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With Calculus of Variations—we know—we indicate a body of mathematical techniques aiming at determining existence and properties of functions on which certain classes of functionals attain optimal values, according to some prescribed criteria, and pertinent results. Our common belief that nature minimizes energy at constant entropy and maximizes entropy at constant energy—a belief not falsified so far by common experiments—is at roots of the continuous interest in calculus of variations, besides aesthetic evaluation of the theory per se. Companion motivation for the attention on such a topic is also the technological interest for designing objects with some optimal property—e.g., shape, strength, conductivity—under some constraints, or to get optimal control of processes, e.g., in mechanics or economy. Problems requiring recourse to calculus of variations techniques to be tackled emerge in several sectors, even in social sciences, but above all in mechanics (be it classical, quantum, or relativistic), condensed matter physics, chemistry, and else. The motion of a three-dimensional rigid body can be viewed as a geodetic curve (the one with minimal length) over the special orthogonal group, while perfect fluids move along geodetic paths over the special group of diffeomorphisms. We can also aim at controlling optimally the motion of certain systems, be them multi-rigid-bodies with flexible mutual constraints or continua suffering distributed strain, as, e.g., rods are. We may ask to find the minimal energy of an atom, a molecule, a thin film, or we may tackle optimality questions connected with chemical reactions, or we aim at printing and connecting microstructures, in order to obtain an artifact with some optimized properties, what we call a metamaterial. Also, we may be interested in optimal transportation of mass, charges, and their like. Energy minimization characterizes equilibrium configurations.When coupled with appropriate monotonicity conditions mimicking irreversible behavior, such a minimization procedure may allow us to describe classes of (rate-independent) dissipative processes, such as plastic flows, damage, some phase transitions, or nucleation of fractures, by adapting Ennio De Giorgi’s idea of minimizing movements. The idea is to partition the time interval into finitely many sub-intervals, presuming to go from

Calculus of Variations in Mechanics and Related Fields