The Euler-Lagrange Equation

Within this contimiiim approach Calm and Flilliard [48] have studied the universal properties of interfaces. While their elegant scheme is applicable to arbitrary free-energy fiinctionals with a square gradient fomi we illustrate it here for the important special case of the Ginzburg-Landau fomi. For an ideally planar mterface the profile depends only on the distance z from the interfacial plane. In mean field approximation, the profile m(z) minimizes the free-energy fiinctional (B3.6.11). This yields the Euler-Lagrange equation... 

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. 

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 ... 

The Euler-Lagrange equations can he formed for the dynamical variables q—Rji, Pji, Zph, Zph and collected into a matrix equation which, when solved, yields the wave function for the compound system at each time step. 

The critical nucleus, which can be found explicitly, is described by a homoclinic trajectory of the Euler-Lagrange equation ew = g w) (see, for instance Bates and Fife, 1993). The fact that this perturbation plays a role of a threshold is clear from Fig.9 which demonstrates extreme sensitivity of the problem to slight variations around the critical nucleus representing particular initial data (see Ngan and Truskinovsky (1996b) for details). 

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 ... 

This means that all moving nuclei (atoms) are treated as classical particles which is a serious approximation, but which was found to work very well (60,61). Applying the Euler-Lagrange equation (Eq. 2) to the Lagrangian C (Eq. 1) leads to the same equations as the well-known Newton s second law (Eq. 3). Or in other words, in classical mechanics the derivative of the Lagrangian is taken with respect to the nuclear positions. 

As the electronic energy is a function of the nuclear position as well as function of the wavefunctions , its derivative is once taken not only with respect to the nuclear position but also with respect to the wavefunction. The Euler - Lagrange equations then read ... 

Therefore the fact that 9 is arbitrary in U(l) theory compels that theory to assert that photon mass is zero. This is an unphysical result based on the Lorentz group. When we come to consider the Poincare group, as in section XIII, we find that the Wigner little group for a particle with identically zero mass is E(2), and this is unphysical. Since 9 in the U(l) gauge transform is entirely arbitrary, it is also unphysical. On the U(l) level, the Euler-Lagrange equation (825) seems to contain four unknowns, the four components of , and the field tensor H v seems to contain six unknowns. This situation is simply the result of the term 7/MV in the initial Lagrangian (824) from which Eq. (826) is obtained. However, the fundamental field tensor is defined by the 4-curl ... 

The inhomogeneous 0(3) field equation (32) is obtained through the Euler-Lagrange equation ... 

This algorithm may be derived from the Euler-Lagrange equations... 

Using the Euler-Lagrange equation (122) with the Lagrangian (126) pro-... 

The equations of motion of the held at the Higgs minimum (the minimum potential energy of the vacuum) are the Euler-Lagrange equations... 

It turns out that V is an exact form. As explained in Section I, there exist two 1-forms in S3 si and such that F = dsi and 3F = df (because the second group of cohomology of S3 is trivial). It is then clear that V = — 4a) ld si A I A Af). As a consequence, the Euler-Lagrange equations of (124) are trivial (just 0 = 0) and the action (125) takes a stationary value for a pair of maps (or of scalar helds), even if they are not dual. 

The middle term is a Proca Lagrangian for a massive photon. Here the mass of this photon is assumed to be larger than the masses of the W and W° bosons. The current / 31( is determined by the charged fermions with masses given by the Yukawa interactions with the Higgs held. These are yet to be explored. Now consider the term in the Euler-Lagrange equation... 

Since this field equation is obtained by the Euler-Lagrange equation the inhomogenous term is the result of... 

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