Introduction to the subject of calculus of variations, examples of variational problems. The basic problem of calculus of variations (the Lagrange function, variational functional, variation, the du BoisReymond Lemma, Euler  Lagrange equations). Jet spaces, total derivatives and contact forms. Differential equations as submanifolds in jet spaces. Vector fields on jet spaces. Prolongation. The symmetries of variational problems (symmetries and generalized symmetries, invariance groups, criteria of invariance, gauge transformations, the first and second Noether's theorems).


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