1. Classical and modern examples of variational problems 1.1 The brachistochrone problem 1.2 A navigation problem 1.3 One-dimensional nonlinear elasticity 2. Classical method in one space dimension 2.1 Definition of a local and global minimizer 2.2 Necessary conditions for local minimum 2.3. Sufficient conditions for local minimum 2.4 Sufficient conditions for global minimum, convex functions 3. Technical preliminaries: Function spaces 3.1 Space of measurable functions, Holder’s inequality, Minkowski’s inequality 3.2 Fundamental Lemma of Calculus of Variations 3.3 The Sobolev Space 3.4 Boundary conditions, Poincare Inequality 4. Global and local minimizers 4.1 Examples when there is no global minimizer 4.2 Definition of weak and strong local minimizer 4.3 Necessary conditions for weak local minimizers , Euler-Lagrange equations, the second variation 4.4 Necessary conditions for strong local minimizers , the Weierstrass condition 4.5 Sufficient conditions for weak local minimizers 4.6 Sufficient conditions for strong local minimizers 5. The direct method of the calculus of variations 5.1 Weak convergence 5.2 Tonelli’s existence theorem 5.3 The brachistochrone problem revisited 6. Multi-dimensional problems, done via some examples (if time allows)