Geodesic
The inverse problem in the calculus of variations: new developments
Communications in Mathematics, Tome 29 (2021) no. 1, pp. 131-149.

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We deal with the problem of determining the existence and uniqueness of Lagrangians for systems of $n$ second order ordinary differential equations. A number of recent theorems are presented, using exterior differential systems theory (EDS). In particular, we indicate how to generalise Jesse Douglas's famous solution for $n=2$. We then examine a new class of solutions in arbitrary dimension $n$ and give some non-trivial examples in dimension 3.
Classification : 37J06, 49N45, 58A15, 70H03
Keywords: Inverse problem in the calculus of variations; Helmholtz conditions; Exterior differential systems; Lagrangian system. 
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Do, Thoan; Prince, Geoff. The inverse problem in the calculus of variations: new developments. Communications in Mathematics, Tome 29 (2021) no. 1, pp. 131-149. http://geodesic.mathdoc.fr/item/COMIM_2021__29_1_a7/