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An Observation on Two Methods of Obtaining Solutions to Variational Problems

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  • D.A. Carlson

    (University of Toledo)

Abstract

In this paper, we compare two methods for obtaining solutions for free problems in the calculus of variations. The first is due to Carathéodory (Ref. 1) and the second due to Leitmann (Ref. 2). Both methods introduce the notion of equivalent variational problems. Using either approach, an auxiliary problem is obtained for which the solution is more easily obtained. We compare both approaches by using each to solve the same class of examples. We conclude our discussion by unifying the two approaches into one and illustrating the potential of this new method through the use of an elementary example.

Suggested Citation

  • D.A. Carlson, 2002. "An Observation on Two Methods of Obtaining Solutions to Variational Problems," Journal of Optimization Theory and Applications, Springer, vol. 114(2), pages 345-361, August.
    • Handle: RePEc:spr:joptap:v:114:y:2002:i:2:d:10.1023_a:1016035718160
    DOI: 10.1023/A:1016035718160
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    References listed on IDEAS

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    1. E. J. Dockner & G. Leitmann, 2001. "Coordinate Transformations and Derivation of Open-Loop Nash Equilibria," Journal of Optimization Theory and Applications, Springer, vol. 110(1), pages 1-15, July.
    2. G. Leitmann, 2001. "Some Extensions to a Direct Optimization Method," Journal of Optimization Theory and Applications, Springer, vol. 111(1), pages 1-6, October.
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    Cited by:

    1. F. O. O. Wagener, 2009. "On the Leitmann Equivalent Problem Approach," Journal of Optimization Theory and Applications, Springer, vol. 142(1), pages 229-242, July.
    2. D. A. Carlson & G. Leitmann, 2004. "An Extension of the Coordinate Transformation Method for Open-Loop Nash Equilibria," Journal of Optimization Theory and Applications, Springer, vol. 123(1), pages 27-47, October.
    3. A. B. Malinowska & D. F. M. Torres, 2010. "The Hahn Quantum Variational Calculus," Journal of Optimization Theory and Applications, Springer, vol. 147(3), pages 419-442, December.
    4. Wagener, F.O.O., 2009. "On conjugate points and the Leitmann equivalent problem approach," CeNDEF Working Papers 09-08, Universiteit van Amsterdam, Center for Nonlinear Dynamics in Economics and Finance.
    5. D. F. M. Torres & G. Leitmann, 2008. "Contrasting Two Transformation-based Methods for Obtaining Absolute Extrema," Journal of Optimization Theory and Applications, Springer, vol. 137(1), pages 53-59, April.
    6. S. G. Ji & S. Y. Shi, 2006. "Periodic Solutions for a Class of Second-Order Ordinary Differential Equations," Journal of Optimization Theory and Applications, Springer, vol. 130(1), pages 125-137, July.

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