Geodesic
Newton's Method, Differential Equations, and the Lagrangian Principle for Necessary Extremum Conditions
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Optimal control, Tome 262 (2008), pp. 156-177

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We show how one can use a modified Newton's method to prove existence and uniqueness theorems for solutions of differential equations and theorems on the continuous and differentiable dependence of these solutions on the initial data and parameters and to derive necessary conditions for an extremum in various extremum problems (from the origins to our days). 
@article{TM_2008_262_a11,
     author = {G. G. Magaril-Il'yaev and V. M. Tikhomirov},
     title = {Newton's {Method,} {Differential} {Equations,} and the {Lagrangian} {Principle} for {Necessary} {Extremum} {Conditions}},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     pages = {156--177},
     publisher = {mathdoc},
     volume = {262},
     year = {2008},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TM_2008_262_a11/}
}
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G. G. Magaril-Il'yaev; V. M. Tikhomirov. Newton's Method, Differential Equations, and the Lagrangian Principle for Necessary Extremum Conditions. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Optimal control, Tome 262 (2008), pp. 156-177. http://geodesic.mathdoc.fr/item/TM_2008_262_a11/