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
Thomsen in  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
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 equations
Vese and Osher  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.