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The Inverse Problem of the Calculus of Variations for Ordinary Differential Equations
Ian Anderson and Gerard Thompson

Memoirs of the American Mathematical Society
1992; 110 pp; softcover
Volume: 98
ISBN-10: 0-8218-2533-X
ISBN-13: 978-0-8218-2533-4
List Price: US$29
Individual Members: US$17.40
Institutional Members: US$23.20
Order Code: MEMO/98/473
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This monograph explores various aspects of the inverse problem of the calculus of variations for systems of ordinary differential equations. The main problem centers on determining the existence and degree of generality of Lagrangians whose system of Euler-Lagrange equations coincides with a given system of ordinary differential equations. The authors rederive the basic necessary and sufficient conditions of Douglas for second order equations and extend them to equations of higher order using methods of the variational bicomplex of Tulcyjew, Vinogradov, and Tsujishita. What emerges is a fundamental dichotomy between second and higher order systems: the most general Lagrangian for any higher order system can depend only upon finitely many constants. The authors present an algorithm, based upon exterior differential systems techniques, for solving the inverse problem for second order equations. A number of new examples illustrate the effectiveness of this approach. The monograph also contains a study of the inverse problem for a pair of geodesic equations arising from a two dimensional symmetric affine connection. The various possible solutions to the inverse problem for these equations are distinguished by geometric properties of the Ricci tensor.


Research mathematicians in differential geometry and the calculus of variations, exterior differential systems, and mathematical physics.

Table of Contents

  • The variational bicomplex for ordinary differential equations
  • First integrals and the inverse problem for second order ordinary differential equations
  • The inverse problem for fourth order ordinary differential equations
  • Exterior differential systems and the inverse problem for second order ordinary differential equations
  • Examples
  • The inverse problem for two dimensional sprays
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