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Lagrangians for differential equations of any order. (English) Zbl 0756.49018

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.

MSC:

49N45 Inverse problems in optimal control
70H03 Lagrange’s equations
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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
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[9] DOI: 10.1063/1.527375 · Zbl 0594.53051
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