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

SIGMA 14 (2018), 089, 13 pages      arXiv:1801.06888      https://doi.org/10.3842/SIGMA.2018.089

On Lagrangians with Reduced-Order Euler-Lagrange Equations

David Saunders
Department of Mathematics, Faculty of Science, The University of Ostrava, 30. dubna 22, 701 03 Ostrava, Czech Republic

Received January 26, 2018, in final form August 23, 2018; Published online August 25, 2018

Abstract
If a Lagrangian defining a variational problem has order $k$ then its Euler-Lagrange equations generically have order $2k$. This paper considers the case where the Euler-Lagrange equations have order strictly less than $2k$, and shows that in such a case the Lagrangian must be a polynomial in the highest-order derivative variables, with a specific upper bound on the degree of the polynomial. The paper also provides an explicit formulation, derived from a geometrical construction, of a family of such $k$-th order Lagrangians, and it is conjectured that all such Lagrangians arise in this way.

Key words: Euler-Lagrange equations; reduced-order; projectable.

pdf (329 kb)   tex (17 kb)

References

  1. Anderson I.M., The variational bicomplex, Technical report, Utah State University, 1989.
  2. Carmeli M., Classical fields: general relativity and gauge theory, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1982.
  3. Krupka D., Variational sequences on finite order jet spaces, in Differential Geometry and its Applications (Brno, 1989), World Sci. Publ., Teaneck, NJ, 1990, 236-254.
  4. Krupková O., Lepagean $2$-forms in higher order Hamiltonian mechanics. I. Regularity, Arch. Math. (Brno) 22 (1986), 97-120.
  5. Krupková O., The geometry of ordinary variational equations, Lecture Notes in Mathematics, Vol. 1678, Springer-Verlag, Berlin, 1997.
  6. Olver P.J., Differential hyperforms I, University of Minnesota Mathematics Report 82-101, 1982, available at http://www-users.math.umn.edu/~olver/a_/hyper.pdf.
  7. Olver P.J., Hyper-Jacobians, determinantal ideals and weak solutions to variational problems, Proc. Roy. Soc. Edinburgh Sect. A 95 (1983), 317-340.
  8. Palese M., Vitolo R., On a class of polynomial Lagrangians, Rend. Circ. Mat. Palermo Suppl. (2001), 147-159, math-ph/0111019.
  9. Saunders D.J., The geometry of jet bundles, London Mathematical Society Lecture Note Series, Vol. 142, Cambridge University Press, Cambridge, 1989.

Previous article  Next article   Contents of Volume 14 (2018)