Encyclopedia of Operations Research and Management Science

2001 Edition
| Editors: Saul I. Gass, Carl M. Harris

Calculus of variations

  • Stephen G. Nash
Reference work entry
DOI: https://doi.org/10.1007/1-4020-0611-X_94

INTRODUCTION

The calculus of variations is the grandparent of mathematical programming. From it we have inherited such concepts as duality and Lagrange multipliers. Many central ideas in optimization were first developed for the calculus of variations, then specialized to nonlinear programming, all of this happening years before linear programming came along.

The calculus of variations solves optimization problems whose parameters are not simple variables, but rather functions. For example, how should the shape of an automobile hood be chosen so as to minimize air resistance? Or, what path does a ray of light follow in an irregular medium? The calculus of variations is closely related to optimal control theory, where a set of “controls” are used to achieve a certain goal in an optimal way. For example, the pilot of an aircraft might wish to use the throttle and flaps to achieve a particular cruising altitude and velocity in a minimum amount of time or using a minimum amount of fuel. We...

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References

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    Bliss, G. A. (1925). Calculus of Variations, Open Court, Chicago.Google Scholar
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    Bolza, O. (1904). Lectures on the Calculus of Variations, University of Chicago Press, Chicago.Google Scholar
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    Courant, R. and Hilbert, D. (1953). Methods of Mathematical Physics, Volume I, Interscience, New York.Google Scholar
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    Goldstine, H. H. (1980). A History of the Calculus of Variations from the 17th through the 19th Century, Springer-Verlag, New York.Google Scholar
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    Gregory, J. and Lin, C. (1992). Constrained Optimization in the Calculus of Variations and Optimal Control Theory, Van Nostrand Reinhold, New York.Google Scholar
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    Hestenes, M. R. (1966). Calculus of Variations and Optimal Control Theory, John Wiley, New York.Google Scholar
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    Kuhn, H. W. (1991). Nonlinear Programming: A Historical Note, in History of Mathematical Programming, J. K. Lenstra, A. H.G. Rinnooy Kan, and A. Schrijver, eds., North-Holland (Amsterdam), 82–96.Google Scholar
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    Lagrange, J. L. (1888–89). Oeuvres de Lagrange, Volumes XI and XII, Gauthier-Villars, Paris.Google Scholar
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    Mayer. A. (1886). “Begründung der Lagrange'schen Multiplicatorenmethode in der Variationsrechnung,” Mathematische Annalen, 26, 74–82.Google Scholar

Copyright information

© Kluwer Academic Publishers 2001

Authors and Affiliations

  • Stephen G. Nash
    • 1
  1. 1.George Mason UniversityFairfaxUSA