We know exactly when the Calculus of Variation was invented. The story started with the brachistochrone problem posed by Johann Bernoulli Johann Bernoulli in Acta Eruditorum in June 1696.
The problem he posed was the following:-
Given two points A and B in a vertical plane, what is the curve traced out by a point acted on only by gravity, which starts
at A and reaches B in the shortest time.
Johann Bernoulli introduced the challenge in the remarkable words:-
I, Johann Bernoulli, address the most brilliant mathematicians in the world. Nothing is more attractive to intelligent people
than a honest, challenging problem, whose possible solution will bestow fame and remain as a lasting monument.
Following the example set by Pascal, Fermat, etc., I hope to gain the gratitude of the whole scientific community by placing
before the finest mathematicians of our time a problem which will test their methods and the strength of their intellect. If
someone communicates to me the solution of the proposed problem, I shall publicly declare him worthy of praise.
In a year, five solutions were obtained by Newton, Jacob Bernoulli, Leibniz and de L'Hôpital in addition to the solution by Johann Bernoulli.
Methods for extremal problems: Global, variational, and algorithmic (overview) Convexity and global methods. Sufficient conditions Symmetrization. Notion of fixed-point limits Euler equations and first invariants. Variational boundary conditions, non-fixed interval. Approximation with penalty Geometric optics Lagrangian mechanics Constraints in variational problems Distinguish min from max (Weierstrass and Jacobi tests) Nonexistence of the solutions: What does it mean? Multivariable problems: Euler-Lagrange eqns. Equilibria