Different length scales are known to be associated with defect core sizes in these two different regimes, and it is also shown below how these can be identified via balances in appropriate scalings of the

Euler-Lagrange equations associated with (1.1).

For the sake of simplicity, the derivations of the

Euler-Lagrange equations are attached in Appendix B.

The first one is based on Bogomolny's trick by assuming the existence of a homotopy invariant term in the energy density that does not contribute to

Euler-Lagrange equations [17].

Thomsen in [3] obtained the first variation and the

Euler-Lagrange equations corresponding to (1).

The equations of motion associated with this Lagrangian are obtained from the

Euler-Lagrange equations [6].

In recent years, under different differentiability, several researchers [2-9] studied the fractional

Euler-Lagrange equations for general fractional variational problems.

For the mechanism of class IV with dwell driven link construct the

Euler-Lagrange equations, substitute equations (9) and (10) in equation (4), will also assume that [F.sub.1] = [T.sub.L] and L = T - V.

The solution of minimization problem requires calculation of

Euler-Lagrange equations, found via partial derivatives of the designed cost function, IP, for given state variables as

Vese and Guyader explore variational models, their corresponding

Euler-Lagrange equations, and numeral implementations for image processing.

The principle of stationary action [delta]S = 0 respectively induced the

Euler-Lagrange equationsVese and Osher [39] proposed a VO model which approximates Meyer's theoretical model; they gave an [L.sup.P] approximation to the norm [[parallel] * [parallel].sub.G], the corresponding

Euler-Lagrange equations, and numerical methods, but the speed of computation is slow.