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Sufficiency and the Jacobi Condition in the Calculus of Variations

Published online by Cambridge University Press:  20 November 2018

Frank H. Clarke  and
Vera Zeidan
Frank H. Clarke
Affiliation:
Université de Montréal, Montréal, Québec
Vera Zeidan
Affiliation:
University of Alberta, Edmonton, Alberta
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Besides stating the problem and the results, we shall give in this section a brief overview of the classical necessary and sufficient conditions in the calculus of variations, in order to clearly situate the contribution of this article.

1.1 The problem. We are given an interval [a, b], two points xa, xb in Rn, and a function L (the Lagrangian) mapping [a, b] × Rn × Rn to R. The basic problem in the calculus of variations, labeled (P), is that of minimizing the functional

over some class X of functions x and subject to the constraints x(a) = xa, x(b) = xb. Let us take for now the class X of functions to be the continuously differentiable mappings from [a, b] to Rn; we call such functions smooth arcs.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 38 , Issue 5 , 01 October 1986 , pp. 1199 - 1209
Copyright
Copyright © Canadian Mathematical Society 1986

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