Euler Lagrange equation

Since the variation of the action has to vanish for all variations of the gauge field this immediately yields the Euler-Lagrange equations for the gauge field A,  

Insertion of the explicit form of em as given by Eq. (3.191) into this Euler-Lagrange equation and directly evaluating the corresponding derivatives yields the equations of motion for the gauge field A ,  

198) just represents the inhomogeneous Maxwell equations in covariant form, cf. Eq. (3.172). We have thus derived the inhomogeneous Maxwell equations as the natural equations of motion for the gauge potential A. The sources, as described by the charge-current density are considered as external variables which do not represent dynamical degrees of freedom, i.e., only the action (or effect) of the sources on the gauge fields is taken into account. 

Application of a variation of the four-dimensional path Xi(t) of the particle to the Lagrangian L2 immediately yields the covariant Euler-Lagrange equations for the charged particle. 

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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 ... 

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... 

Next we impose the orthonormality constraint on the wave functions by means of Lagrange multipliers, sy, and obtain the n one-electron Euler-Lagrange equations ... 

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 ... 

For further details with respect to notation and the derivation of the corresponding Euler-Lagrange equations we refer to [11], By replacing the meson fields by their expectation values one obtains an effective Dirac equation for... 

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. 

Gear, C. W., Leimkuhler, B., and Gupta, G. K., Automatic integration Euler-Lagrange equations with constraints, Journal of Computational and Applied Mathematics 12 and 13, 77-90 (1985). 

Euler-Lagrange Equation for Intra-Orbit Optimization of p(r) 206... 

Euler-Lagrange Equations for the Intra-Orbit Optimization of... 

As indicated in Fig. 7, the next step after either an explicit or an implicit energy density functional orbit optimization procedure. For this purpose, one introduces the auxiliary functional Q[p(r) made up of the energy functional [p(r) 9 ]. plus the auxiliary conditions which must be imposed on the variational magnitudes. Notice that there are many ways of carrying out this variation, but that - in general - one obtains Euler-Lagrange equations by setting W[p(r) = 0. 

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). 

The total energy E of the system is also a functional of the density distribution, E = [] (r)]. Therefore, if the form of this functional is known, the ground-state electron density distribution n t) can be determined by its Euler-Lagrange equation. However, except for the electron gas of almost constant density, the form of the functional [ (r)] cannot be determined a priori. 

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 ... 

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