Communications in Mathematical Physics

, Volume 362, Issue 1, pp 107–128 | Cite as

The Lagrangian Order-Reduction Theorem in Field Theories

  • Olga Rossi


It is known that for every system of variational second order PDEs affine in second derivatives the Tonti Lagrangian is locally reducible to an equivalent first order Lagrangian (Order reduction Theorem). In this paper, a new proof is presented, based on investigation of closed forms related with variational equations, and an explicit formula for the first order Lagrangians arising by order reduction is found. The presented approach extends and completes the Order reduction Theorem by a geometric content and physical meaning of order reducibility: all variational second order PDEs affine in second derivatives admit a first-order covariant Hamiltonian formulation (Hamilton–De Donder equations), i.e. (under certain regularity conditions) carry a multisymplectic structure which is determined directly from the Euler–Lagrange expressions.


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  1. 1.
    Anderson I., Duchamp T.: On the existence of global variational principles. Am. J. Math. 102, 781–867 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Douglas J.: Solution to the inverse problem of the calculus of variations. Trans. Am. Math. Soc. 50, 71–128 (1941)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Garcia, P.L.: The Poincaré–Cartan invariant in the calculus of variations. In: Symposia Mathematica, vol. XIV, pp. 219–246 (1974)Google Scholar
  4. 4.
    Goldschmidt H., Sternberg S.: The Hamilton–Cartan formalism in the calculus of variations. Ann. Inst. Fourier 23, 203–267 (1973)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Helmholtz H.: Ueber die physikalische Bedeutung des Prinzips der kleinsten Wirkung. J. für die Reine u. Angew. Math. 100, 137–166 (1887)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Krupka, D.: On the local structure of the Euler–Lagrange mapping of the calculus ofvariations. In: Proceedings of the Conference on Differential Geometry and Its Application, Nové Město na Moravě (Czechoslovakia), 1980, pp. 181–188. Charles Univ. Prague (1982); arXiv:math-ph/0203034
  7. 7.
    Krupka, D.: Lepagean forms in higher order variational theory. In: Benenti, S., Francaviglia, M., Lichnerowicz, A.(eds.) Modern Developments in Analytical Mechanics I: Geometrical Dynamics, Proceedings of the IUTAM-ISIMM Symposium, Torino, Italy 1982, Accad. Sci. Torino, Torino, 1983, pp. 197–238 (1983)Google Scholar
  8. 8.
    Krupka, D.: Variational sequences on finite order jet spaces. In: Janyška, J., Krupka, D. (eds.) Differential Geometry and Its Applications Proceedings of the Conference on, Brno, Czechoslovakia, 1989, pp. 236–254. World Scientific, Singapore, (1990)Google Scholar
  9. 9.
    Krupka D.: Introduction to Global Variational Geometry. Atlantis Press, Amsterdam (2015)CrossRefzbMATHGoogle Scholar
  10. 10.
    Krupka, D., Štěpánková, O.: On the Hamilton form in second order calculus of variations. In: Modugno, M. (ed.) Geometry and Physics (Florence, 1982), pp. 85–102. Pitagora, Bologna (1983)Google Scholar
  11. 11.
    Krupková O.: The Geometry of Ordinary Variational Equations, Lecture Notes in Mathematics 1678. Springer, Berlin (1997)Google Scholar
  12. 12.
    Krupková O.: Hamiltonian field theory. J. Geom. Phys. 43, 93–132 (2002)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Krupková, O., Prince, G.E.: Second order ordinary differential equations in jet bundles and the inverse problem of the calculus of variations. In: Handbook on Global Analysis, pp. 841–908. Elsevier(2008)Google Scholar
  14. 14.
    Krupková O., Smetanová D.: Legendre transformation for regularizable Lagrangians in field theory. Lett. Math. Phys. 58, 189–204 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Krupková O., Smetanová D.: Lepage equivalents of second-order Euler–Lagrange forms and the inverse problem of the calculus of variations. J. Nonlinear Math. Phys. 16, 235–250 (2009)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Palese M., Rossi O., Winterroth E., Musilová J.: Variational sequences, representation sequences and applications in physics. SIGMA 12, 45 (2016)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Rossi, O. Geometry of variational partial differential equations and Hamiltonian systems. In: Geometry of Jets and Fields, Banach Center Publications, Vol. 110, pp. 219–237. Inst. of Math., Polish of Academy of Sciences, Warszawa (2016)Google Scholar
  18. 18.
    Rossi, O.: Lepage manifold. In: Falcone G. (ed.) Lie Groups, Differential Equations and Geometry: Surveys and Advances, pp. 287–316. Springer, Berlin (2017)Google Scholar
  19. 19.
    Rossi O., Saunders D.: Lagrangian and Hamiltonian duality. J. Math. Sci. 218, 813–816 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Sarlet W., Thompson G., Prince G.E.: The inverse problem of the calculus of variations: the use of geometrical calculus in Douglas’s analysis. Trans. Am. Math. Soc. 354(7), 2897–2919 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Tonti, E.: Variational formulation of nonlinear differential equations I, IIBull. Acad. R. Belg. Cl. Sci. 55 137–165 262–278 (1969)Google Scholar
  22. 22.
    Vainberg M.M.: Variational methods in the theory of nonlinear operators. GITL, Moscow (1959) (in Russian)Google Scholar
  23. 23.
    Vanderbauwhede A.L.: Potential operators and variational principles. Hadron. J. 2, 620–641 (1979)MathSciNetzbMATHGoogle Scholar

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Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of ScienceUniversity of OstravaOstravaCzech Republic
  2. 2.Department of Mathematics and StatisticsLa Trobe UniversityMelbourneAustralia

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