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Isoperimetric Problems in the Calculus of Variations

Published online by Cambridge University Press:  20 November 2018

William Karush
William Karush*
Affiliation:
The University of Chicago
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We are concerned with establishing sufficiency theorems for minima of simple integrals of the parametric type in a class of curves with variable end points and satisfying isoperimetric side conditions. The results which are obtained involve no explicit assumptions of normality. Such results can be derived by transforming our problem to a problem of Bolza and using the latest developments in the theory of that problem. More recently [6] an indirect method of proof has been published. Our object is to present a direct method of proof without transformation of the problem which is based upon a generalization of the classical theory of fields.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 4 , 1952 , pp. 257 - 280
Copyright
Copyright © Canadian Mathematical Society 1952

References

1. Birkhoff, G. D. and Hestenes, M. R., Natural isoperimetric conditions in the calculus of variations, Duke Math. J., vol. 1 (1935), 198–286.Google Scholar
2. Bliss, G. A., Normality and abnormality in the calculus of variations, Trans. Amer. Math. Soc, vol. 43 (1938), 365–376.Google Scholar
3. Bliss, G. A., Lectures on the calculus of variations (Chicago, 1946).Google Scholar
4. Hestenes, M. R., A note on the Jacobi condition for parametric problems in the calculus of variations, Bull. Amer. Math. Soc, vol. 40 (1934), 297–302.Google Scholar
5. Hestenes, M. R., Generalized problem of Bolza in the calculus of variations, Duke Math. J., vol. 5 (1939), 309–324.Google Scholar
6. Hestenes, M. R., Sufficient conditions for the isoperimetric problem of Bolza in the calculus of variations, Trans. Amer. Math. Soc, vol. 60 (1946), 93–118.Google Scholar
7. Hestenes, M. R. and Reid, W. T., A note on the Weierstrass condition in the calculus of varialions, Bull. Amer. Math. Soc, vol. 45 (1939), 471–473.Google Scholar