On the inverse problem of the calculus of variations for ordinary differential equations. (English) Zbl 0786.58012

The author reproves some well known results, obtained by different methods and rather complicated tools, using techniques based on the Lepagean \(2\)-form and on the Poincaré lemma from contact forms in higher-order mechanics.
The problems studied in this setting are the following: first, the local and global inverse problem of the calculus of variations, this is the determination of the necessary and sufficient conditions that a system of higher order partial differential equations must satisfy in order to be the Euler-Lagrange equations of some Lagrangian.
Second, the explicit construction of Lagrangians once the previous conditions hold.
And third, the possibility of lowering the order of the Lagrangian, and the possibility of finding a minimal-order Lagrangian.
Reviewer: J.Monterde


58E30 Variational principles in infinite-dimensional spaces
49N45 Inverse problems in optimal control
70H03 Lagrange’s equations
Full Text: EuDML