2.4. Recommended reading 45 2.4. Recommended reading Goldstine (1980), Fraser (1994, 2005a), and Thiele (2007) analyze Eu- ler’s early contributions to the calculus of variations. Euler’s idea of using a polygonal curve to approximate the solution of a variational problem was revived in the 20th century by Russian mathematicians working on direct methods of solution. In a direct method, you con- struct a sequence of approximating functions, determine the unknown values and coefficients in each function using minimization, and let the sequence of functions converge to the solution. Euler’s approach suggests the direct method of finite differences (Elsgolc, 1961). Other direct methods include the Ritz method, which is frequently and inap- propriately (Leissa, 2005) called the Rayleigh-Ritz method, the Kan- torovich method, and the Galerkin method. See Forray (1968) for an introduction to these direct methods. The theory of the differentiation of functionals has its origins in the work of Volterra (1887). See Gelfand and Fomin (1963), Hamil- ton and Nashed (1982, 1995), Kolmogorov and Yushkevich (1998), and Lebedev and Cloud (2003) for more on variational derivatives. There are errors in the statements and proofs of the existence of the variational derivative for the simplest problem of the calculus of vari- ations in Volterra (1913) and Gelfand and Fomin (1963). These errors were pointed out and corrected by Bliss (1915) and Hamilton (1980). Lagrange announced his new approach to the calculus of vari- ations in a 1755 letter to Euler his results appeared in print seven years later (Lagrange, 1762). Fraser (1985) reviews the lengthy corre- spondence between Joseph Lagrange and Leonhard Euler and traces the development of Lagrange’s approach to the foundations of the calculus of variations. It has been said that the fundamental lemma is like a watch- dog that guards the entrance gates to the entire classical domain of the calculus of variations (Dresden, 1932). Many early writers took the conclusion of the fundamental lemma as self-evident while others erred in their proof of this lemma. Huke (1931) traces the long and fascinating history of the fundamental lemma and of the lemma of du Bois-Reymond.
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