SIGMA 14 (2018), 089, 13 pages arXiv:1801.06888
On Lagrangians with Reduced-Order Euler-Lagrange Equations
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
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.
Euler-Lagrange equations; reduced-order; projectable.
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