Geodesic
Investigation of variational problems by direct methods
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 48 (2008) no. 3, pp. 373-386 Cet article a éte moissonné depuis la source Math-Net.Ru

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A direct method is proposed for solving variational problems in which an extremal is represented by an infinite series in terms of a complete system of basis functions. Taking into account the boundary conditions gives all the necessary conditions of the classical calculus of variations, that is, the Euler–Lagrange equations, transversality conditions, Erdmann–Weierstrass conditions, etc. The penalty function method reduces conditional extremum problems to variational ones in which the isoperimetric conditions described by constraint equations are taken into account by Lagrangian multipliers. The direct method proposed is applied to functionals depending on functions of one or two variables. 
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V. G. Butov. Investigation of variational problems by direct methods. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 48 (2008) no. 3, pp. 373-386. http://geodesic.mathdoc.fr/item/ZVMMF_2008_48_3_a2/

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