now define a Lagrange function as the original function minus a constant times the constraint function. 

Application of the Lagrange method, used above, leads to the following set of homogeneous simultaneous linear equations as the condition for minimum energy ... 

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 Cauchy-Lagrange method of constant multipliers yields... 

The minimization of Wlost (=Elost) entails the minimization of the integral in Eq. (4.182) with the constraint of constant production rate J (mol/h) and use of the Euler-Lagrange method... 

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

Kohnen, G., Ruger, M. and Sommerfeld, M. (1994), Convergence behavior for numerical calculations by the Eular/ Lagrange method for strongly coupled phases. ASME Symp. On Numerical Methods in Multiphase Flows FED, 185, 191. 

The arbitrary Euler-Lagrange method. It consists of moving the finite element mesh nodes with time. This method works well as long as the mesh is not too distorted. In practice, remeshing is usually required after a few time steps. 

This is the basic principle of the Lagrange method for free energy minimization. In practice, the computation can become quite complex and intricate. Part of the problem lies in the non-linear relationship between the chemical potential pi and the concentration in equation (19.60). 

Add a multiple of the orthonormaJity constraints using the Lagrange method. 

The advantage here is that the second-order Schrodinger differential equation has been transformed into a first-order partial differential equation, involving an unknown function iff and two independent variables z and t. It can be solved by the well-known Lagrange method. So equation (59) can be written as... 

Libersky LD, Petschek AG. SPH with strength of materials. In Trease HE, Eritts ME, Crowley WP, editors. Advances in the free Lagrange method. Berlin Springer Verlag 1993. 

Once an exchange-correlation functional has been selected, the computational problem is very similar to that encountered in wave mechanics HF theory determine a set of orthogonal orbitals that minimizes the energy. Since the 7[p] (and xc[p]) functional depends on the total density, a determination of the orbitals involves an iterative sequence. The orbital orthogonality constraint may be enforced by the Lagrange method (Section 12.5), again in complete analogy with wave mechanics HF methods (eq. (3.34)). 

The Lagrange method increases the number of variables by one for each constraint, which is counterintuitive since introduction of a constraint should decrease the number of variables by one. For simple objective and constraint functions, the reduction can be obtained by solving the constraint condition for one of the variables, and substituting it into the object function. 

It should be emphasized once again the important point confining considerably the application field of the calculus of variations. When determining extrema of the target functional the Euler-Lagrange method does not take into account the possibility for the existence of limitations imposed on the control parameters and phase coordinates. 

When a block diagram is manually derived using equations derived by Newton or Lagrange methods, there are some complexities to be considered. While the... 

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