Geodesic
Necessary Conditions in a Problem of Calculus of Variations
Publications de l'Institut Mathématique, _N_S_45 (1989) no. 59, p. 143

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Problem of the calculus of variations with Bolza functionals is considered. Constraints are of both types: equalities and inequalities. The Lagrange multipler rule type theorem, which gives necessary conditions for weak optimality, is proved. When applied to the simplest problem of the calculus of variations , this theorem gives that every smooth minimizing function must satisfy the well known Euler equation and also the differential equation $ (d/dt) (L_{\dot x}\dot x-L)=-L_t. $ It should be emphasized that both differential equations are obtained under the only condition that integrand $L$ is continuously differentiable.
Classification : 49B10
Keywords: Bolza functional, weak optimality, Lagrange multipliers 
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     author = {Vladimir Jankovi\'c},
     title = {Necessary {Conditions} in a {Problem} of {Calculus} of {Variations}},
     journal = {Publications de l'Institut Math\'ematique},
     pages = {143 },
     publisher = {mathdoc},
     volume = {_N_S_45},
     number = {59},
     year = {1989},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/PIM_1989_N_S_45_59_a21/}
}
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Vladimir Janković. Necessary Conditions in a Problem of Calculus of Variations. Publications de l'Institut Mathématique, _N_S_45 (1989) no. 59, p. 143 . http://geodesic.mathdoc.fr/item/PIM_1989_N_S_45_59_a21/