Variational methods

History

The calculus of variations began with the work of Isaac Newton, such as with Newton's minimal resistance problem, which he formulated and solved in 1685, and later published in his Principia in 1687, which was the first problem in the field to be formulated and correctly solved, and was also one of the most difficult problems tackled by variational methods prior to the twentieth century. This problem was followed by the brachistochrone curve problem raised by Johann Bernoulli (1696), which was similar to one raised by Galileo Galilei in 1638, but he did not solve the problem explicitly nor did he use the methods based on calculus. Bernoulli solved the problem using the principle of least time in the process, but not calculus of variations. In 1697 Newton solved the problem using variational techniques, and as a result, he pioneered the field with his work on the two problems. The problem would immediately occupy the attention of Jacob Bernoulli and the Marquis de l'Hôpital, but Leonhard Euler first elaborated the subject, beginning in 1733. Joseph-Louis Lagrange was influenced by Euler's work to contribute greatly 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.