Lagrange small variations

This needs to be minimized with respect to all possible variations of l r) compatible with the normalization condition, equation (15.34). Minimization can be accomplished via the introduction of the Lagrange multiplier Sq such that for small variations in... 

When Lagrange multipliers are known in correspondence with the solution, it is possible to perform a sensitivity analysis of the solution subject to small variations in the constraint conditions (Buzzi-Ferraris and Manenti, 2010a). 

Now, consider independently small changes in e(r), / and x- These variations are now allowed to be independent because of the use of the Lagrange multiplier, x(0 to impose the constraint equation. 

As in ordinary mechanics, the Euler - Lagrange equations demand p = F, which arises from a variational problem. Now we want to transform the position q into another new position. We may express this as the position g is a function of some parameter s, i.e., q = q s). In the same way the velocity transforms when the position is changed, q = q s). When the Lagrangian is invariant for such a transformation, we address this as a kind of symmetry, here the translational homogeneity of space. This means, changing the parameter 5 by a small value, the Lagrangian should remain the same ... 

To check if any active constraint is to be changed into a passive one, we can exploit ( ) the aforementioned feature of performing a sensitivity analysis of the solution by means of Lagrange multipliers against small constraint variations. 

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