Introduction to the Calculus of Variations

Part 1 Introduction to the problem: examples; definition of the most important concepts; the question of the existence of solutions. Part 2 The Euler differential equation: derivation of the Euler differential equation; integration of the Euler equation under special assumptions on the function f; further examples and problems; new admissibility conditions and the lemma of Du Bois-Reymond; the Erdmann Corner conditions. Part 3 Sufficient conditions for variational problems with convex integrands: convex functions; a sufficient condition. Part 4 The necessary conditions of Weierstrass and Legendre: the Weierstrass necessary conditions; the Legendre necessary condition; examples and problems. Part 5 The necessary condition of Jacobi: the second variation and the accessory problem; Jacobi's necessary condition; one-parameter families of extremals and the geometric significance of conjugate points; concluding remarks about conjugate points. Part 6 The Hilbert independence integral and sufficiency conditions: fields of extremals; the Hilbert integral and the Weierstrass sufficient condition; further sufficient conditions; proof of the path-independence of the Hilbert integral. Part 7 Variational problems with variable boundaries: problems having a free boundary point; variational problems with boundaries a point and a curve; sufficient conditions for problems with moveable endpoints. Part 8 Variational problems depending of several functions: conditions of the first variation; further necessary and sufficient conditions. Part 9 The parametric problem: statement of the problem; necessary and sufficient conditions; some particulars of the parametric problem and proof of the second Erdmann Corner condition. Part 10 Variational problems with multiple integrals: statement of hte problem and examples; the Euler differential equations; sufficient conditions. Part 11 Variational problems with side conditions: the Lagrange multiplier rule; the isoperimetric problem; variational problems having an equation as a side condition; Lagrange's problem. Part 12 Introduction to direct methods.