Calculus of variations

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Source: Wikipedia. Pages: 53. Chapters: Fermat's principle, Noether's theorem, Signorini problem, Caccioppoli set, Action, Euler-Lagrange equation, Isoperimetric inequality, Variational inequality, Principle of least action, Hamilton's principle, Maupertuis' principle, History of variational principles in physics, Transportation theory, Obstacle problem, Inverse problem for Lagrangian mechanics, Direct method in the calculus of variations, Hilbert's nineteenth problem, Envelope theorem, Brunn-Minkowski theorem, Fundamental lemma of calculus of variations, Lagrange multipliers on Banach spaces, Mountain pass theorem, Lagrangian system, Minkowski-Steiner formula, Plateau's problem, Ekeland's variational principle, Mosco convergence, Energy principles in structural mechanics, Noether identities, Beltrami identity, Nehari manifold, Morse-Palais lemma, Noether's second theorem, Path of least resistance, Minkowski's first inequality for convex bodies, G-convergence, Palais-Smale compactness condition, Homicidal chauffeur problem, Pseudo-monotone operator, Variational bicomplex, Saint-Venant's theorem, Tonelli's theorem, Dirichlet's principle, Dirichlet's energy, First variation, Hu Washizu principle, Legendre-Clebsch condition, Chaplygin problem, Weierstrass-Erdmann condition, Variational vector field, List of variational topics. Excerpt: Noether's (first) theorem states that any differentiable symmetry of the action of a physical system has a corresponding conservation law. The theorem was proved by German mathematician Emmy Noether in 1915 and published in 1918. The action of a physical system is the integral over time of a Lagrangian function (which may or may not be an integral over space of a Lagrangian density function), from which the system's behavior can be determined by the principle of least action. Noether's theorem has become a fundamental tool of modern theoretical physics and the calculus of variations. A generalization of the seminal formulations on constants of motion in Lagrangian and Hamiltonian mechanics (1788 and 1833, respectively), it does not apply to systems that cannot be modeled with a Lagrangian, for example, dissipative systems with continuous symmetries need not have a corresponding conservation law. For illustration, if a physical system behaves the same regardless of how it is oriented in space, its Lagrangian is rotationally symmetric, from this symmetry, Noether's theorem shows the angular momentum of the system must be conserved. The physical system itself need not be symmetric, a jagged asteroid tumbling in space conserves angular momentum despite its asymmetry - it is the laws of motion that are symmetric. As another example, if a physical experiment has the same outcome regardless of place or time (having the same outcome, say, somewhere in Asia on a Tuesday or in America on a Wednesday), then its Lagrangian is symmetric under continuous translations in space and time, by Noether's theorem, these symmetries account for the conservation laws of linear momentum and energy within this system, respectively. (These examples are just for illustration, in the first one, Noether's theorem added nothing new - the results were known to follow from Lagrange's equations and from Hamilton's equations.) Noether's theorem is important, both because of the insight it gives into