The multiplier problem of the calculus of variations for scalar ordinary differential equations



In the long-standing inverse problem of the calculus of variations one is asked to find a Lagrangian and a multiplier so that a given differential equation, after being multiplied with the multiplier, becomes the Euler–Lagrange equation for the Lagrangian. An answer to this problem for the case of a scalar ordinary differential equation of order \(2n, n\ge 2,\) is proposed.

Mathematics Subject Classification

49N45 53B50 (35K25) 



The author is deeply indebted to his M.Phil. advisor Prof. Kai-Seng Chou for suggesting this problem, as well as carefully reviewing the first draft of this article and giving numerous valuable suggestions. The author also thanks Prof. Kai-Seng Chou for his continuous support, while and after pursuing his master degree at The Chinese University of Hong Kong. The author also thanks the anonymous referee for the very useful comments.


  1. 1.
    Anderson, I., Duchamp, T.E.: On the existence of global variational principles. Am. J. Math. 102, 781–868 (1980)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Anderson, I., Duchamp, T.E.: Variational principles for second-order quasi-linear scalar equations. J. Differ. Equ. 51, 1–47 (1984)CrossRefMATHGoogle Scholar
  3. 3.
    Anderson, I., Thompson, G.: The inverse problem of the calculus of variations for ordinary differential equations. Mem. Am. Math. Soc. 98, 1–110 (1992)MathSciNetMATHGoogle Scholar
  4. 4.
    Chan, H.-T. H.: Convergence of bounded solutions for nonlinear parabolic equations. M.Phil. Thesis, Chinese University of Hong Kong (2012)Google Scholar
  5. 5.
    Darboux, G.: Leçon sur la théorie générale des surfaces III, p. 535. Gauthier-Villars, Paris (1894)Google Scholar
  6. 6.
    Doubrov, B., Zelenko, I.: Equivalence of variational problems of higher order. Differ. Geom. Appl. 29, 255–270 (2011)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Fels, M.E.: The inverse problem of the calculus of variations for scalar fourth-order ordinary differential equations. Trans. Am. Math. Soc. 348, 5007–5029 (1996)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Juráš, M.: The inverse problem of the calculus fo variations for sixth- and eighth-order scalar ordinary differential equations. Acta Appl. Math. 66, 25–39 (2001)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Juráš, M.: Towards a solution of the inverse problem of the calculus of variations for scalar ordinary differential equations. Differ. Geom. Appl. (Opava, 2001) 3, 425–434 (2001)MathSciNetMATHGoogle Scholar
  10. 10.
    Lovász, L.: Combinatorial Problems and Exercises, p. 635. North-Holland Publishing Co., Amsterdam (1993)MATHGoogle Scholar
  11. 11.
    Nucci, M.C., Arthurs, A.M.: On the inverse problem of calculus of variations for fourth-order equations. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 466, 2309–2323 (2010)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Olver, P.J.: Applications of Lie Groups to Differential Equations, p. 500. Springer, New York (1993)CrossRefGoogle Scholar
  13. 13.
    Saunders, D.J.: Thirty years of the inverse problem in the calculus of variations. Rep. Math. Phys. 66, 43–53 (2010)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Zelenyak, T.I.: Stabilization of solutions of boundary value problems for a second-order parabolic equation with one space variable. Differ. Equ. 4, 17–22 (1968)MathSciNetMATHGoogle Scholar

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Authors and Affiliations

  1. 1.University of British ColumbiaVancouverCanada

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