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 ... 

Therefore S can be constructed by solving the Lagrange equations for the nuclear variables. Alternatively it can be obtained from the Hamilton equations introducing the generalized momenta P = dS/dQ and solving the equations... 

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 ... 

Since the ground-state electron density minimizes the energy, subject to the normalization constraint, Jp(r)dr — N = 0, theEuler-Lagrange equation (see Equation 4.23) becomes... 

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... 

Lagrange equation for relative movement of isolated system of two interacting material points with masses mi and m2 in coordinate x can be presented as follows ... 

The calculation of the torsional accelerations, i.e. the second time derivatives of the torsion angles, is the crucial point of a torsion angle dynamics algorithm. The equations of motion for a classical mechanical system with generalized coordinates are the Lagrange equations... 

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). 

In the development of the various treated arguments, we adopted infinitesimal calculation procedures, vectorial notations (gradient, Laplacian), and some mathematical functions (Lagrange equations, Hamiltonians, error functions) that should already be known to the reader but that perhaps would be useful to review in a concise way. 

This sort of equation, known as a Lagrange equation of motion, is generally valid for systems with unconstrained degrees of freedom. If a system has n degrees of freedom, n second-order Lagrange equations will exist (functions of qj, qj, and time quadratic in qj). 

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. 

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