# A Comprehensive Guide to the Calculus of Variations by Charles Fox

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 Jakob Bernoulli and the Marquis de l'HÃ´pital, but Leonhard Euler first elaborated the subject, beginning in 1733. Lagrange was influenced by Euler's work to contribute significantly to the theory. After Euler saw the 1755 work of the 19-year-old Lagrange, Euler dropped his own partly geometric approach in favor of Lagrange's purely analytic approach and renamed the subject the calculus of variations in his 1756 lecture Elementa Calculi Variationum.[3][4][b]

## An Introduction To The Calculus Of Variations Fox Pdf 14

In the 20th century David Hilbert, Oskar Bolza, Gilbert Ames Bliss, Emmy Noether, Leonida Tonelli, Henri Lebesgue and Jacques Hadamard among others made significant contributions.[5] Marston Morse applied calculus of variations in what is now called Morse theory.[6] Lev Pontryagin, Ralph Rockafellar and F. H. Clarke developed new mathematical tools for the calculus of variations in optimal control theory.[6] The dynamic programming of Richard Bellman is an alternative to the calculus of variations.[7][8][9][c]

Calculus of variations is concerned with variations of functionals, which are small changes in the functional's value due to small changes in the function that is its argument. The first variation[l] is defined as the linear part of the change in the functional, and the second variation[m] is defined as the quadratic part.[22]