Geodesic
On a generalization of Helmholtz conditions
Communications in Mathematics, Tome 17 (2009) no. 1, pp. 11-21.

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Helmholtz conditions in the calculus of variations are necessary and sufficient conditions for a system of differential equations to be variational ‘as it stands’. It is known that this property geometrically means that the dynamical form representing the equations can be completed to a closed form. We study an analogous property for differential forms of degree 3, so-called Helmholtz-type forms in mechanics ($n=1$), and obtain a generalization of Helmholtz conditions to this case.
Classification : 58A10, 58A20, 58E30, 70G45
Keywords: Lagrangian; Euler-Lagrange form; dynamical form; Helmholtz-type form; Helmholtz form; Helmholtz conditions 
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     author = {Mal{\'\i}kov\'a, Radka},
     title = {On a generalization of {Helmholtz} conditions},
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Malíková, Radka. On a generalization of Helmholtz conditions. Communications in Mathematics, Tome 17 (2009) no. 1, pp. 11-21. http://geodesic.mathdoc.fr/item/COMIM_2009__17_1_a2/