Euler-Lagrange functional method

The free energy is at its minimum when the liquid crystal is at equilibrium. One can obtain the minimum by means of the Euler-Lagrange functional method. 

An alternative procedure is the dynamic programming method of Bellman (1957) which is based on the principle of optimality and the imbedding approach. The principle of optimality yields the Hamilton-Jacobi partial differential equation, whose solution results in an optimal control policy. Euler-Lagrange and Pontrya-gin s equations are applicable to systems with non-linear, time-varying state equations and non-quadratic, time varying performance criteria. The Hamilton-Jacobi equation is usually solved for the important and special case of the linear time-invariant plant with quadratic performance criterion (called the performance index), which takes the form of the matrix Riccati (1724) equation. This produces an optimal control law as a linear function of the state vector components which is always stable, providing the system is controllable. 

We present how to treat the polarization effect on the static and dynamic properties in molten lithium iodide (Lil). Iodide anion has the biggest polarizability among all the halogen anions and lithium cation has the smallest polarizability among all the alkaline metal cations. The mass ratio of I to Li is 18.3 and the ion size ratio is 3.6, so we expect the most drastic characteristic motion of ions is observed. The softness of the iodide ion was examined by modifying the repulsive term in the Born-Mayer-Huggins type potential function in the previous workL In the present work we consider the polarizability of iodide ion with the dipole rod method in which the dipole rod is put at the center of mass and we solve the Euler-Lagrange equation. This method is one type of Car-Parrinello method. 

If we now look back at Vcin der Waals treatment, sec. 2.5. several steps can be recognized. In a sense he anticipated the present method. His function to be minimized is [2.5.19] he eliminated boundary condition [A1.3] by working grand canonically and [2.5.25] is his Euler-Lagrange equation. From this. F could be written as [2.5.30] ). 

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 ... 

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. 

Some functions depend on more than a single variable. To find extrema of such functions, it is necessary to find where all the partial derivatives are zero. To find extrema of multivariate functions that are subject to constraints, the Lagrange multiplier method is useful. Integrating multivariate functions is different from integrating single-variable functions multivariate functions require the concept of a pathway. State functions do not depend on the pathway of integration. The Euler reciprocal relation relation can be used to distinguish state functions from path-dependent functions. In the next three chapters, we will combine the First and Second Laws with multivariate calculus to derive the principles of thermodynamics. 

Because of a vivid interest in spray combustion, quite a few papers deal with the effect of finely dispersed particles or droplets on the turbulence characteristics of jets and sprays. The interaction of particle dynamics and turbulence in these flows has prompted very fundamental stochastic approaches involving filtering and averaging techniques, probability density functions, and quadrature-based moment methods (Fevrier et al, 2005 Fox, 2012 Labourasse et al, 2007) which are beyond the scope of this chapter. Riber et al (2009) and Senoner et al (2009) obtained LES results for recirculating and evaporating two-phase flows, respectively, by both Euler-Lagrange and Euler—Euler methods and compared them mutually and with experimental data. 

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