## The development of sufficient conditions in the calculus of variations ... |

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### Contents

THE FIELD METHOD OF WEIERSTRASS FOR SIMPLE PROBLEMS | 28 |

minimum | 34 |

The existence of a minimizing extremal joining two points sufficiently near to each other | 35 |

15 other sections not shown

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### Common terms and phrases

admissible arcs joining admissible sets analogue Bliss Bolza Buler calculus of variations Caratheodory Chapter class of arcs closed extremal comparison arc covers a region curve curvilinear coordinates defined derived differential equation double integrals existence theorem expansion method expansion proof extremal arc extremaloid family of extremals ficiency field method fundamental sufficiency theorem furnishes a strong geodesic gral Hahn Hamilton-Jacobi theory Hilbert integral holds inequality integrand Introduction isoperimetric problem Jacobi condition joining the points Kneser Lagrange problem Legendre Legendre's lemma Levi Levi's theorem Lindeberg Mayer field method of Weierstrass minimal surfaces minimizes an integral minimizing arc necessary condition neighborhood non-parametric notations Osgood parametric form parametric problem prob problem of Lagrange problem with variable proved published respect to admissible satisfies the conditions Scheeffer second variation simply covered slope functions strong relative minimum suffi sufficiency proofs sufficient conditions tion Tonelli transformation tremal variable end variable end-points weak relative minimum writers