We begin the talk with the story of Dido and the Brachistochrone problem. We show how these two problems leads to the two must fundamental problems of the calculus of variations. The Brachistochrone problem leads to the basic problem of calculus of variations and that leads to the Euler-Lagrange equation. We show the link between the Euler-Lagrange equations and the laws of classical mechanics.
We also discuss about the Legendre conditions and Jacobi conjugate points which leads to the sufficient conditions for weak local minimum points.
The Dido's problem leads to the problem of Lagrange in which Lagrange introduces his multiplier rule. We also speak a bit about the problem of Bolza and further also discuss about how the class of extremals can be enlarged and the issue of existence of solutions in calculus of variations, the Tonelli's direct methods and some more facts on the quest for multiplier rules.