1.1 The fundamental problem and lemma of calculus of variations1.2 The Legendre test1.3 The Euler–Lagrange differential equation1.4 Minimal path problems1.4.1 Shortest curve between two points1.4.2 The brachistochrone problem1.4.3 Fermat’s principle1.4.4 Particle moving in a gravitational field1.5 Open boundary variational problems1.6 Exercises