Variational principles Euler

Writing the Euler-Lagrange equations in terms of the single-particle wave functions (tpi) the variation principle finally leads to the effective singleelectron equation, well-known as the Kohn-Sham (KS) equation ... 

Euler-Lagrange equations, electron nuclear dynamics (END), time-dependent variational principle (TDVP) basic ansatz, 330-333 free electrons, 333-334 Evans-Dewar-Zimmerman approach, phase-change rule, 435... 

For a family of trajectories all starting at the value X(to) and at t=t all arriving at X(t), there is one trajectory that renders the action stationary. The classical mechanical trajectory of a given dynamical system is the one for which 5S=0, i.e. the action becomes stationary. The equation of motion is obtained from this variational principle [59], The corresponding Euler-Lagrange equations are obtained d(3L/3vk)/dt = 9L/dXk. In Cartesian coordinates these equations become Newton s equations of motion for each nucleus of mass Mk ... 

Define the functional U(Q [v[/])=q integration over the electronic configuration space is indicated as a sub-index. The variational principle applied [6] to the (spin-free) function space v /(q) leads to the Euler-Lagrange equation ... 

Variational principles for classical mechanics originated in modem times with the principle of least action, formulated first imprecisely by Maupertuis and then as an example of the new calculus of variations by Euler (1744) [436], Although not stated explicitly by either Maupertuis or Euler, stationary action is valid only for motion in which energy is conserved. With this proviso, in modem notation for generalized coordinates,... 

Using the variation principle the Euler-equation can be obtained ... 

It should be emphasized, that this density functional for the ground-state energy differs from the (HK) functional for the energy, Ev[p, in which the external potential is fixed (not related to the trial density p). Only for the exact ground-state density Ev[p0] = E[p0. The variational principle of equation (16), also defined for the fixed v(r), determines the minimum of Ev[p subject to the density normalization constraint iV[p] = J p(r)dr = N. The associated Euler equation for the optimum (ground-state) density p(r) = p /V0. v r] = p0(r) reads ... 

The variation principle is more fundamental than the differential equa tion to which it is equivalent the differential equation is simply the Euler-Lagrange equation generated by application of the variation principle within the boundaries of applicability. 

We now show, as announced before, that the field equations (1) are the Euler-Lagrange equations of the variation principle... 

Orbit Variational Principle and Euler-Lagrange Equation... 

One of the most important consequences of the HK2 conveys the rewriting of the variational principle (4.384) in the light of above iV-contingency conditions of the trial densities as the working Euler type equation ... 

Show that the Euler-Lagrange equations are identical with the inhomogeneous transport equations, when the variation principle (73) is a free-task varying only the intensive parameters. (Such type of non-exact principles of thermodynamics are called the partial variational principles p4],... 

Supposing there are missing connections between the forces and currents, the variational principle (159) is the necessary condition for the extremum at the free variation of the intensives and fluxes we get the Euler-Lagrange equations belonging to the variation principle (159) are the following... 

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

We seek a variational principle which will characterize the time-independent state of this diffusion. Let us first consider the principle originally suggested by Prigogine.i da = 0. The Euler... 

We now have to find a variational principle who.se Euler equation is... 

Using the variational principle, an Euler-Lagrange equation can be derived for the liquid-vapor profile Cl( )... 

Variational calculus, Dreyfus (1962), may be employed to obtain a set of differential equations with certain boundary condition properties, known as the Euler-Lagrange equations. The maximum principle of Pontryagin (1962) can also be applied to provide the same boundary conditions by using a Hamiltonian function. 

Before pursuing the variation of the atomic action integral, it is helpful to first recover the statement of the principle of stationary action in the Schrodinger representation for the total system. If one sets the boundary of the region Cl at infinity in eqn (8.118) to obtain the variation of the total system action integral 2 [ ]> and restricts the variation so that ST vanishes at the time end-points and the end-points themselves are not varied, then only the terms multiplied by the variations in the first integral on the right-hand side remain. The Euler equation obtained by the requirement that this restricted... 

J. -L. Lagrange proposed his original solution for general problems and named it the variational method (1754). Euler introduced this variation method in his paper entitled Principle of the variation method (1766) (Ekeland 2009). 

Chapter 3 gives the fundamental mathematical principles of the calculus of variations used for the optimization of dynamic systems. Classical results of the Euler equation for functional extrema and those of constrained optimization given by the Euler-Lagrange equation are developed. 

Due to the Onsager s principle of last dissipation of energy this functional is maximal with respect to variations of the fluxes j,. The necessary condition of the maximum are the constitutive equations, which are presented as Euler-Lagrange equations ... 

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