Geodesic
Some geometric aspects of the calculus of variations in several independent variables
Communications in Mathematics, Tome 18 (2010) no. 1, pp. 3-19.

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This paper describes some recent research on parametric problems in the calculus of variations. It explains the relationship between these problems and the type of problem more usual in physics, where there is a given space of independent variables, and it gives an interpretation of the first variation formula in this context in terms of cohomology.
Classification : 35A15, 58A10, 58A20
Keywords: calculus of variations; parametric problems 
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Saunders, David. Some geometric aspects of the calculus of variations in several independent variables. Communications in Mathematics, Tome 18 (2010) no. 1, pp. 3-19. http://geodesic.mathdoc.fr/item/COMIM_2010__18_1_a1/