Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 20 (2024), 013, 18 pages      arXiv:2309.01594
Contribution to the Special Issue on Symmetry, Invariants, and their Applications in honor of Peter J. Olver

Lepage Equivalents and the Variational Bicomplex

David Saunders
Lepage Research Institute, Prešov, Slovakia

Received October 05, 2023, in final form January 30, 2024; Published online February 09, 2024

We show how to construct, for a Lagrangian of arbitrary order, a Lepage equivalent satisfying the closure property: that the Lepage equivalent vanishes precisely when the Lagrangian is null. The construction uses a homotopy operator for the horizontal differential of the variational bicomplex. A choice of symmetric linear connection on the manifold of independent variables, and a global homotopy operator constructed using that connection, may then be used to extend any global Lepage equivalent to one satisfying the closure property. In the second part of the paper we investigate the rôle of vertical endomorphisms in constructing such Lepage equivalents. These endomorphisms may be used directly to construct local homotopy operators. Together with a symmetric linear connection they may also be used to construct global vertical tensors, and these define infinitesimal nonholonomic projections which in turn may be used to construct Lepage equivalents. We conjecture that these global vertical tensors may also be used to define global homotopy operators.

Key words: jet bundle; Poincaré-Cartan form; Lepage equivalent of a Lagrangian; variational bicomplex.

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  1. Alonso Blanco R.J., $\mathcal D$-modules, contact valued calculus and Poincaré-Cartan form, Czech. Math. J. 49 (1999), 585-606.
  2. Anderson I.M., The variational bicomplex, Technical Report, Utah State University, 1989.
  3. Betounes D., Global shift operators and the higher order calculus of variations, J. Geom. Phys. 10 (1993), 185-201.
  4. Betounes D.E., Extension of the classical Cartan form, Phys. Rev. D 29 (1984), 599-606.
  5. Carathéodory C., Über die Variationsrechnung bei mehrfachen Integralen, Acta Sci. Math. (Szeged) 4 (1929), 193-216.
  6. Crampin M., Sarlet W., Cantrijn F., Higher-order differential equations and higher-order Lagrangian mechanics, Math. Proc. Cambridge Philos. Soc. 99 (1986), 565-587.
  7. Crampin M., Saunders D.J., The Hilbert-Carathéodory and Poincaré-Cartan forms for higher-order multiple-integral variational problems, Houston J. Math. 30 (2004), 657-689.
  8. Crampin M., Saunders D.J., On null Lagrangians, Differential Geom. Appl. 22 (2005), 131-146.
  9. Crampin M., Saunders D.J., Homotopy operators for the variational bicomplex, representations of the Euler-Lagrange complex, and the Helmholtz-Sonin conditions, Lobachevskii J. Math. 30 (2009), 107-123.
  10. Ferraris M., Francaviglia M., Global formalisms in higher order calculus of variations, in Proceedings of the Conference on Differential Geometry and its Applications, Part 2, University of J.E. Purkynĕ, Brno, 1984, 93-117.
  11. García P.L., Muñoz J., On the geometrical structure of higher order variational calculus, Atti Accad. Sci. Torino Cl. Sci. Fis. Mat. Natur. 117 (1983), 127-147.
  12. Horák M., Kolář I., On the higher order Poincaré-Cartan forms, Czechoslovak Math. J. 33 (1983), 467-475.
  13. Krupka D., A map associated to the Lepagian forms on the calculus of variations in fibred manifolds, Czechoslovak Math. J. 27 (1977), 114-118.
  14. Krupka D., Krupková O., Saunders D., The Cartan form and its generalizations in the calculus of variations, Int. J. Geom. Methods Mod. Phys. 7 (2010), 631-654.
  15. Libermann P., Introduction to the theory of semi-holonomic jets, Arch. Math. (Brno) 33 (1997), 173-189.
  16. Olver P.J., Applications of Lie groups to differential equations, Grad. Texts Math., Vol. 107, Springer, New York, 1986.
  17. Olver P.J., Equivalence and the Cartan form, Acta Appl. Math. 31 (1993), 99-136.
  18. Palese M., Rossi O., Zanello F., Geometric integration by parts and Lepage equivalents, Differential Geom. Appl. 81 (2022), 101866, 26 pages, arXiv:2010.16135.
  19. Saunders D.J., An alternative approach to the Cartan form in Lagrangian field theories, J. Phys. A 20 (1987), 339-349.
  20. Saunders D.J., The geometry of jet bundles, London Math. Soc. Lecture Note Ser., Vol. 142, Cambridge University Press, Cambridge, 1989.
  21. Takens F., A global version of the inverse problem of the calculus of variations, J. Differential Geometry 14 (1979), 543-562.
  22. Tsujishita T., On variation bicomplexes associated to differential equations, Osaka Math. J. 19 (1982), 311-363.
  23. Tulczyjew W.M., The Euler-Lagrange resolution, in Differential Geometrical Methods in Mathematical Physics, Lecture Notes in Math., Vol. 836, Springer, Berlin, 1980, 22-48.
  24. Urban Z., Volná J., The fundamental Lepage form in two independent variables: A generalization using order-reducibility, Mathematics 10 (2022), 1211, 14 pages, arXiv:2204.01663.
  25. Vinogradov A.M., On the algebro-geometric foundations of Lagrangian field theory, Sov. Math. Dokl. 18 (1977), 1200-1204.
  26. Voicu N., Garoiu S., Vasian B., On the closure property of Lepage equivalents of Lagrangians, Differential Geom. Appl. 81 (2022), 101852, 19 pages, arXiv:2102.12955.

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