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The calculus of variations may be said to begin with Newton's minimal resistance problem in 1687, followed by the brachistochrone curve problem raised by Johann Bernoulli (1696). [2] It immediately occupied the attention of Jacob Bernoulli and the Marquis de l'Hôpital , but Leonhard Euler first elaborated the subject, beginning in 1733.
PDF According to WorldCat, the book is held in 419 libraries [2] Direct methods in the calculus of variations; Springer-Verlag, New-York (1989), 2nd ed. (2007). According to WorldCat, the book is held in 625 libraries [3] PDF. Introduction to the calculus of variations; Imperial College Press, London (2004), 2nd ed. (2009), 3rd ed (2014 ...
In mathematics, specifically in the calculus of variations, a variation δf of a function f can be concentrated on an arbitrarily small interval, but not a single point. Accordingly, the necessary condition of extremum ( functional derivative equal zero) appears in a weak formulation (variational form) integrated with an arbitrary function δf .
In mathematics, the direct method in the calculus of variations is a general method for constructing a proof of the existence of a minimizer for a given functional, [1] introduced by Stanisław Zaremba and David Hilbert around 1900. The method relies on methods of functional analysis and topology. As well as being used to prove the existence of ...
The Beltrami identity, named after Eugenio Beltrami, is a special case of the Euler–Lagrange equation in the calculus of variations. The Euler–Lagrange equation serves to extremize action functionals of the form [] = [, (), ′ ()],
Harold Calvin Marston Morse (March 24, 1892 – June 22, 1977) was an American mathematician best known for his work on the calculus of variations in the large, a subject where he introduced the technique of differential topology now known as Morse theory.
He published his Variationsrechnung und Partielle Differentialgleichungen Erster Ordnung (Calculus of Variations and First-order Partial Differential Equations) in 1935. [10] More recently, Carathéodory's work on the calculus of variations and the Hamilton-Jacobi equation has been taken into the theory of optimal control and dynamic programming.
[1] [2] The maximum principle was formulated in 1956 by the Russian mathematician Lev Pontryagin and his students, [3] [4] and its initial application was to the maximization of the terminal speed of a rocket. [5] The result was derived using ideas from the classical calculus of variations. [6]
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