On Lagrangians with Reduced-Order Euler-Lagrange Equations

Symmetry Integrability and Geometry-Methods and Applications

article

David Saunders¹ 

[1] Department of Mathematics, Faculty of Science, The University of Ostrava

关键词: Euler–Lagrange equations;    reduced-order;    projectable;   

DOI  :  10.3842/SIGMA.2018.089

来源: National Academy of Science of Ukraine

【 摘 要 】

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.

【 授权许可】

Unknown