Euler-Lagrange relation

The proof of this principle starts from the Euler-Lagrange relation for the chemical potential, applied to the atom and respectively to the molecule and then writing the difference, so obtaining the atoms-in-molecule chemical potential difference ... 

In this case, the conditions of extremum of the functional given by eq A.3 with respect to ij), E, and P, considered as independent functions (the Euler—Lagrange equations), lead to the Maxwell equations and the equation that relates the polarization to the field. It should be noted that the above equations imply that p is independent of E and P. Of course, this assumption is not valid in the presence of an electrolyte. 

Clearly, the assets of a useful, in itself noncontradictory, and physically based CNM analysis are the internal vibrational motions and their properties as well as the amplitudes that relate internal modes to normal modes. As shown in the previous section, the adiabatic internal modes an are the appropriate candidates for internal modes. Adiabatic modes are based on a dynamic principle, they are calculated by solving the Euler-Lagrange equations, they are independent of the composition of the set of internal coordinates to describe a molecule, and they are unique in so far as they provide a strict separation of electronic and mass effects [18,19]. Therefore, they fulfil the first requirement for a physically based CNM analysis. 

With relation (4.171) in (4.165), a new Euler-Lagrange equation of the electronic system is obtained, involving the chemical potential... 

For small values of E, the dielectric coupling (oc E ) can be ignored, and the Euler-Lagrange equations corresponding to 0 z) and (f> z) can be used to get a very simple relation for the maximum twist angle (0) occurring close to the homeotropically aligned plate ... 

Classical methods of calculus of variations are attractive from the point of view of the opportunity to obtain solutions in analytical form. But this is feasible in simple cases, which often are far from the demands of the state-of-art practice. In complicated cases, at a large number of optimization parameters, numerical approaches are used to solve the appropriate Euler-Lagrange equations. The main obstacle arising here is related to the fact that the numerical solution of the system of differential equations may turn out to be more complicated than the solution fi-om the very beginning of the optimization problem by numerical methods of mathematical programming. 

This is the most prominent and important example of a velocity-dependent potential U. The evaluation of the Euler-Lagrange equations (2.53) together with the relations between electromagnetic fields and potentials in Eq. (2.127)... 

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. 

It is possible to determine the relations among the kinetic coefficients from the Euler-Lagrange- equations belonging to the variation of i-th coordinate jL and of the heat and a-th chemical component current densities... 

The Euler-Lagrange variational principle leads to the relations... 

Substituting this into the Euler-Lagrange equation (11.10), we find the following relation ... 

On the basis of all information gathered, it is fair to conclude that one of the major drivers behind the occurrence, the shape, and the dynamics of these mesoscale structures is the fluid—particle interaction force that plays a dominant role, both in stability analyses and in CFD simulations of any type. This role is related to the difference in inertia of the two phases and, as a result, to the temporally and spatially varying difference in velocities of dispersed phase (particles) and carrier (or continuous) phase. Cluster and strand formation seem to be closely related to the continuous chaotic accelerations in a turbulent carrier fluid (in the Euler—Lagrange approach) or a turbulent continuous phase (in the Euler-Euler or two-fluid approach). An interesting explanation for cluster formation is the sweep-stick mechanism proposed by Goto and Vasillicos (2008). 

Inhomogeneous or multiphase reaction systems are characterised by the presence of macroscopic (in relation to the molecular level) inhomogeneities. Numerical calculations of the hydrodynamics of such flows are extremely complicated. There are two opposite approaches to their characterisation [63, 64] the Euler approach, with consideration of the interfacial interaction (interpenetrating continuums model) and the Lagrange approach, of integration by discrete particle trajectories (droplets, bubbles, and so on). The presence of a substantial amount of discrete particles in real systems makes the Lagrange approach inapplicable to study motion in multicomponent systems. Under the Euler approach, a two-phase flow is described... 

The ATB model is based on the rigid body dynamics which uses Euler s equations of motion with constraint relations of the type employed in the Lagrange method. The model has been successfully used to study the articulated human body motion under various types of body segment and joint loads. The technology of robotic telepresence will provide remote, closed-loop, human control of mobile robots. 

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