The Calculus of Variations

Abstract

Abstract It is to Queen Dido of Carthage, in the ninth century b.c., that the oldest problem in the Calculus of Variations has been attributed. The ancient Greeks certainly formulated variational problems. Most of the more celebrated analysts of the last three centuries have made substantial contributions to the mathematical theory and, in particular, the names of Bernoulli, Newton, Leibnitz, de l’Hôpital, Euler, Lagrange, Legendre, Dirichlet, Riemann, Jacobi, Weierstrass and Hilbert are attached to important phenomena or results. Nowadays, the Calculus of Variations not only has numerous applications in rigorous pure analysis and in physical applied mathematics (notably in classical and quantum mechanics and potential theory), but it is also a basic tool in operations research and control theory. We shall in this chapter use the notation f ∈ C m(A) to indicate that f is a real-valued function, defined and m-times continuously differentiable on A, where A ⊆ ℝn and m, n are positive integers.