Fields of Extremals and Sufficient Conditions for the Simplest Problem of the Calculus of Variations in n-Variables

  • Dean A. Carlson
  • George Leitmann
Conference paper
Part of the Springer Optimization and Its Applications book series (SOIA, volume 33)


In a 1967 note, Leitmann observed that coordinate transformations may be used to deduce extrema (minimizers or maximizers) of integrals in the simplest problem of the calculus of variations. Subsequently, in a series of papers, starting in 2001, he revived this approach and extended it in a variety of ways. Shortly thereafter, Carlson presented an important generalization of this approach and connected it to Carathéodory’s equivalent problem method. This in turn was followed by a number of joint papers addressing applications to dynamic games, multiple integrals, and other related topics.

For the simplest vector-valued variables problem of the calculus of variations, making use of the classical notion of fields of extremals, we employ Leitmann’s direct method, as extended by Carlson, to present an elementary proof ofWeierstrass’ sufficiency theorem for strong local and global extrema.


Optimization Theory Coordinate Transformation Lagrange Equation Differential Game Simple Problem 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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Copyright information

© Springer-Verlag New York 2009

Authors and Affiliations

  1. 1.Mathematical ReviewsAmerican Mathematical SocietyAnn ArborUSA
  2. 2.Department of Mechanical EngineeringUniversity of California at BerkeleyBerkeleyUSA

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