Variational calculus functional analysis

Structural analysis, initially developed on an intuitive basis, later became identified with variational calculus, in which the Ritz procedure is used to minimize a functional derived mathematically or arrived at directly from physical principles. By substituting the final solutions into the variational statement of the problem and minimizing the latter, the FEM equations are obtained. Example 15.2 gives a very simple demonstration of this procedure. 

Dieudonnd, J. (1981). History of Functional Analysis (North-Holland, Amsterdam). [147] Goldstine, H.H. (1980). A History of the Calculus of Variations from the 17th through the 19th Century (Springer-Verlag, Berlin). 

E is said to be a functional of the functional E = E[W] takes a value that depends on the functional form of W, not merely on particular numerical values of a set of independent variables (adjustable parameters). The analysis of the behaviour of E[W] with respect to variation of W is functional analysis and parallels the study of the behaviour of f(x) with respect to variation of x (i.e. analysis in the usual sense) thus, for example, it is possible to discuss variations in terms of functional derivatives SEISfP. We shall, however, make little use of such concepts, in spite of their formal value (see e.g. Feynman and Hibbs, 1965), since variation functions are most commonly defined in terms of a finite number of parameters, to which the ordinary rules of differential calculus can be applied. The other problems we meet can also be solved by elementary methods. 

The determination of the entropy of mixing functional with bonding constraints is obtained by introducing an additional set of fields — fugacity fields — akin to the procedures of Legrange multipliers in multivariable calculus. The result of this analysis is the set of variational equations... 

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