A General Problem in the Calculus of Variations with Applications to Paths of Least Time

by Magnus Rudolph Hestenes

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The present paper is concerned with paths of least time for an airplane. This problem when formulated analytically leads us to a problem in the calculus of variations of a type which has not been adequately treated in the literature. However, the problem can be transformed into a problem, commonly called the problem of Bolza. The purpose of the present paper is to collect known results for the problem of Bolza and interpret them in terms of the new problem here formulated. This is done in Sections 2, 3, 4 and 5. In sections 10 and 11 we treat the case when additional constraints are imposed. Applications to paths of least time are found in Sections 8, 9 and 11.

The derivation of the results on the problem of Bolza which we have used can be found in a book by G. A. Bliss entitled "Lectures on the Calculus of Variations", The University of Chicago Press. The results given by Bliss are stated in a somewhat different form than those here given. They are however equivalent and are related by the transformation given by Bliss in the introductory sections on the problem of Bolza.

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