
SIGMA 14 (2018), 089, 13 pages arXiv:1801.06888
https://doi.org/10.3842/SIGMA.2018.089
On Lagrangians with ReducedOrder EulerLagrange 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 EulerLagrange equations generically have order $2k$. This paper considers the case where the EulerLagrange equations have order strictly less than $2k$, and shows that in such a case the Lagrangian must be a polynomial in the highestorder 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:
EulerLagrange equations; reducedorder; projectable.
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