Geodesic
On Lagrangians with Reduced-Order Euler–Lagrange Equations
Symmetry, integrability and geometry: methods and applications, Tome 14 (2018) Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

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.
Keywords: Euler–Lagrange equations; reduced-order; projectable. 
@article{SIGMA_2018_14_a88,
     author = {David Saunders},
     title = {On {Lagrangians} with {Reduced-Order} {Euler{\textendash}Lagrange} {Equations}},
     journal = {Symmetry, integrability and geometry: methods and applications},
     year = {2018},
     volume = {14},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SIGMA_2018_14_a88/}
}
TY  - JOUR
AU  - David Saunders
TI  - On Lagrangians with Reduced-Order Euler–Lagrange Equations
JO  - Symmetry, integrability and geometry: methods and applications
PY  - 2018
VL  - 14
UR  - http://geodesic.mathdoc.fr/item/SIGMA_2018_14_a88/
LA  - en
ID  - SIGMA_2018_14_a88
ER  - 
%0 Journal Article
%A David Saunders
%T On Lagrangians with Reduced-Order Euler–Lagrange Equations
%J Symmetry, integrability and geometry: methods and applications
%D 2018
%V 14
%U http://geodesic.mathdoc.fr/item/SIGMA_2018_14_a88/
%G en
%F SIGMA_2018_14_a88
David Saunders. On Lagrangians with Reduced-Order Euler–Lagrange Equations. Symmetry, integrability and geometry: methods and applications, Tome 14 (2018). http://geodesic.mathdoc.fr/item/SIGMA_2018_14_a88/

[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 | MR | Zbl

[3] Krupka D., “Variational sequences on finite order jet spaces”, Differential Geometry and its Applications (Brno, 1989), World Sci. Publ., Teaneck, NJ, 1990, 236–254 | MR | Zbl

[4] Krupková O., “Lepagean $2$-forms in higher order Hamiltonian mechanics. I. Regularity”, Arch. Math. (Brno), 22 (1986), 97–120 | MR | Zbl

[5] Krupková O., The geometry of ordinary variational equations, Lecture Notes in Mathematics, 1678, Springer-Verlag, Berlin, 1997 | DOI | MR | Zbl

[6] Olver P. J., Differential hyperforms I, University of Minnesota Mathematics Report 82-101, 1982 http://www-users.math.umn.edu/õlver/a_/hyper.pdf | MR

[7] Olver P. J., “Hyper-Jacobians, determinantal ideals and weak solutions to variational problems”, Proc. Roy. Soc. Edinburgh Sect. A, 95 (1983), 317–340 | DOI | MR | Zbl

[8] Palese M., Vitolo R., “On a class of polynomial Lagrangians”, Rend. Circ. Mat. Palermo Suppl., 2001, 147–159, arXiv: math-ph/0111019 | MR | Zbl

[9] Saunders D. J., The geometry of jet bundles, London Mathematical Society Lecture Note Series, 142, Cambridge University Press, Cambridge, 1989 | DOI | MR | Zbl