Geodesic
The Euler--Jacobi equation in variational calculus
Matematičeskie zametki, Tome 20 (1976) no. 6, pp. 847-858.

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The use of the method of the Euler–Jacobi equation is considered in the study of a quadratic functional defined on a cone. Such functionals occur in the variation of optimal-control problems. Several concepts are introduced with the aid of which the Euler–Jacobi equation is extended and the application of this method is justified also in the case that the equation is not a linear differential equation. 
@article{MZM_1976_20_6_a6,
     author = {A. V. Dmitruk},
     title = {The {Euler--Jacobi} equation in variational calculus},
     journal = {Matemati\v{c}eskie zametki},
     pages = {847--858},
     publisher = {mathdoc},
     volume = {20},
     number = {6},
     year = {1976},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1976_20_6_a6/}
}
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A. V. Dmitruk. The Euler--Jacobi equation in variational calculus. Matematičeskie zametki, Tome 20 (1976) no. 6, pp. 847-858. http://geodesic.mathdoc.fr/item/MZM_1976_20_6_a6/