Hojman, Sergio; Pardo, Francisco; Aulestia, Luis; de Lisa, Francisco Lagrangians for differential equations of any order. (English) Zbl 0756.49018 J. Math. Phys. 33, No. 2, 584-590 (1992). The authors consider a system of \(n\) equations of \(R\)th order: \(q_ i^{(R)}-f_ i(t,q_ 1,\dots,q_ n^{(R-1)})=0\) (\(i=1,\dots,n\)). The first purpose is to show when the system is equivalent to a Euler- Lagrange system of the form \(G_ i\tilde L=0\), with \[ G_ i=\sum^ R_{\alpha=0}(-1)^ \alpha{d^ \alpha\over dt^ \alpha}({\partial\over\partial q_ i^{(\alpha)}}), \text{ and }\tilde L=\sum^ n_{i=1}\mu_ i(t,q_ 1,\dots,q_ n^{(R-1)})(q_ i^{(R)}-f_ i). \] The authors prove that the equivalence holds if and only if the coefficients \(\mu_ i\) satisfy suitable conditions. This first result is then used to study the symmetries of the Lagrangian and the relations between symmetries and constants of motion. Reviewer: M.Degiovanni (Brescia) Cited in 1 Document MSC: 49N45 Inverse problems in optimal control 70H03 Lagrange’s equations Keywords:calculus of variations; inverse problems; Euler-Lagrange system PDFBibTeX XMLCite \textit{S. Hojman} et al., J. Math. Phys. 33, No. 2, 584--590 (1992; Zbl 0756.49018) Full Text: DOI References: [1] DOI: 10.1063/1.525162 · Zbl 0475.70023 [2] DOI: 10.1103/PhysRevD.28.1333 [3] DOI: 10.1088/0305-4470/17/12/012 [4] DOI: 10.1063/1.525062 · Zbl 0522.70024 [5] DOI: 10.1063/1.526352 · Zbl 0567.70017 [6] DOI: 10.1007/BF02729024 [7] DOI: 10.1007/BF02743927 · Zbl 0077.37202 [8] DOI: 10.1016/0003-4916(82)90334-7 · Zbl 0501.70020 [9] DOI: 10.1063/1.527375 · Zbl 0594.53051 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.