This textbook on the calculus of variations leads the reader from the basics to modern aspects of the theory. Starting with classical issues like Euler-Lagrange equations, important geometric and topological aspects are developed and the basics of optimal control theory are given. Modern concepts from the calculus of variations are also introduced, namely relaxation, Gamma convergence, bifurcation theory and minimax methods based on the Palais-Smale condition. The prerequisites are only basic results from calculus of one and several variables. After having studied this book, the reader will be well equiped to read research papers in the calculus of variations.

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